EXAMPLES OF REACTION-DIFFUSION EQUATIONS IN BIOLOGICAL SYSTEMS:MARINE PROTECTED AREAS AND QUORUM SENSING

By

JESSICA LANGEBRAKE

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2013

c⃝ 2013 Jessica Langebrake

2

For Matt

3

ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. De Leenheer, and my co-advisor, Dr. Hagen,

for their guidance and endless patience. I would like to thank Dr. Osenberg for giving

me perspective on the application of mathematics to biology. I would like to thank Gabe

Dilanji for being a great collaborator; for maintaining his excitement through all those

hours he spent with me in the lab and for helping me see the beauty inherent in quorum

sensing. I would also like to thank my friends and family for their support, especially my

brother, Chris, sister-in-law, Heather, and parents, Beth and Larry. Lastly, I would like to

thank my husband, Matt, for his constant encouragement and understanding.

4

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 Reaction-Diffusion Models in Biological Systems . . . . . . . . . . . . . . 121.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2 Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.4 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . 16

2 DIFFERENTIAL MOVEMENT AND MOVEMENT BIAS MODELS FOR MARINEPROTECTED AREAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Stability of the Steady State Solution . . . . . . . . . . . . . . . . . . . . . 282.4 Qualitative Analysis of the Steady State Solution . . . . . . . . . . . . . . 292.5 A Movement Bias Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.6 Constant Diffusion and Smooth Mortality Rate . . . . . . . . . . . . . . . 342.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 QUORUM SENSING BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 Quorum Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 LuxR-LuxI System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 A SPATIALLY EXPLICIT QUORUM SENSING MODEL . . . . . . . . . . . . . . 46

4.1 Experimental Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.4 Results of Lane Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.1 Diffusion of a Dye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 LuxR-LuxI System Response to AHL Diffusion . . . . . . . . . . . 59

4.5 Discussion and Model Simulations . . . . . . . . . . . . . . . . . . . . . . 624.6 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5

4.6.1 Bacterial cultures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.6.2 Well-plate measurements . . . . . . . . . . . . . . . . . . . . . . . 664.6.3 Lane apparatus and imaging . . . . . . . . . . . . . . . . . . . . . 67

5 SIGNAL PROPAGATION IN A QUORUM SENSING SYSTEM . . . . . . . . . . 69

5.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Traveling Wave Solution of (5–2), (5–3) . . . . . . . . . . . . . . . . . . . 725.3 The Existence of a Traveling Wave Solution to a Class of Reaction-Diffusion

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.2 The Wave Speed c . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.3

∫ a20F (U,VG(U))dU = 0 . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.4∫ a20F (U,VG(U))dU > 0 . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3.4.1 The Sets P1 and P2 . . . . . . . . . . . . . . . . . . . . . 915.3.4.2 P1 = ∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3.4.3 P2 = ∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3.4.4 P1 and P2 are Open and Disjoint . . . . . . . . . . . . . . 1065.3.4.5 The Existence of a Heteroclinic Connection . . . . . . . . 1085.3.4.6 The Existence of a Traveling Wave . . . . . . . . . . . . . 111

5.3.5∫ a20F (U,VG(U))dU < 0 . . . . . . . . . . . . . . . . . . . . . . . . 112

5.3.6 Statement of Existence Theorem . . . . . . . . . . . . . . . . . . . 113

6 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

APPENDIX

A PROOF OF THEOREM 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

B SKETCH OF THE PROOF OF THEOREM 2.4 . . . . . . . . . . . . . . . . . . 121

C CONTINUITY OF THE STABLE MANIFOLD WITH RESPECT TO PARAMETERS123

C.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123C.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123C.3 Fix the Dimension of Stable Manifold . . . . . . . . . . . . . . . . . . . . . 125C.4 P(c) Varies Continuously with Respect to c . . . . . . . . . . . . . . . . . 126C.5 Construction of the Stable Manifold . . . . . . . . . . . . . . . . . . . . . . 129

C.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129C.5.2 Existence of the Stable Manifold . . . . . . . . . . . . . . . . . . . 130

C.5.2.1 ϕ : Cb → C . . . . . . . . . . . . . . . . . . . . . . . . . . 131C.5.2.2 ϕ : Cb → Cb . . . . . . . . . . . . . . . . . . . . . . . . . . 135C.5.2.3 ϕ is a Contraction . . . . . . . . . . . . . . . . . . . . . . 136C.5.2.4 Existence of the Stable Manifold . . . . . . . . . . . . . . 136

C.6 θ(t, a, c) is Continuous with Respect to c . . . . . . . . . . . . . . . . . . 139C.7 The Stable Manifold is Continuous with Respect to c . . . . . . . . . . . . 147

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7

LIST OF TABLES

Table page

4-1 A summary of variables and parameters used in the model (4–9) - (4–21). . . . 55

5-1 A summary of variables and parameters used in the model (5–2),(5–3) . . . . . 72

8

LIST OF FIGURES

Figure page

2-1 MPAs distributed evenly along an infinite coastline. . . . . . . . . . . . . . . . . 22

2-2 Simulation of model (2–1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2-3 Plots of biologically significant quantities . . . . . . . . . . . . . . . . . . . . . . 31

2-4 Plots of bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2-5 Plots of biologically significant quantities for strong bias . . . . . . . . . . . . . 35

3-1 Aliivibrio fischeri MJ11 LuxR-LuxI diagram . . . . . . . . . . . . . . . . . . . . 43

4-1 Chapter 4 methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4-2 Experimental configuration for lane experiments . . . . . . . . . . . . . . . . . 48

4-3 Experimental data from well-plate and fits to model (4–9)-(4–21) . . . . . . . . 56

4-4 Diffusion of fluorescein dye in agar lane . . . . . . . . . . . . . . . . . . . . . . 59

4-5 Response of the sensor strain (E. coli + pJBA132) to diffusing AHL . . . . . . . 60

4-6 Bioluminescence response of luxI-deficient A. fischeri VCW267 to diffusingAHL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4-7 Patterns of expression predicted for the E. coli + pJBA132 sensor strain inresponse to diffusing AHL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5-1 Nullclines of system (5–4), (5–5) . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5-2 Plots of VF (u) and VG(u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5-3 The region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9

Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

EXAMPLES OF REACTION-DIFFUSION EQUATIONS IN BIOLOGICAL SYSTEMS:MARINE PROTECTED AREAS AND QUORUM SENSING

By

Jessica Langebrake

August 2013

Chair: Patrick De LeenheerCochair: Stephen J. HagenMajor: Mathematics

Reaction-diffusion models are widely used to describe physical phenomena,

with applications as varied as epidemic spread and self-regulated pattern formation

in animal embryos. In this document, we present three reaction-diffusion models.

The first is a model of fish movement into and out of a marine protected area (MPA),

an area of coastline wherein fishing is restricted or prohibited. MPAs are promoted

as a tool to protect over-fished stocks and increase fishery yields. Previous models

suggested that adult mobility modified effects of MPAs by reducing densities of fish

inside reserves, but increasing yields (i.e. increasing densities outside of MPAs).

Empirical studies contradicted this prediction: as mobility increased, the relative density

of fishes inside MPAs (relative to outside) increased or stayed constant. To attempt to

explain these empirical results through modeling, we examined the effects of differential

movement inside versus outside the reserve as well as the effects of a movement bias

at the boundary of the reserve. We found that differential movement could not explain

empirical findings, but a movement bias model could.

The second and third models describe quorum sensing systems. In a quorum

sensing system, bacteria synthesize small diffusible chemicals called autoinducers.

Once a critical concentration of autoinducer is reached, the bacterial colony undergoes a

shift in gene expression.

10

The second model describes a colony of genetically modified bacteria that respond

to but cannot produce autoinducer. Our model contains a minimal set of components

necessary to describe experimentally observed patterns of cell response to a diffusing

autoinducer signal in a spatially extended system. Our model incorporates diffusion of

the signal, logistic growth of the bacteria and a cooperative (Hill function) response to

the signal. We observe and predict cell response to the diffusing signal over distances

of ∼ 1 cm on time scales of ∼ 10 h. Our model and experiments display patterns that

are qualitatively dissimilar from simple diffusion: the observed response is surprisingly

insensitive to the distance the signal has traveled.

The third model describes an intact quorum sensing system in the bacterium

Aliivibrio fischeri. Our model describes only the autoinducer signal concentration and

the autoinducer synthase concentration, and incorporates diffusion of the autoinducer

signal and auto-feedback in the production of this signal. This model is able to describe

a quorum sensing shift in gene expression on the colony level, which appears as a

traveling wave. We give a proof of the existence of a traveling wave solution to a class

of models that includes our quorum sensing model. We also use the conditions of this

theorem to determine parameter ranges over which our quorum sensing model admits a

traveling wave solution.

11

CHAPTER 1BACKGROUND

1.1 Reaction-Diffusion Models in Biological Systems

A reaction-diffusion equation is an equation of the form

du

dt= D

d2u

dx2+ f (u)

where D is the diffusion constant, x ∈ R represents position and t ∈ R represents time.

D d2udx2

is the diffusion term and f (u) is the reaction term. u(x , t) is a continuous function

that represents a quantity of interest, such as chemical concentration or population

density.

Scientists use systems of reaction-diffusion equations to describe a plethora of

biological systems. Models exist to describe tumor growth [32], epidemic spread [65],

animal dispersal [64], and even self-regulated pattern formation in an animal embryo.

[52] In this text, we present three reaction-diffusion models. The first, appearing in

Chapter 2, is a model describing the movement of fish into and out of a marine protected

area, an area of coastline where fishing is restricted or prohibited. Chapters 4 and 5

contain reaction-diffusion models that describe quorum sensing systems. In a quorum

sensing system, bacteria synthesize small diffusible chemicals called autoinducers.

Once a critical concentration of autoinducer is reached, the bacterial colony undergoes a

shift in gene expression. The model presented in Chapter 5 describes an intact quorum

sensing system, while the model in Chapter 4 describes a colony of genetically modified

bacteria that respond to but cannot produce autoinducer.

1.2 Mathematical Preliminaries

Here, we give several definitions, lemmas and theorems that will be used later in the

text.

1.2.1 Linear Algebra

The following theorem can be found in Friedberg et al. (2003).[34]

12

Theorem 1.1 (Primary Decomposition Theorem). Let T be a linear operator on an

n−dimensional vector space V with characteristic polynomial

f (t) = (−1)n(ϕ1(t))n1(ϕ2(t))

n2 · · · (ϕk(t))nk ,

where the ϕi(t)’s (1 ≤ i ≤ k) are distinct irreducible monic polynomials and the ni ’s are

positive integers. Then

V = Kϕ1 ⊕ Kϕ2 ⊕ · · · ⊕ Kϕk

where Kϕi = {x ∈ V |(ϕi(T ))p(x) = 0 for some positive integer p} (1 ≤ i ≤ k).

1.2.2 Complex Analysis

The following definitions and theorem can be found in Brown and Churchill

(2009).[7]

Definition 1. A contour, or piecewise smooth arc, is an arc consisting of a finite number

of smooth arcs joined end to end. If z = z(t), a ≤ t ≤ b, represents a contour, z(t)

is continuous and z ′(t) is piecewise continuous. When only the initial and final values

of z(t) are the same, a contour C is called a simple closed contour. Such a curve is

positively oriented when it is in the counterclockwise direction.

Definition 2. A function f of the complex variable z is analytic at a point z0 if it has a

derivative at each point in some neighborhood of z0.

Definition 3. A point z0 is called a singular point of a function f if f fails to be analytic

at z0 but is analytic at some point in every neighborhood of z0. A singular point z0 is said

to be isolated if, in addition, there is a deleted neighborhood 0 < |z − z0| < ϵ of z0

throughout which f is analytic.

13

Definition 4. When z0 is an isolated singular point of a function f , there is a positive

number R2 such that f is analytic at each point z for which 0 < |z − z0| < R2. Conse-

quently, f (z) has a Laurent series representation

f (z) =

∞∑n=0

an(z − z0)n +

∞∑n=1

bn

(z − z0)n(1–1)

(0 < |z − z0| < R2), where the coefficients an and bn have the following integral

representations:

an =1

2π i

∫C

f (z)

(z − z0)n+1dz (n = 0, 1, 2, ... )

bn =1

2π i

∫C

f (z)

(z − z0)−n+1dz (n = 1, 2, ... )

and where C is any positively oriented simple closed contour around z0 that lies in the

punctured disk 0 < |z − z0| < R2. In particular, when n = 1, the expression for bn

becomes ∫C

f (z)dz = 2π ib1.

The complex number b1, which is the coefficient of 1z−z0 in expansion (1–1), is called the

residue of f at the isolated singular point z0, and we will write

b1 = Resz=z0f (z).

Theorem 1.2 (Cauchy’s Residue Theorem). Let C be a simple closed contour, de-

scribed in the positive sense. If a function f is analytic inside and on C except for a finite

number of singular points zk (k = 1, 2, ... , n) inside C , then∫C

f (z)dz = 2π in∑

k=1

Resz=zk f (z)

where Resz=zk f (z) denotes the residue of f (z) at z = zk .

1.2.3 Analysis

The following two theorems can be found in Rudin (1976).[78]

14

Theorem 1.3 (Implicit Function Theorem). If x = (x1, x2, ... , xn) ∈ Rn and y =

(y1, y2, ... , ym) ∈ Rm, let (x , y) denote the point (or vector) (x1, x2, ... , xn, y1, y2, ... , ym) ∈

Rn+m. Let f (x , y) be a C 1 mapping of an open set E ⊂ Rn+m into Rn, such that

f (x0, y0) = 0 for some point (x0, y0) ∈ E . Assume that Dfx is invertible. Then there

exist open sets U ⊂ Rn+m and W ⊂ Rm with (x0, y0) ∈ U and y0 ∈ W having the following

property:

To every y ∈ W corresponds a unique x such that

(x , y) ∈ U and f (x , y) = 0.

If this f is defined to be g(y), then g is a C 1 mapping of W into Rn, g(y0) = x0,

f (g(y), y) = 0 (y ∈ W ),

and

Dg(y0) = −(Dfx)−1Dfy .

Theorem 1.4 (Dominated Convergence Theorem). Let µ be a measure, E be a measur-

able set and {fn} be a sequence of measurable functions such that

fn(x) → f (x)

for each x ∈ E as n → ∞. If there exists a function g that is integrable on E such that

|fn(x)| ≤ g(x)

for all n and for each x ∈ E , then f is integrable and

limn→∞

∫E

fndµ =

∫E

fdµ.

The following lemma can be found in Logemann and Ryan (2004).[56]

15

Lemma 1 (Barbalat’s Lemma). Suppose f (t) ∈ C 1(a,∞) and limt→∞ f (t) = α where

α ∈ R. If f ′ is uniformly continuous, then limt→∞ f ′(t) = 0.

1.2.4 Ordinary Differential Equations

The following definitions, lemmas and theorems can be found in Chicone (2006).[13]

Lemma 2. If A is an n × n matrix, then etA is a matrix whose components are (finite)

sums of terms of the form

p(t)eαt sin βt and p(t)eαt cos βt

where α and β are real numbers such that α + iβ is an eigenvalue of A, and p(t) is a

polynomial of degree at most n − 1.

Theorem 1.5. Suppose that A is an n × n (real) matrix. The following statements are

equivalent.

(1) There is a norm ∥·∥a on Rn and a real number λ > 0 such that for all v ∈ Rn andall t ≥ 0, ∥∥etAv∥∥

a≤ e−λt ∥v∥a .

(2) If ∥·∥g is an arbitrary norm on Rn, then there is a constant C > 0 and a realnumber λ > 0 such that for all v ∈ Rn and all t ≥ 0,∥∥etAv∥∥

g≤ Ce−λt ∥v∥g .

(3) Every eigenvalue of A has negative real part.

Moreover, if −λ exceeds the largest of all the real parts of the eigenvalues of A, then λ

can be taken to be the decay constant in (1) or (2).

In the following two definitions and theorem, let (X , ∥·∥) be a normed vector space

and define the induced metric d on X by d(x , y) = ∥x − y∥.

Definition 5. A point x0 ∈ X is a fixed point of a function ϕ : X → X if ϕ(x0) = x0.

16

Definition 6. Suppose that ϕ : X → X , and λ is a real number such that 0 ≤ λ < 1. The

function ϕ is called a contraction with contraction constant λ if

d(ϕ(x),ϕ(y)) ≤ λd(x , y)

for all x , y ∈ X .

Theorem 1.6 (Contraction Mapping Theorem). If the function ϕ is a contraction on the

complete metric space (X , d), then ϕ has a unique fixed point x∗ ∈ X .

Theorem 1.7 (Gronwall’s Inequality). Suppose that a < b are real numbers and let α,ϕ

and ψ be nonnegative, continuous functions defined on the interval [a, b]. Moreover,

suppose that α is differentiable on (a, b) with nonnegative continuous derivative. If

ϕ(t) ≤ α(t) +

∫ t

a

ψ(s)ϕ(s)ds

for all t ∈ [a, b], then

ϕ(t) ≤ α(t)e∫ t

aψ(s)ds

for all t ∈ [a, b].

Lemma 3. Let U ⊆ Rn and J ⊆ R be open sets such that the open interval (α, β) is

contained in J. Also, let x0 ∈ U. If f : J × U → Rn is a C 1 function and the maximal

interval of existence of the solution t → ϕ(t) of the initial value problem _x = f (t, x),

x(t0) = x0 is α < t0 < β with β < ∞, then for each compact set K ⊂ U there is some

t ∈ (α, β) such that ϕ(t) /∈ K . In particular, either |ϕ(t)| becomes unbounded or ϕ(t)

approaches the boundary of U as t → β.

Definition 7. Let J ⊆ R and U ⊆ Rn be open. A function ϕ : J × U → Rn given by

(t, x) → ϕ(t, x) is called a flow if ϕ(0, x) ≡ x and ϕ(t + s, x) = ϕ(t,ϕ(s, x)) whenever

both sides of the equation are defined. We will sometimes use t → ϕt(x) in place of

t → ϕ(t, x).

17

Definition 8. Suppose that ϕt is a flow on Rn and p ∈ Rn. A point x in Rn is called an

omega limit point (ω-limit point) of the orbit through p if there is a sequence of numbers

t1 ≤ t2 ≤ t3 ≤ · · · such that limi→∞ ti = ∞ and limi→∞ ϕti (p) = x . The collection of all

such omega limit points is denoted ω(p) and is called the omega limit set (ω-limit set) of

p.

Lemma 4. Consider _x = f (x), x ∈ Rn. Let x(t) be a solution such that limt→∞ x(t) = �x .

Then f (�x) = 0.

Proof. Since limt→∞ x(t) = �x , {�x} is the ω-limit set of x . Since ω-limit sets are invariant

under the flow and this ω-limit set consists of a single point, we have that f (�x) = 0.

Lemma 5. If the map f : J × × � → Rn in the differential equation _x = f (t, x ,λ) is

continuously differentiable, t0 ∈ J ⊆ R, x0 ∈ ⊆ Rn, and λ0 ∈ � ⊆ Rm, then there are

open sets J0 ⊆ J, 0 ⊆ and �0 ⊆ � such that (t0, x0,λ0) ∈ J0 × 0 × �0, and a unique

C 1 function σ : J0 × 0 × �0 → Rn given by (t, x ,λ) → σ(t, x ,λ) such that t 7→ σ(t, x ,λ)

is a solution of the differential equation _x = f (t, x ,λ) and σ(0, x ,λ) = x . In particular,

t 7→ σ(t, x0,λ0) is a solution of the initial value problem _x = f (t, x ,λ0), x(t0) = x0.

18

CHAPTER 2DIFFERENTIAL MOVEMENT AND MOVEMENT BIAS MODELS FOR MARINE

PROTECTED AREAS

2.1 Introduction

1 Overfishing has reduced marine fish stocks and degraded habitats [79, 80]. As a

consequence, fisheries management has become a major economic and environmental

challenge. Marine reserves (or marine protected areas, MPAs) are frequently advocated

as an efficient management tool to restore habitats and protect over-harvested stocks

[15, 41, 44, 79, 80]. MPAs offer two potential benefits. First, they can locally increase

the densities of harvested species [16, 41], but see [71]. Secondly, they can increase

fishing yields outside of the marine reserve via spillover and/or larval export [39, 76, 79]

(spillover is defined as the net movement of adult fish from the reserve to the fishing

grounds, which results in a biomass export).

Despite the evidence supporting local benefits of MPAs, uncertainties remain

[44, 71, 79]. For example, theoretical studies have suggested that the local effectiveness

of an MPA decreases as adult mobility increases [38, 59, 63, 73, 95]. Empirical data

do not support this theoretical expectation. For example, in a recent meta-analysis

of Mediterranean MPAs, Claudet et al. (2010) [16] calculated the relative densities of

fish inside vs. outside MPAs, and compared the results for species with low, medium

or high adult mobility. Contrary to the theoretical expectation, they found that more

mobile species showed greater increases in density inside of MPAs (relative to outside).

To explain their surprising results, Claudet et al. suggested that mobile species could

benefit more from MPAs than expected if they biased their movement in favor of the

1 Reproduced with permission from J. Langebrake, L. Riotte-Lambert, C. W.Osenberg, and P. De Leenheer. Differential movement and movement bias models formarine protected areas. J. Math. Biol., 64(4):667696, 2012.

19

reserve. Such a bias could result if the MPA altered habitat availability or quality [80] and

the target species preferred this modification.

In contrast to the expected negative relationship between increased local effects

(one goal of MPAs) and mobility, models generally indicate that fishing yields (the

second goal) should increase with fish mobility as a result of increased spillover.

[38, 54, 63] There are no available empirical data to evaluate whether this expectation

also is contradicted.

The conflict between empirical and theoretical predictions about the relationship

between mobility and local effects of MPAs, as well as the importance of spillover for

producing increased fisheries yields, suggests that we need to examine the effects of

mobility in new ways. The main purpose of this chapter is to propose several models

that could reconcile model predictions and empirical results.

We will start by introducing a model that examines how differential movement inside

versus outside the MPA can affect the efficacy of MPAs. To date there have been only

limited studies of this phenomenon. For example, Rodwell et al. (2003) [77] developed

a two patch model where adult movement was described by an annual transfer from

the most populated patch to the other, i.e. from the reserve to the fishing grounds. We

model fish movement as a diffusion process and assume that the diffusion parameter

is smaller inside the reserve than outside. Mathematically, the model is a boundary

value problem with piecewise constant parameters in different spatial regions. On each

region the steady state equation is linear so that it can be solved explicitly. The solutions

need to be matched at the interface of the regions, a technique that is well-known, see

for instance [8, 81]. Then we investigate how differential diffusion affects the expected

benefits of MPAs. We focus on four measures: abundance of fish in the fishing grounds

(i.e. the amount of fish in the fished area), total abundance (i.e. the amount of fish

contained in the MPA and fished area combined), the local effect (i.e., log of the ratio of

the density inside vs. outside of the MPA, which is a common measure of the effect of

20

an MPA), and fisheries yield (i.e., the amount of fish caught by fishers per unit of time).

We will show that if the two diffusion parameters are scaled as mobility increases, yet

their ratio remains constant, the measures vary in a way that is in accordance with the

theoretical predictions from traditional models.

Next we introduce a model that incorporates a movement bias towards the MPA that

is localized to the MPA boundary. It arises as the limit of a random walk model where the

random walk is only truly random if the random walker is not located on the boundary,

but biased if he is. We show that there is a critical value for the bias parameter that

controls the dependence of the four measures on increased mobility. For small bias, the

results are in line with what traditional models predict, but for large bias values, once

again, we are able to reconcile theory and data. The results for this model only depend

on the bias value, and they remain valid whether or not we assume differential diffusion

inside and outside the MPA.

Finally, we propose a simplified model with homogeneous diffusion everywhere,

but with smooth -as opposed to piecewise constant- mortality rates. We show that once

more, it is possible to unite data and theory, at least on the level of one of our measures,

namely the abundance of fish in the fishing grounds.

Our results suggest that explanations of data depend on the underlying model

assumptions in a very subtle way. They seem to indicate that various explanations are

possible and that further research is required to elucidate this problem.

The rest of this chapter is organized as follows. In Section 2.2 we present our model

and show that it has a unique steady state. We examine the stability of the steady state

in Section 2.3. Section 2.4 introduces various measures that quantify the effect of the

MPA, and we investigate how increased mobility affects these measures. In Section 2.5

we investigate a movement bias model, and in Section 2.6 we consider a model with

homogeneous diffusion and smooth mortality. We conclude our chapter in Section 2.7

with a discussion. Proofs of two of our results are in Appendices A and B.

21

2.2 Model

Marine Protected Areas (MPAs) are portions of coastline in which fishing is

restricted or disallowed. It has been theorized that establishing MPAs periodically

along a coastline will increase the overall population of fish as well as increase total

fishing yield.

In this chapter, we examine the case where MPAs are distributed evenly and

periodically along a straight coastline. The coastline can therefore be split up into

several (or, in fact, infinitely many) identical sections, each containing an MPA surrounded

by unprotected waters, called fishing grounds. To examine this situation, we allow one

section of coastline to be represented by the interval [−1+2k , 1+ 2k ] and the MPA to be

[−l + 2k , l + 2k ], where 0 < l < 1 and k ∈ Z, as illustrated in Figure 2-1.

Figure 2-1. MPAs distributed evenly along an infinite coastline.

Assuming a completely open system, we let R represent a positive, constant

recruitment rate. To describe the difference in conditions inside and outside the MPA, we

assign the positive diffusion coefficient inside the MPA to be Di and the positive diffusion

coefficient outside the MPA to be Do . We do not specify any relationship between Di

and Do as it is not required for the following analysis. It may, however, be reasonable to

choose Di ≤ Do ; this inequality reflects that fish diffuse more sowly in the MPA than in

the fishing grounds, possibly because the protection afforded by the MPA increases the

likelihood that fish will remain in the MPA for longer periods of time.

The protection afforded by the MPA also creates an important difference in mortality

rates inside and outside the MPA. Namely, the mortality rate inside the MPA is lower

than outside because the fish are only dying of natural causes inside the MPA while

22

additional fish are being removed by fishermen outside the MPA. Thus, we let the

mortality rate inside the MPA be µi and the mortality rate outside the MPA be µo where

µi < µo .

Allowing the density of fish at time t and position x to be denoted by n(x , t), we have the

following model on an infinite coastline:

nt = (D(x)nx)x + R − µ(x)n (2–1)

where

D(x) =

Do x ∈ (−1 + 2k ,−l + 2k) ∪ (l + 2k , 1 + 2k)

Di x ∈ (−l + 2k , l + 2k)

and

µ(x) =

µo x ∈ (−1 + 2k ,−l + 2k) ∪ (l + 2k , 1 + 2k)

µi x ∈ (−l + 2k , l + 2k)

Using MATLAB, we can plot the solution to this system as time progresses. In

Figure 2-2, we can see the progression of the system (graphed on the interval [0, 1], for

reasons that will become clear below) through time and see the plots become more and

more similar to the steady state solution calculated later in this section.

We are interested in finding steady state solutions to (2–1). Steady state solutions

are functions n(x) that are independent of t, non-negative, continuous and that satisfy

0 = (D(x)n′)′ + R − µ(x)n,

where the prime ′ stands for d/dx . We additionally require that n(x) have continuous flux

and be periodic. That is, the flux −D(x)n′(x) must be continuous and

n(x + 2) = n(x) for all x ∈ R. (2–2)

23

A t = 0.1 B t = 0.5 C t = 2

D t = 5 E t = 10 F Steady State Solution

Figure 2-2. Simulation of model (2–1). The above graphs were produced in MATLABusing the parameters l = 3/16 km, Do = 2 km2

year, Di = 0.04 km2

year, R =

0.5 thousands of fish(year)(km)

, µi = 0.25 1year

, µo = 0.5 1year

and Initial Conditionn(x , 0) = 1 thousands of fish

kmfor all x .

Note that since D(x) is discontinuous at the MPA boundaries, requiring continuous flux

implies that n′(x) must also be discontinuous there.

We restrict our search for steady state solutions to only those that are symmetric

with respect to x = 0, that is, functions n(x) such that

n(x) = n(−x) for all x not on the MPA boundaries. (2–3)

These requirements create additional conditions that n(x) must satisfy. In order to

have continuous density n(x) and continuous flux −D(x)n′(x), we must force the left-

and right-hand limits of these functions to match at the boundaries between the MPA

and unprotected waters. Thus, we have the following matching conditions:

n−(l + 2k) = n+(l + 2k), n−(−l + 2k) = n+(−l + 2k) (2–4)

24

Din′−(l + 2k) = Don

′+(l + 2k), Don

′−(−l + 2k) = Din

′+(−l + 2k)

for every k ∈ Z. Here, the subscripts − and + indicate the left and right limit respectively.

We will show that the problem can be substantially simplified. Instead of solving the

steady state equation on R, it will suffice to solve the problem on [0, 1] with Neumann

boundary conditions.

To see this, note first that taking the derivative with respect to x in (2–3) yields that :

n′(x) = −n′(−x) for all x not on the MPA boundaries. (2–5)

In particular, setting x = 0 implies that:

n′(0) = 0 (2–6)

Similarly, taking derivatives in (2–2) and setting x = −1 shows that

n′(1) = n′(−1)

But together with (2–5), evaluated at x = 1, this implies that

n′(1) = 0 (2–7)

Thus, for every solution to our steady state problem, there is no flux in the points x = 0

and x = 1.

Let’s assume for now (we will actually prove this below) that we can find a

non-negative function n(x) satisfying:

0 = (D(x)n′)′ + R − µ(x)n, x ∈ [0, 1] (2–8)

that is continuous in [0, 1], differentiable in [0, 1], except perhaps in x = l , where the

following matching conditions hold:

n−(l) = n+(l) and Din′−(l) = Don

′+(l) (2–9)

25

and with Neumann boundary conditions

n′(0) = n′(1) = 0 (2–10)

Then it is not hard to see that the function n(x) can be extended to R and that the

resulting extension is a solution to our original steady state problem that satisfies all the

constraints we imposed. Indeed, first we extend the function n(x) defined on [0, 1] to

[−1,+1] by defining

n(−x) = n(x).

It is easily verified that this extension satisfies the steady state equation on [−1, 0]. Also,

by the very definition of this extension, it automatically satisfies the symmetry constraint

(2–3) on the interval [−1, 1], and the matching conditions (2–4) at x = −l . Secondly, we

extend this extended function n(x), which is now defined on [−1, 1], periodically to R, by

defining:

n(x + 2k) = n(x),

for all k ∈ Z. It is easily verified that the resulting extension is a solution to our original

problem.

What remains to be proved is the following:

Theorem 2.1. The boundary-value problem (2–8) with (2–9) and (2–10) has a unique

non-negative solution n(x) which is continuous in [0, 1] and continuously differentiable in

[0, 1] \ {l}.

Proof. Solving the equation on [0, l) and (l , 1] and using the Neumann boundary

conditions (2–10), we find that:

n(x) =

c cosh(αix) +

Rµi, x ∈ [0, l)

d cosh(αo(x − 1)) + Rµo, x ∈ (l , 1]

, (2–11)

26

where c and d are constants determined below, and where we have introduced the

following positive parameters:

αi =

õiDi

, αo =

õoDo

. (2–12)

To find c and d we use the matching condition (2–9):

c cosh(αi l) +R

µi= d cosh(αo(1− l)) +

R

µo

cDiαi sinh(αi l) = −dDoαo sinh(αo(1− l)),

or, using matrix notation:cosh(αi l) − cosh(αo(1− l))

sinh(αi l)DoαoDiαi

sinh(αo(1− l))

c

d

=

R(

1µo

− 1µi

)0

This set of equations has a unique solution if and only if the determinant of the matrix on

the left, is nonzero. We calculate this determinant:

� = det

cosh(αi l) − cosh(αo(1− l))

sinh(αi l)DoαoDiαi

sinh(αo(1− l))

=

DoαoDiαi

cosh(αi l) sinh(αo(1− l)) + sinh(αi l) cosh(αo(1− l)), (2–13)

and see that it is always positive, since both terms of the sum always are. The set of

linear equations therefore has a unique solution:c

d

=1

�R

(1

µi− 1

µo

)−DoαoDiαi

sinh(αo(1− l))

sinh(αi l)

(2–14)

In particular, we see that c < 0 and d > 0. Also, notice that plugging these values of

c and d back into (2–11), we find that the unique steady state solution is a decreasing

function of x (because its derivative is negative everywhere except in x = l where it is

not defined, but where n is continuous).

27

Finally, we need to verify that n(x) ≥ 0 for all x ∈ [0, 1]. But since n(x) is decreasing,

its minimal value is achieved at x = 1, where n(x) equals d + Rµo

which is positive. Thus,

n(x) ≥ 0 for all x ∈ [0, 1] as required.

2.3 Stability of the Steady State Solution

The simulation in Figure 2-2 suggests that the steady state n(x) determined

analytically in (2–11) with (2–14) is asymptotically stable. Stability properties of steady

states are often established using a linearization argument. In this Section we will study

the eigenvalue problem that arises when the system is linearized at the steady state. We

will show that all the eigenvalues are negative, providing further evidence of the stability

of the steady state.

Linearizing model (2–1) at the steady state yields the following eigenvalue problem:

λw = (D(x)w ′)′ − µ(x)w , w ′(0) = w ′(1) = 0 (2–15)

Solutions of this problem are eigenvalue-eigenfunction pairs (λ,w(x)) with w(x) =

0. We denote the operator on the right-hand side of the equation (2–15) by L[w ].

Its domain consists of functions w that are continuous on [0, 1], with continuously

differentiable flux −D(x)w ′, and satifying Neumann boundary condition in x = 0 and

x = 1. Integration by parts shows that this operator is self-adjoint, i.e. (L[u], v) =

(u,L[v ]) for all u and v in the domain of L, where (u, v) denotes the inner product∫ 1

0uvdx . Consequently, the eigenvalues λ of L are real. We will prove that in fact every

eigenvalue must be negative. To see this, assume that there is an eigenvalue λ ≥ 0 and

corresponding eigenfunction w(x) = 0, satisfying (2–15). Using the Neumann boundary

condition, the solution w(x) takes the following form:

w(x) =

A cosh(γix), x ∈ [0, l)

B cosh(γo(x − 1)), x ∈ (l , 1]

,

28

where A and B are constants determined below. The parameters γi and γo are:

γi =

√µi + λ

Di

and γo =

√µo + λ

Do

,

and they are positive because λ ≥ 0. To determine A and B we match the values of

w(x) and of the fluxes D(x)w ′(x) at x = l :

A cosh(γi l) = B cosh(γo(l − 1))

ADiγi sinh(γi l) = BDoγo sinh(γo(l − 1))

Since A and B cannot be zero (otherwise w(x) would be zero), we can divide both

equations, which yields:

Diγi tanh(γi l) = Doγo tanh(γo(l − 1)).

But this equation cannot holds since γi and γo are positive, so that the left-hand side is

always positive, whereas the right-hand side is always negative since l < 1. Thus, there

cannot be an eigenvalue λ ≥ 0. The absence of a nonnegative eigenvalue suggests

stability.

2.4 Qualitative Analysis of the Steady State Solution

In order to analyze the steady state solution qualitatively, we introduce and examine

four quantities:

1. Fishing Grounds Abundance (FGA):

Io =

∫ 1

l

n(x)dx (2–16)

where n(x), the steady state solution to (2–8) given in (2–11) and (2–14)represents the density of fish in the fishing grounds. The FGA is the total amountof fish in the fishing grounds, at the state state.

2. Yield

Y =

∫ 1

l

(µo − µi) n(x)dx , (2–17)

29

where µo − µi represents the fishing rate. Hence, the Yield represents the numberof fish caught by fishermen in the fishing grounds per unit of time, at the steadystate.

3. The Total Abundance

A =

∫ 1

0

n(x)dx , (2–18)

which represents the total number of fish in both the MPA and in the fishinggrounds combined at the steady state.

4. The Log Ratio

L = ln

(1l

∫ l0n(x)dx

11−l

∫ 1

ln(x)dx

), (2–19)

the natural log of the ratio of the average abundance of fish in the MPA and in thefishing grounds evaluated at the steady state.

We are interested in what happens as both Di and Do increase, yet their ratio Di

Do

remains constant. For convenience, we define

Di = D, Do =1

βD (2–20)

and we let D vary, while all other parameters, including β, remain constant. We

summarize the behavior of Io , Y , A and L as functions of D as follows:

Theorem 2.2. Assume that (2–20) holds for some constant β > 0. As the diffusion

coefficient D increases, the FGA Io and Yield Y are non-decreasing, whereas both the

Total Abundance A and Log Ratio L are non-increasing.

The proof can be found in Appendix A. Figure 2-3 includes graphs of these

quantities for chosen parameters. We also investigated what happens if instead of

the ratio, the difference of Do and Di remains constant, yet both increase linearly with D.

The conclusions of Theorem 2.2 remain the same, but since the proof is very similar, it

has been omitted.

30

A Fishing Grounds AbundanceIo

B Yield Y

C Total Abundance A D Log Ratio L

Figure 2-3. The above graphs were produced in MATLAB using the parametersl = 3/16 km, R = 0.5 thousands of fish

(year)(km), µi = 0.25 1

year, µo = 0.5 1

year. The values

for β differ in the three different color plots, where β = 0.5, 1, 2 for the blue,red and green plots, respectively.

2.5 A Movement Bias Model

2 In Ovaskainen and Cornell (2003)[72] a model is proposed that incorporates a

movement bias towards the MPA. This model is obtained as the limit of a biased random

walk model. The bias occurs when the random walker is situated on the boundary of

the MPA, because taking the next step towards the MPA is preferred. It is assumed

that the probability to move to the right is (1 + z)/2, and the probability to move to the

left is (1 − z)/2, where z takes a value in (−1, 1) as a measure of the degree of bias.

Note that z = 0 corresponds to a case without bias. When the random walker is not

on the boundary, the probability of moving left or right is 1/2. The steady state problem

2 In the original publication, this section contained an error in the matching conditions(2–22). It has been corrected here.

31

corresponding to the movement bias model from Ovaskainen and Cornell (2003),

applied in the setup of an MPA, is as follows:

0 = (D(x)n′)′ + R − µ(x)n, 0 < x < 1, n′(0) = n′(1) = 0, (2–21)

with matching conditions:

√Di(1 + z)n−(l) =

√Do(1− z)n+(l) and Din

′−(l) = Don

′+(l) (2–22)

A steady state function is any non-negative function that is continuously differentiable,

except perhaps in x = l , that satisfies (2–21) and the matching condition (2–22). Note

that if z = 0, then necessarily n is discontinuous at x = l by the first matching condition

in (2–22). In what follows, we assume that the bias is towards the MPA, or equivalently

that:

− 1 < z < 0. (2–23)

Since z is negative, it follows from the first matching condition in (2–22) that n−(l) is

larger than n+(l) whenever Do ≥ Di , that is, the limiting values on the MPA boundary

are always higher when the approach occurs within the MPA. We also note that the

only difference between the steady-state problem considered in Theorem 2.1, and the

one considered here, is in the first matching condition. Nevertheless, this condition has

serious impact on the behavior of the four measures considered earlier. But first, we

investigate the shape of the graph of the steady state:

Theorem 2.3. There is a unique non-negative solution n(x) for (2–21) with (2–22).

Moreover, there is a critical bias value

z∗ = −µo −

√Do

Diµi

µo +√

Do

Diµi, (2–24)

such that:

1. If z∗ < z < 0 (weak bias towards the MPA), then n(x) is decreasing in [0, 1].

32

2. If z = z∗ (critical bias), then n(x) is piecewise constant with a jump discontinuityat x = l , given by the first matching condition in (2–22).

3. If −1 < z < z∗ (strong bias towards the MPA), then n(x) is increasing in theMPA, and increasing outside the MPA.

The three cases are illustrated in Figure 2− 4.

A Weak Bias B Critical Bias C Strong Bias

Figure 2-4. The above graphs were produced in MATLAB using the parametersl = 3/16 km, Do = 0.01 km2

year, Di = 0.0002 km2

year, R = 0.5 thousands of fish

(year)(km), µi =

0.0236 1year

, µo = 0.5 1year

. The plots for weak, critical and strong bias use thevalues z = −5× 10−6,−0.5,−0.99999, respectively.

Proof. From (2–21) considered in the MPA and outside the MPA separately, we find that:

n(x) =

c1 cosh(αix) +

Rµi, x ∈ [0, l)

d1 cosh(αo(x − 1)) + Rµo, x ∈ (l , 1]

(2–25)

where αi and αo were defined in (2–12), and where c1 and d1 are obtained using the

matching condition (2–22):c1d1

=1

�R(z)

−DoαoDiαi

sinh(αo(1− l))

sinh(αi l)

(2–26)

where

R(z) =

(1

µi−√Do

Di

1− z

1 + z

1

µo

)R (2–27)

and

� =DoαoDiαi

sinh(αo(1− l)) cosh(αi l) +

√Do

Di

1− z

1 + zcosh(αo(1− l)) sinh(αi l) (2–28)

33

which is positive because of (2–23). Therefore, the sign of c1 and d1 is determined by

the sign of R(z), and the latter changes sign when z crosses z∗. This in turn implies the

three distinctive cases for the shapes of the graphs of n(x). Finally, we need to check

that n(x) is non-negative in all three cases. For z = z∗ this is obvious because in this

case, n(x) is either equal to R/µi or to R/µo , and both values are positive. If z∗ < z , then

it suffices to check that n(1) ≥ 0, since n(x) is decreasing. But n(1) = d1 + R/µo , and d1

has the same sign as R(z) which is positive in this case. Similarly, if z < z∗, it suffices to

check that n(0) ≥ 0. But n(0) = c1 + R/µi and c1 has the same sign as −R(z), which is

positive as well.

Assuming that both Di and Do increase while their ratio remains constant, it

turns out that, provided that the bias is weak, the monotonicity properties of the FGA,

Yield, Total Abundance and Log Ratio remain the same as in the case of the unbiased

model discussed in Theorem 2.2. Interestingly however, when the bias is strong, the

monotonicity is reversed.

Theorem 2.4. Assume that (2–20) holds for some constant β > 0. As the diffusion co-

efficient D increases, the FGA Io and Yield Y are non-decreasing (non-increasing),

whereas both the Total Abundance A and Log Ratio L are non-increasing (non-

decreasing), provided that z∗ < z < 0 (−1 < z < z∗), where z∗ is given by (2–24).

The proof can be found in Appendix B. Figure 2-5 includes graphs of these

quantities for chosen parameters.

2.6 Constant Diffusion and Smooth Mortality Rate

In this section we show that a different model than the two models discussed before

may also explain the experimental data in Claudet et al. (2010)[16], provided the MPA

size is sufficiently small. We consider a situation with constant diffusion D > 0 in the

entire domain [0, 1], and with nonconstant, positive, smooth and increasing mortality rate

µ(x), say in C∞[0, 1] with dµ/dx > 0. This reflects that the values of the mortality rate

are higher outside than inside the MPA due to fishing.

34

A Fishing Grounds AbundanceIo

B Yield Y

C Total Abundance A D Log Ratio L

Figure 2-5. The above graphs for the case of strong bias were produced in MATLABusing the parameters l = 3/16 km, β = 0.02, R = 0.5 thousands of fish

(year)(km), µi =

0.0236 1year

, µo = 0.5 1year

, z = −0.99999.

We consider the steady state problem:

Dn′′ + R − µ(x)n = 0, 0 < x < 1, n′(0) = n′(1) = 0. (2–29)

Using the methods in Section 3.5 from Cantrell and Cosner (2003)[9] and cited

references therein, it can be shown that (2–29) has a unique smooth positive solution

n(x) with the following properties:

limD→0

n(x) =R

µ(x), and lim

D→∞n(x) =

R∫ 1

0µ(x)dx

(2–30)

where the limits exist in L∞[0, 1].

We investigate what happens to the FGA Io =∫ 1

ln(x)dx as D varies from 0 to ∞.

Contrary to what we found for the solution of model (2–8), (2–9) and (2–10), the FGA

35

Io(D) is not necessarily nondecreasing with D as in Theorem 2.2, at least for sufficiently

small MPA sizes:

Theorem 2.5. There exists l∗ ∈ (0, 1) such that if l < l∗, there holds that:

limD→0

Io(D) > limD→∞

Io(D) (2–31)

Proof. In order to establish (2–31), it suffices to show that∫ 1

l

1

µ(x)dx >

1− l∫ 1

0µ(x)dx

(2–32)

holds, by (2–30). To that end, we define the following smooth auxiliary function:

F (l) :=

(∫ 1

0

µ(x)dx

)(∫ 1

l

1

µ(x)dx

)− (1− l)

We have that:

1. F (0) > 0. Indeed, this condition follows from an application of the Cauchy-Schwarzinequality in L2[0, 1] to the functions

õ(x) and 1/

õ(x) (the inequality is strict

becauseõ(x) and 1/

√µ(x) are linearly independent since by assumption µ(x)

is not a constant function).

2. F (1) = 0, which is immediate from the definition of F .

3. dFdl(1) > 0. Indeed, we have that:

dF

dl(l) =

(∫ 1

0

µ(x)dx

)(− 1

µ(l)

)+ 1,

and since µ(x) is increasing on [0,1], there holds that µ(x) < µ(1) for x < 1, so that

dF

dl(1) =

(∫ 1

0

µ(x)dx

)(− 1

µ(1)

)+ 1 < µ(1)

(− 1

µ(1)

)+ 1 = 0.

These three facts imply the existence of some l∗ in (0, 1) such that F (l∗) = 0, and

F (l) > 0 for all l ∈ [0, l∗). But this implies that for these values of l , (2–32), and hence

(2–31) holds.

36

2.7 Discussion

Previous models suggested that increasing mobility (e.g., as reflected in an

increasing diffusion parameter) would: (1) reduce the local effect of an MPA (i.e., reduce

the relative disparity in density inside vs. outside of the MPA); and (2) increase the yield

(i.e., increase the catch by fishers in the unprotected region). Spillover, the movement

of adults from the MPA into the fished region, contributes to both phenomena. Empirical

data contradict the first expectation: more mobile fishes show a greater relative density

inside of MPAs compared to more sedentary species.We hypothesize that this might

be the result of spill in, driven by differential movement of fish into the MPA: i.e., if fish

diffuse at different rates inside versus outside the MPA (as reflected by the parameter

β in our model), then we hypothesize that increased overall movement (reflected in the

parameter D) would lead to a greater buildup of fish inside the MPA.

We described movement and movement bias via a diffusion process and a

discontinuity in diffusion parameters. We were able to show that our model had a

unique, nonnegative, continuous steady state solution. At this steady state, as fish

mobility (D) increased, abundances in the fishing grounds and yields increased,

whereas total abundances and log-response ratios decreased (Figure 2-3). These

qualitative results were independent of the ratio β of the diffusion constants in and

outside the MPA. This result is consistent with past models (using one diffusion

parameter, even if these models did not calculate explicitly the log ratio, a measure

commonly used in empirical studies), but it is inconsistent with the empirical results

of Claudet et al. (2010).[16] In a study of MPAs in the Florida keys, Eggleston and

Parsons (2008)[26] observed spill-in of lobster to MPAs, presumably resulting from

greater movement of lobsters in the fished regions and less movement inside the MPAs.

Thus, differential diffusion (as defined in our model), in the absence of movement bias,

cannot explain these interesting empirical results, nor can other existing models with

even simpler diffusion dynamics. However, the incorporation of a strong movement bias

37

(Figure 2-5, z < z∗) does resolve this paradox by reversing the expected relationships

between mobility and log ratio and yield. Our description of movement bias was one of

a variety of theoretical options available based on Ovaskainen and Cornell (2003).[72]

Because there are no empirical studies of movement patterns inside and outside of

MPAs and at their boundary, we could not motivate our approach from relevant data.

Other assumptions likely affected our results. For example, in our first two models

we assumed fishing mortality was homogeneous within the fished region (and through

time),with no reallocation of fishing effort after the closure of areas to fishing pressure.

Although this is a common assumption made for MPA models [38, 63, 73], it also is well

known that reallocation of fishing effort (temporal heterogeneity) can influence efficacy

of MPAs [54], as can ”fishing the line” [50], that is, increased fishing along the borders of

an MPA.

In all our models we assumed a constant recruitment rate, originating from a

completely open and well mixed larval pool coming from an external system unaffected

by the MPA [92]. As a consequence, the influx of new recruits was independent of

local adult density or habitat. This assumption simplified our analytic approach. Other

theoretical studies also assume open recruitment [38, 77]. Others assume a totally

closed system, usually described by logistic growth terms with density-dependence

arising at particular points in space (e.g., via reproduction or survival) [6, 54, 59, 73, 95].

Assumptions about the recruitment (i.e., reaction) term can affect predicted responses

to MPAs for organisms with different rates of movement. For example, theoretical

work in Lou (2006) [57], shows that for a reaction diffusion equation with Neumann

boundary condition and logistic reaction term, the total abundance A at steady state,

is not monotone in terms of the spatially uniform diffusion constant D, and increases

over certain ranges of D, but decreases over others. Both approaches (open and closed

systems) are extreme versions of real systems and more appropriate models likely

should use dispersal kernels or other distance-limited dispersal modes [63]. More

38

research is needed to determine if our constant recruitment assumption will alter the

conclusions about the qualitative effects of adult movement. Similarly, more analysis is

required to establish whether the form of density-dependence affects predictions about

other aspects of MPAs, including the relationship between yield and MPA size.

We have also presented a movement bias model in which the bias only occurs on

the MPA boundary and nowhere else. In many other movement bias models, the bias

occurs everywhere. For instance, there is a recent body of work on advection diffusion

models in the theoretical ecology literature [10, 11]. Advection may occur through

different mechanisms. An obvious one is when ocean currents move fish populations,

but more sophisticated ways are possible such as movement of fish in the direction of a

resource gradient. Populations will crowd in regions where there are lots of resources

when movement due to this advective source dominates diffusion. This is comparable

to the blow-up phenomenon in chemotaxis systems [66] like the KellerSegel model [49],

which incorporates movement of cells in the direction of a chemical substance that they

secrete themselves.

Finally, following the suggestion of an anonymous reviewer, we investigated a

simple model with smooth parameters. We assumed that diffusion, and recruitment

are spatially uniform, but that mortality is nonuniform and monotonically increasing so

that the mortality rate is higher outside than inside the MPA. It turns out that the fishing

grounds abundance is not necessarily increasing with increased mobility, provided

that the MPA size is small enough. This model therefore provides yet another possible

explanation for the empirical data. More targeted field research will be needed to

elucidate which of these models is the more accurate one, or what modifications the

models should be subject to.

Acknowledgments

We thank Ben Bolker for his invaluable discussions and the Ocean Bridges Program

(funded by the French-American Cultural Exchange) and the QSE3 IGERT Program

39

(NSF award DGE-0801544) for facilitating this collaboration, which was initiated during

Louise Riotte-Lambert’s internship at the University of Florida.

We are also very grateful to two anonymous reviewers whose suggestions allowed

us to make significant improvements to an earlier version of the paper. The first reviewer

suggested that we try to establish the results in Section 2.5. The result in Section 2.6 is

entirely credited to the second reviewer.

40

CHAPTER 3QUORUM SENSING BACKGROUND

3.1 Quorum Sensing

Quorum Sensing (QS) is a means by which bacteria can control gene expression.

In a QS system, diffusible chemicals called autoinducers are synthesized and accumulate

in the local environment. When a critical autoinducer concentration is reached, it triggers

a population-wide shift in gene expression.

One of the first descriptions of QS was as a method of bioluminescence regulation

in the bacteria Aliivibrio fischeri (formerly Vibrio fischeri [89]) and Vibrio harveyi in the

1970’s.[67, 68, 74] Quorum sensing was then thought to be a mechanism by which

bacteria could detect their population density, and that a population-wide reaction would

take place only when a certain density, or quorum, was reached. [36] QS is now realized

to be much more versatile, allowing bacteria to regulate symbiotic interactions and

potentially detect changes in their environment.[25, 42, 75]

There are many examples of QS systems. One such is found in Pseudomonas

aeruginosa, a bacterium that chronically infects the lungs of most cystic fibrosis

patients. P. aeruginosa has at least two QS systems, one of which regulates a multitude

of virulence factors, including the creation of a biofilm, an extracellular matrix of

polysaccharides that encases bacteria and protects them from antimicrobial treatments.

[18] Another example is Sinorhizobium meliloti, a nitrogen-fixing bacteria that forms

a symbiotic relationship with some legumes. S. meliloti controls the establishment of

symbiosis with Medicago sativa through a QS system. [40]

The QS system in A. fischeri was one of the first to be recognized and is still an

area of fervent research. Regulation of bioluminescence in A. fischeri is controlled by

at least three QS circuits, the AinS-AinR, LuxS-LuxP/Q and LuxR-LuxI systems.[58, 62]

The LuxR-LuxI system is used as a paradigm for QS systems in many Gram-negative

bacteria, including P. aeruginosa, described above. LuxR-LuxI homologues exist

41

to regulate genes involved in bioluminescence (as in the case of Vibrio harveyi),

symbiosis, pathogenesis, biofilm formation, genetic competence, motility and antibiotic

production.[24, 35, 62, 74] As LuxR-LuxI homologues are relatively common in nature,

we chose this system as the basis for the models we present in the following chapters.

A more complete description of the LuxR-LuxI system found in A. fischeri appears in

Section 3.2.

3.2 LuxR-LuxI System

Aliivibrio fischeri is a Gram-negative bacteria found both free-living in marine

environments and as a symbiont with the Hawaiian bobtail squid (Euprymna scolopes).

As a symbiont, A. fischeri inhabits the light organ of E. scolopes, a nutrient-rich

environment. In return, A. fischeri equips E. scolopes with counterillumination via

QS-regulated bioluminescence. Light emitted by symbiotic A. fischeri hides the shadow

of the host squid, which provides E. scolopes an additional defense against predation.

[93]

Bioluminescence in A. fischeri is regulated by at least three QS circuits, the

AinS-AinR, LuxS-LuxP/Q and LuxR-LuxI systems. The LuxR-LuxI system is the core

of A. fischeri QS-regulated bioluminescence, as we describe below. The AinS-AinR

and LuxS-LuxP/Q systems influence bioluminescence by regulating production of a

sRNA that transcriptionally represses litR transcript. The transcriptional regulator LitR

enhances luxR expression without altering expression of the other lux genes. [62]

In the LuxR-LuxI system (Figure 3-1), the luxR gene encodes the transcription

factor LuxR. The luxICDABEG operon encodes the LuxI enzyme as well as components

necessary for synthesis of the luciferase, the light-producing enzyme, and production

of its substrates. LuxI catalyzes the synthesis of the acyl-homoserine lactone (AHL)

3-oxo-C6, an autoinducer that transcriptionally activates LuxR. The LuxR/3-oxo-C6

complex activates the expression of the luxICDABEG operon, creating a positive

feedback loop. [24, 35, 62]

42

Figure 3-1. MJ11 is a wildtype A. fischeri with an intact lux operon for synthesis (viaLuxI) and detection (via LuxR) of AHL signal and production ofbioluminescence. Reproduced with permission from Dilanji et al.[22].Copyright 2012 American Chemical Society.

This positive feedback loop acts as a switch, flipping genetic expression from

an unactivated state to an activated state. In the case of the LuxR-LuxI system

in A. fischeri, the activated state is characterized by bioluminescence, which the

unactivated state lacks. Since AHLs are freely diffusible through the cell membrane,

their concentrations are locally approximately equal extracellularly and intracellularly.[35]

Hence, we expect that if one bacterium is experiencing an AHL level high enough to be

activated, then so are its nearest neighbors. As more bacteria become activated, we will

see a switch on the colony level from the unactivated state to the activated state.

3.3 Previous Models

QS systems have been modeled extensively using differential equations and

computational models, mostly for spatially homogeneous systems.[12, 37, 46, 70, 91]

For example, Garde et al. (2010) present a kinetic differential equations model of the

QS system found in Aeromonas hydrophila. The model system is an Escherichia coli

strain that has been genetically modified to contain the genes necessary for autoinducer

detection, but not those necessary for production. When this E. coli strain is activated

(due to high autoinducer concentrations), the synthesis of green fluorescent protein

(GFP) drastically increases. The concentration of GFP (measured via fluorescence)

then gives a measure of bacterial response to AHL. A scheme like this is a common

method of experimentally studying QS systems. Such genetically modified strains of

43

bacteria (commonly E. coli) are called sensor strains. Garde et al. explicitly model the

concentrations of transcription factor proteins, activated transcription factors, activated

receptor sites, nonmature GFP and mature GFP. They use Michaelis-Menten kinetics to

describe the decay of GFP, as we do in Chapter 4. Unlike our model, however, Garde

et al. assume that the signal (autoinducer) molecules are well-mixed and they neglect

auto-feedback by making a quasi-steady state approximation in the equation describing

the concentration of activated transcription factors. [37] These simplifications eliminate

the complex spatial patterns our model elicits, as discussed in Chapter 4.

When authors describe a spatially explicit QS system, the models are typically

very complex and are not analyzed analytically.[43] A common theme among many

of these models is the description of a QS system that regulates biofilm production.

[4, 14, 23, 33, 51, 90] Two examples of other applications of spatially extended models

are Netotea et al. (2009) and Melke et al. (2010). Netotea et al. develop an agent-based

computational model of QS regulated swarming in P. aeruginosa.[69] Melke et al.

formulate a model of QS that allows for the diffusion of autoinducer. They simulate

bacterial QS response under several different environmental geometries and are able to

elicit QS activation in a sparsely populated yet confining geometry.[60]

In a spatially extended system, the QS modulated colony-wide change in gene

expression may appear as a propagating wave, as we explore in Chapter 5. Some

authors have delved into this phenomenon, both experimentally and through modeling.

As we do in Chapter 5, both Danino et al. (2010) and Ward et al. (2003) explicitly

incorporate the diffusion of autoinducer molecules in their models. Danino et al.

constructed microfluidic devices in which a modified E. coli strain exhibited an oscillating

fluorescence response under an autoinducer flow. Under low flow rates, they observed

a spatially propagating wave of fluorescence. Danino et al. formulate a complimentary

system of delay differential equations to describe their experiments and give several

computational simulations. [20] Ward et al. model a QS system that incorporates biofilm

44

production. Their simulations demonstrate a propagating wave of up-regulation through

the colony. They further investigate this wave by examining the scenario where an

up-regulated biofilm is artificially introduced to a significantly larger down-regulated

biofilm. The scenario is further restricted by the assumption that bacteria do not produce

any autoinducer until they are up-regulated. Though they do not mathematically prove

the existence of a traveling wave solution to this simplified model, certain necessary

conditions are explored.[90] In Chapter 5, we introduce a simple QS model intended to

represent the LuxR-LuxI QS system seen in A. fischeri. Our model is able to describe a

QS shift in gene expression, which appears as a traveling wave. The advantage to our

model is that we are able to mathematically prove the existence of this traveling wave.

More recently, stochastic QS models, which compare large-scale and single-cell

dynamics, have appeared.[94] These have accompanied the emergence of experimental

studies that examine the QS dynamics of a single cell using microfluidic devices.[61, 82]

45

CHAPTER 4A SPATIALLY EXPLICIT QUORUM SENSING MODEL

1 As discussed in Section 3.3, much of the previous QS modeling effort has

centered around either spatially homogeneous systems or biofilms. However, all

bacteria, not just those that produce a biofilm, live in a spatially extended world. In

this chapter, we present a simple, spatially explicit QS model based on the LuxR-LuxI

system (see Section 3.2) with an aim to examine spatial patterns in QS.

We constructed a mathematical model for the activation of the quorum sensing

circuit in the sensor strain E. coli + pJBA132 in response to diffusing AHL. This sensor

strain is Escherichia coli MT102 harboring plasmid pJBA132, constructed by Andersen

et al. [2] and containing the sequence luxR-PluxI-gfp(ASV). (Figure 4-2 (A)) The strain

was provided by Dr. Fatma Kaplan. E. coli + pJBA132 contains the sequence (luxR)

necessary to synthesize the protein LuxR. The LuxR/AHL complex binds to the promoter

region PluxI and up-regulates transcription of the sequence (gfp(ASV)) necessary to

synthesize GFP. Since E. coli + pJBA132 synthesizes GFP in place of LuxI, it will

respond to AHL but cannot synthesize AHL. The concentration of GFP (measured

via fluorescence) gives a measure of the response of E. coli + pJBA132 to AHL. The

sequence gfp(ASV) encodes a variant of GFP with a short half life (≤ 1 h), which

prevents GFP from accumulating indefinitely during the measurements.[3]

Our modeling efforts are closely entwined with complementary experiments.

In Section 4.1, we detail the experiment that our model attempts to capture. In this

experiment, diffusing AHL induces GFP production in E. coli + pJBA132. The GFP

then decays over time. In Section 4.2, we develop our model. Next, in Section 4.3, we

1 Reproduced in part with permission from G. E. Dilanji, J. B. Langebrake, P.De Leenheer, and S. J. Hagen. Quorum activation at a distance: Spatiotemporalpatterns of gene regulation from diffusion of an autoinducer signal. J. Am. Chem. Soc.,134(12):56185626, 2012.

46

estimate parameters for our model using a spatially homogeneous E. coli + pJBA132

experiment. We are then ready to test our parameterized model predictions against

spatially extended data, which we describe in Section 4.4. The first of four sets of

experiments (Section 4.4.1) explores the spatial pattern created by a diffusing dye.

The other three spatially extended experiments (Section 4.4.2) investigate the spatial

patterns elicited by three strains of bacteria (E. coli + pJBA132, A. fischeri strain

VCW267 (-luxI) and A. fischeri wild-type strain MJ11) when exposed to a diffusing AHL

signal. A. fischeri strain MJ11 is a wild-type strain with an intact lux operon for AHL

synthesis and response. VCW267 is a mutant A. fischeri that lacks the AHL synthase

(LuxI). (Figure 4-2 (A)) Lastly, in Section 4.5, we compare our parameterized model

predictions against all four sets of spatially extended experiments and provide some

discussion. (Figure 4-1)

Figure 4-1. Chapter 4 methodology. Beginning in the upper left (Modeling) box, themodeling in this chapter progresses via the solid arrows. The left-hand(Experiment) boxes give input to modeling boxes via dashed and dottedarrows. The dashed arrows signify data input while the dotted arrow signifiesexperimental setup input.

47

Figure 4-2. Experimental configuration: (A) Strains used in this study. MJ11 is a wildtypeA. fischeri with an intact lux operon for synthesis (via LuxI) and detection(via LuxR) of AHL signal and production of bioluminescence; VCW267 is amutant A. fischeri lacking the AHL synthase (LuxI); The pJBA132 “sensor”strain of E. coli has a gfp reporter under control of the luxI promoter, butlacks luxI; (B) The light dome provides highly uniform, diffuse excitation lightfor imaging GFP fluorescence of bacteria embedded in agar. The sameoptical configuration allows us to measure bioluminescence and opticaldensity of the samples in situ; (C) Bacteria/agar mixture is loaded into aframe containing four parallel, independent lanes. A droplet of autoinducerdeposited at the terminus of each lane diffuses down the lane, generating apattern of QS activation; (D) Representative fluorescence images showingimages collected from a typical E. coli + pJBA132 experiment.

48

4.1 Experimental Configuration

We model an experiment performed by Gabriel Dilanji wherein AHL (concentration

C(x , t) nM) is loaded into the terminus of a long (32 mm) agar lane populated by E. coli

+ pJBA132. The AHL is then allowed to diffuse down the lane with diffusion constant D

(mm2/h). The bacteria are held stationary by the agar and therefore do not diffuse. [19]

(Figure 4-2 (B)) We will model this lane as one-dimensional (x , pointing down the lane

and where x = 0 is the source) under the assumption that the lane is homogeneous in

the transversal direction.

For the duration of the experiment, we measure both cell population density

(n(t), cells per cm3) and the concentration of fluorescent GFP per unit volume of agar

(G(x , t)). The cell density is assumed to be proportional to the experimentally-measured

optical density of the agar and thus is measured in OD units. As we detect GFP through

its fluorescence (per camera pixel), G(x , t) is measured in units of counts per pixel.

Since E. coli + pJBA132 responds to exogenous AHL by synthesizing GFP, the

spatio-temporal pattern of fluorescence in the lane is analogous to the pattern of QS

up-regulation.

We perform similar lane experiments with a fluorescent dye as well as two strains

of Aliivibrio fischeri, MJ11 and VCW267 (-luxI). A. fischeri strain MJ11 is a wild-type

strain with an intact lux operon for AHL synthesis and response. VCW267 is a mutant

A. fischeri that lacks the AHL synthase (LuxI). (Figure 4-2 (A)) As both of these strains

bioluminesce in the presence of AHL, we measure luminescence (counts per pixel) in

lieu of fluorescence.

In addition to these lane experiments, we performed a well-plate experiment

wherein E. coli + pJBA132 was grown in agar in the presence of varied concentrations

of exogenous AHL (0 - 500 nM). Each of sixteen wells contained homogeneously

distributed AHL at a fixed concentration C . We measured each of cell density (n) and

GFP fluorescence (G) in each well over a period of ∼ 25 h. This experiment was used to

49

parameterize the model described in Section 4.2. We detail the parameterization of this

model in Section 4.3.

For additional details regarding these experiments, see Methods, Section 4.6.

4.2 Mathematical Model

To model the interaction between spatial diffusion of the AHL (acyl homoserine

lactone autoinducer) and the expression of GFP, we consider each of cell population

density (cells per cm3), AHL concentration and GFP concentration. The number of

bacterial cells per unit volume in the agar lane is denoted by n(t), which is presumed

to be proportional to the experimentally-measured optical density of the agar and is

therefore measured in OD units. In our experiments, the initial cell density is the same

everywhere and the cells are immobilized in agar [19]. Hence, n is independent of

one-dimensional space x and is a function only of time t. We describe the change in cell

density with respect to time as a logistic function:

dn

dt= nα

(1− n

K

)(4–1)

where K is the carrying capacity and α/ ln(2) is the intrinsic growth rate (doublings per

hour) [21].

In a well-plate experiment (see Methods, Section 4.6), exogenous AHL is provided

in the growth medium and is well-mixed. As the sensor strain E. coli + pJBA132 cannot

produce AHL, the AHL concentration C is then constant. However in a lane experiment,

exogenous AHL is supplied at the lane terminus (x = 0) and diffuses outward with time.

Then C is a function of space and time, C(x , t). As AHL is chemically stable for our

experimental conditions and time scales [27], C(x , t) evolves according to the diffusion

equation,∂C

∂t= D

∂2C

∂x2(4–2)

where D is the diffusion constant.

50

We also consider the concentration of the fluorescent protein GFP per unit volume

of agar. Following expression of the gfp gene, the GFP polypeptide is not fluorescent

until it has undergone a maturation process that involves folding, cyclization, dehydration

and oxidation [88]. We model this process with three forms of GFP: U1 represents

the newly synthesized, non-fluorescent polypeptide, U2 represents a folded but

non-fluorescent protein, and G is the mature fluorescent protein. As G is detected

through its fluorescence (per camera pixel) the units of measurement for all three

forms are counts/pixel. The GFP is an unstable variant (GFP(ASV)[3]) and we model

the degradation of each of its forms (U1, U2, G ) as a competitive, Michaelis-Menten

process[37, 55]:

g(V ) =k1V

k2 + U1 + U2 + G(4–3)

where V can be any of U1, U2 or G . Hence, all forms of GFP are degraded at equivalent

rates and the total rate of protein degradation is

k1U1

k2 + U1 + U2 + G+

k1U2

k2 + U1 + U2 + G+

k1G

k2 + U1 + U2 + G=

k1T

k2 + T(4–4)

where T = U1 + U2 + G . It is now easy to see that k1 is the maximum degradation rate

(h−1) of GFP while k2 (counts/pixel) is the Michaelis constant of (4–4).

Since GFP is synthesized as U1, the time derivative ∂U1/∂t depends explicitly

on the rate of gfp expression in response to AHL. This rate depends on both the

concentration of AHL (C ) and on the growth stage, as well-plate experiments show

synthesis slowing as n → K . We use a Hill function (4–5) to model the AHL-dependence

of U1 synthesis. The Hill function provides cooperative switching from the synthesis-off

to the synthesis-on states, via two parameters a and m:

f (C) =Cm(x , t)

am + Cm(x , t)(4–5)

Here a is the half-activation constant (units of nM) and m is the Hill coefficient (dimensionless).

51

To model the growth-rate dependence of GFP production our model also includes

the logistic growth function (4–1) in the rate of GFP production:

αn(1− n

K

).

Finally the model also requires a proportionality factor γ (counts/pixel) that

determines the overall rate of GFP production. Then the production of U1 proceeds

at a rate

γf (C)αn(1− n

K

).

We describe the transformation of U1 into U2 with the constant per capita rate m1

(h−1), and the transformation of U2 into G with the constant per capita rate m2 (h−1).

Combining all of these processes, we have

∂U1

∂t= γf (C)αn

(1− n

K

)−m1U1 − g(U1) (4–6)

∂U2

∂t= m1U1 −m2U2 − g(U2) (4–7)

∂G

∂t= m2U2 − g(G) (4–8)

where f (C) is defined in (4–5) and g(V ) is defined in (4–3).

Now that we have the system of (4–1), (4–2), (4–6), (4–7), and (4–8), we specify

initial conditions. The initial cell density is constant everywhere and we denote it by n0.

n(0) = n0

In the lane (diffusing signal) experiments, the cells grow in a narrow channel of length

L (mm). AHL is initially deposited at time t = 0 onto a region of length ν (mm) at one

terminus (x = 0) of the lane. The amount of AHL initially loaded is characterized by the

concentration it produces when fully diffused (t → ∞) throughout the lane, C∞ (nM).

52

Hence the initial condition for C(x , t) is

C(x , 0) =

C∞

Lν

0 ≤ x ≤ ν

0 ν < x ≤ L

There is no GFP present at the beginning of the experiment:

U1(x , 0) = 0

U2(x , 0) = 0

G(x , 0) = 0

Finally, we must include a reflective boundary condition for C at the edges of the lane

[0,L]:

∂C

∂x(0, t) =

∂C

∂x(L, t) = 0 for all t > 0

In summary, our model is:

dn

dt= nα

(1− n

K

)(4–9)

∂C

∂t= D

∂2C

∂x2(4–10)

∂U1

∂t= γf (C)αn

(1− n

K

)−m1U1 − g(U1) (4–11)

∂U2

∂t= m1U1 −m2U2 − g(U2) (4–12)

∂G

∂t= m2U2 − g(G) (4–13)

n(0) = n0 (4–14)

C(x , 0) =

C∞

Lν

0 ≤ x ≤ ν

0 ν < x ≤ L

(4–15)

U1(x , 0) = 0 (4–16)

53

U2(x , 0) = 0 (4–17)

G(x , 0) = 0 (4–18)

∂C

∂x(0, t) =

∂C

∂x(L, t) = 0 for all t > 0 (4–19)

where

f (C) =Cm(x , t)

am + Cm(x , t)(4–20)

and

g(V ) =k1V

k2 + U1 + U2 + G. (4–21)

Table 4-1 lists all of the parameters and variables and their estimated values. The

method for estimation is described in the following section.

4.3 Parameter Estimation

All parameter estimation was performed with MATLAB R⃝ [86].

We obtained parameters for our model by analyzing data from experiments

conducted in a standard 48-well-plate. E. coli + pJBA132 was grown in agar in the

presence of different concentrations of exogenous AHL (3-oxo-C6-HSL, see Methods,

Section 4.6), while optical density and fluorescence were recorded over a period of

∼ 25 h. We fit data from sixteen wells, with AHL concentrations ranging from 0 nM to

500 nM. Since the AHL was mixed into the agar at the start of this measurement, (4–9)

- (4–19) reduce to a space-independent system (G(x , t) → G(t), U1(x , t) → U1(t),

etc.) where C is constant within each well. The experiment does not provide the AHL

diffusion constant D. However, for the 3-oxo-C6-HSL autoinducer used in the present

study and an aqueous medium, the literature suggests D ≃ 5.5× 10−6 cm2/s = 2 mm2/h

[45, 84]. We do not fit the data at later times on the growth curve, t ≥∼ 13 h, where the

degradation of GFP slows and the slope of OD(t) indicates weak or negative growth

that is inconsistent with the logistic (4–1).

54

Table 4-1. A summary of variables and parameters used in the model (4–9) - (4–21).

Variable / Definition Values / UnitsParametern cell concentration optical density (dimensionless)α intrinsic cell growth rate 0.8 h−1

K cell carrying capacity 0.9 (well-plate),0.4 (lane) optical density (dimensionless)

C AHL concentration nMD AHL diffusion constant 2 mm2/hU1 unfolded GFP counts/pixelγ proportionality factor 6.9× 107 counts/pixelf cooperative switch function unitlessa half-activation coefficient 1.5 nMm Hill coefficient 1.66 unitlessm1 folding rate of GFP 0.67 h−1

g(V ) degradation of GFP in form V counts/(pixel · h)k1 maximum degradation rate 2.4× 105 counts/(pixel · h)k2 Michaelis constant 6.2× 104 counts/pixelU2 folded but non-fluorescent GFP counts/pixelm2 maturation rate of GFP 29 h−1

G fluorescent GFP counts/pixeln0 initial cell concentration optical density (dimensionless)C∞ fully-diffused concentration of AHL nML length of agar lane 32 mmν length of AHL loading region 2 mm

We performed some adjustments to the well-plate data prior to fitting. As wells

containing different AHL concentrations did not reach peak growth rate at exactly the

same time, we applied small horizontal shifts to the OD(t) (and corresponding G(t))

curves until all dn/dt data reached their peak at the same time t. This offset was on the

order of 0.04-0.65 h, which is small with respect to the duration of the GFP production

phase of interest.

Furthermore the OD(t) values are measured relative to OD(0), not absolute. This

creates an ambiguity in the initial OD for each well, or equivalently in n0. Therefore we

estimated n0 for each well by requiring that each n(t) (and its dn/dt) fit a logistic growth

model at early times: (4–1) requires that dn/dt → 0 as n → 0 at early times. Therefore

55

Figure 4-3. Experimental data from well-plate and fits to model. Curves are offsetvertically for clarity. (A) GFP fluorescence, (B) ∂G/∂t, (C) OD(t), and (D)dOD(t)/dt are shown for E. coli + pJBA132 growing in LB agar in awell-plate in the presence of the indicated exogenous AHL concentrations.Data (solid) were fit to the model of (4–9) - (4–19) (dashed).

we added to each OD(t) curve a small offset sufficient to give OD → 0 as dOD/dt → 0.

That is, we use the logistic growth model to set the (unknown) offset OD(0) in the data.

We then use the OD(t) as the experimental values for n(t).

We then obtained the growth rate α by fitting (4–9) to all 16 optical density curves,

using a global nonlinear least squares method, which consists of a Nelder-Mead

simplex (direct search). The resulting growth rate α ≃ 0.8 h−1 is typical under our

growth conditions. [37, 87] We find a higher carrying capacity K ≃ 0.9 (OD units) in

the well-plate than in the lane experiments, where we find K ≃ 0.4, as the well-plate

samples are deeper than the agar lanes. (Figure 4-3)

56

We then fit (4–11) - (4–13) to obtain the remaining parameters. We first defined a

set of scaled variables

U1 =U1

γ, U2 =

U2

γ, G =

G

γ, k1 =

k1

γ, k2 =

k2

γ

in order to eliminate the constant factor γ in the model equations. This reduces the

number of parameters to be fit:

∂U1

∂t= f (C)αn

(1− n

K

)−m1U1 −

k1U1

k2 + U1 + U2 + G(4–22)

∂U2

∂t= m1U1 −m2U2 −

k1U2

k2 + U1 + U2 + G(4–23)

∂G

∂t= m2U2 −

k1G

k2 + U1 + U2 + G(4–24)

where f (C) is defined in (4–20). Recall that C , and hence f (C), is constant for each

well. We used literature data to make an initial guess for the Hill parameters a and m.

(The guess was later refined as described below.)

We then used a global nonlinear least squares routine to fit (4–22)-(4–24) and

found values for m1, m2, k1 and k2. The routine employed a Nelder-Mead simplex (direct

search) method, minimizing the sum of square errors between the GFP data and the

rescaled model prediction γ∗G , where γ∗ is the best constant solution to the equation

γ∗G = Gdata. Least square errors were found for each of 16 well-plate data sets with

nonzero AHL concentration and summed to give the global error.

We refined our estimate for a and m by repeating the above fit (for m1, m2, k1 and

k2) for many values of a and m and looking for a global minimum to the sum of squares.

We found the maturation rates to be m1 ≃ 0.67 h−1 and m2 ≃ 29 h−1. These

values imply a GFP maturation time m−11 + m−1

2 ≃ 1.5 h. Thus for growth in agar we

obtain slightly slower maturation for the unstable green fluorescent protein GFP(ASV)

maturation time than the ≃ 0.67 h that was observed in liquid medium.[37]

57

To compare our degradation parameters k1 ( = γ∗k1 ≃ 2.4 × 105 counts/(pixel · h) )

and k2 ( = γ∗k2 ≃ 6.2 × 104 counts/pixel ) to literature values measured in units of h−1,

we can examine the typical degradation rates present in our model system. From the

total reporter protein T = U1 + U2 + G present in the model, the effective decay rate at a

given time t is

k1T (t)

k2 + T (t)

(1

T (t)

)=

k1

k2 + T (t)

This yields a (t-dependent) decay rate 0.09-3.7 h−1, which spans literature values of

∼ 0.3− 0.4 h−1.[3, 37, 55]

4.4 Results of Lane Experiments

The experiments described herein are attributed to Gabriel Dilanji.

To experimentally explore spatio-temporal patterns in QS regulation, we performed

four sets of lane experiments. In these experiments, bacteria were embedded in a

long agar lane, then either fluorescent dye or exogenous AHL was loaded into the

lane terminus (x = 0) and allowed to diffuse down the lane. As the dye diffused, we

measured the fluorescence of the lane. As the exogenous AHL diffused, we measured

both cell density (optical density) and either fluorescence or luminescence, depending

on the nature of the QS response of the embedded bacteria.

4.4.1 Diffusion of a Dye

The simplest pattern we may expect to see in the lane experiments is that of

one-dimensional diffusion, as governed by (4–10). To examine this pattern and to create

a benchmark for analyzing our subsequent results, we loaded 1 µL of a fluorescein

solution (∼ 0.2 µM in water) into the terminus (x = 0) of a long agar lane populated by

E. coli + pJBA132. This fluorescent dye is then allowed to diffuse down the lane. The

resulting pattern in C(x , t) (measured via fluorescence) is shown in Figure 4-4 (A-C).

We see strong interdependence of space and time: the concentration of dye changes

rapidly close to the dye droplet and at early times, while it changes more slowly farther

58

from the droplet and at later times. This is characteristic behavior of diffusive spreading,

described by x2 ∼ 2Dt. The same patterns are observed regardless of whether bacteria

are present in the agar lane.

Figure 4-4. Diffusion of fluorescein dye in agar lane. Fluorescence is plotted as afunction of (A) time t and (B) distance x from dye droplet. The contour linesin a map of fluorescence vs. x and t show the x2 ∼ 2Dt behavior that ischaracteristic of simple diffusive spreading; (D) Simulation of (4–10) by thefinite difference method in MATLAB, based on D =1.5× 10−6 cm2 /s andusing the same dye concentration and boundary conditions as in (C).

4.4.2 LuxR-LuxI System Response to AHL Diffusion

Figure 4-5 shows the response G(x , t) of the sensor strain E. coli + pJBA132 to

a diffusing AHL signal. We observed a spatially propagating response that extended

on the order of ∼ 1 cm over the course of ∼ 10 h. This distance is significant when

compared to the size of a single E. coli bacterium, which is ∼ 2− 4 µm in length.

The pattern of activation we observed (Figure 4-5) is qualitatively different from

standard diffusion (Figure 4-4). The observed fluorescence response is remarkably

self-similar when examined at fixed points in space or at fixed times. Each of these

slices of the surface G(x , t) are qualitatively similar, but vary in magnitude. When

we consider a fixed point in space x , we see a stronger response closer to the lane

terminus (x = 0). If we consider a fixed point in time, the magnitude of the response

59

increases until t ≃ 10 h, then the signal begins to fade. This pattern is caused by the

cell’s nonlinear activation response, which tapers off when cells enter the stationary

phase. That G(x , t) displays qualitative patterns that are independent of position and

time is characteristically dissimilar from diffusion.

Figure 4-5. Response of the sensor strain (E. coli + pJBA132) to diffusing AHL. (A) and(D) show G(x , t), the spatiotemporal pattern of reporter fluorescence,following deposition of an AHL (3-oxo-C6- HSL) droplet at the terminus(x = 0) of an agar lane at t = 0. The amount of AHL introduced wassufficient to produce a final (fully diffused) concentration of C∞ = 0.4 nM(A-C) or 4nM (D-F) throughout the lane. (B) and (E) show slices throughG(x , t) at fixed distances x , while (C) and (F) show slices though G(x , t) atfixed times t.

Figure 4-6 shows the response of luxI-deficient A. fischeri strain VCW267 to

diffusing AHL. VCW267 contains the genes necessary for detection of AHL and for

bioluminescence, but not those necessary for the production of the AHL synthase LuxI.

(See Methods, Section 4.6; Figure 4-2) We see a qualitatively similar response as

in the experiments with E. coli + pJBA132 (Figure 4-5), though the sensitivity of the

60

response is reduced. We again observe signaling distances on the order of ∼1 cm and

independence of the response G(x , t) with respect to position and time.

Figure 4-6. Bioluminescence response of luxI-deficient A. fischeri VCW267 to diffusingAHL. (A) and (D) show the bioluminescence vs. x and t, following depositionof a 3-oxo-C6-HSL droplet at the terminus (x = 0) of an agar lane at t = 0.The amount of AHL introduced was sufficient to produce a final (fullydiffused) concentration of C∞ = 400 nM (A,B,C) or 2 µM (D,E,F). (B) and (E)show slices through the data at fixed distances x , while (C) and (F) showslices at fixed t.

Lastly, we examined the response of wild-type A. fischeri strain MJ11 to diffusing

AHL. MJ11 contains an intact LuxR-LuxI system, and thus is able to detect and

synthesize AHL, and to bioluminesce. (See Methods, Section 4.6; Figure 4-2) The

ability of MJ11 to synthesize AHL allows for auto-feedback in this process. In fact, in our

experiments, the bacterial production of AHL overwhelmed the exogenously introduced

AHL (C∞ = 0,1,6, or 60 nM) by t ≃ 5 h, at which time the entire lane luminesced brightly.

However, for small x and early t, the patterns of response were qualitatively similar to

61

those described above for E. coli + pJBA132 and for A. fischeri VCW267. (Data not

shown)

4.5 Discussion and Model Simulations

QS is an inherently spatial process as it depends on the production and transport

of small autoinducer molecules. Now that there is a wealth of QS modeling in spatially

homogeneous systems, which neglect autoinducer transport by assuming that the

medium is well-mixed (as discussed in Section 3.3), we are well-equipped to explore the

realm of spatial inhomogeneities. This is of practical importance as natural QS systems

often occur in highly heterogeneous environments, such as the rhizosphere, the area of

soil very close to the root of a plant. [42] We have already mentioned (in Section 3.1) an

example of this very circumstance: S. meliloti is a nitrogen-fixing bacteria that occurs

both free-living in the soil and as a symbiont with some legumes. The establishment

of this symbiosis occurs partially in the rhizosphere and is facilitated by a complex

QS system.[40] Though our model does not attempt to incorporate environmental

heterogeneity, it does allow for spatial heterogeneity in autoinducer concentration.

Our mathematical model (4–9)-(4–21) gives a simple spatially-explicit description of

a QS circuit. We modeled the spatio-temporal patterns of gene regulation due to a QS

circuit via the diffusion of an autoinducer. We modeled the transcriptional response to

this autoinducer with a nonlinear (Hill) function modulated by logistic bacterial population

growth(dndt

). Our measure of response is the magnitude of fluorescence of GFP, which

exists in two immature (non-fluorescent) and one mature (fluorescent) state, each

degraded by a Michaelis-Menten process.

As a baseline against which to compare our model simulations and experiments,

we first examined the simple diffusion of a dye. (Figure 4-4 (A-C) gives the experimental

results.) Figure 4-4 (D) shows a numerical simulation of the diffusion equation (4–10).

The initial concentration of dye C(x , t = 0) is modeled as a step function to emulate

our experimental setup. The calculated C(x , t) pattern is very similar to that obtained

62

experimentally and displays the same x2 ∼ 2Dt behavior, a characteristic of diffusive

spread. Comparing this numerical solution to the data allows us to estimate D ≃

1.5× 10−6 cm2 /s for diffusion of fluorescein in 0.75% agar. The good agreement

between the simulation and experiment suggests that the agar lane may indeed be

approximated by one spatial dimension.

Figure 4-7. Patterns of expression predicted for the E. coli + pJBA132 sensor strain inresponse to diffusing AHL. The patterns were generated by simulation usingthe model (4–9)-(4–21), assuming a final AHL concentration of C∞ = 0.4 nM(A) or 4nM (B,C). (D) shows the concentration of diffusing AHL, C(x , t) (inµM), assuming a final AHL concentration of C∞ = 4 nM and a diffusionconstant D = 2 mm2/h.

We used the parameters found in Section 4.3 along with our literature estimate

for the diffusion constant of the AHL 3-oxo-C6-HSL (summarized in Table 4-1) to

simulate the spatio-temporal response (G(x , t)) of the sensor strain E. coli + pJBA132

to diffusing exogenous AHL (C(x , t)). We simulated two different AHL loading scenarios,

one in which the final AHL concentration C∞ = 0.4 nM (Figure 4-7 (A)) and another

in which C∞ = 4 nM (Figure 4-7 (B-D)). These simulations correspond to two

lane experiments described in Section 4.4.2 and shown in Figure 4-5. Both of our

simulations are qualitatively and quantitatively similar to the experimental data. In both

63

experiments and both simulations, the peak in response occurs at ∼ 10 − 12 h. In both

the experiment and the simulation, the C∞ = 0.4 nM loading resulted in a response

extending ∼4mm and the C∞ = 4 nM loading resulted in a response extending ∼1 cm.

As in the corresponding experiments, our model prediction of G(x , t) is qualitatively

independent of space or time, though the magnitude of the response is not. This

excellent agreement between model and experiments suggests that we are able to

capture the essential elements of the LuxR-LuxI system response by incorporating a

diffusive signal, non-linear cell response to AHL, and logistic cell growth in our model.

When compared to lane experiments conducted with A. fischeri VCW267 (-luxI),

our model simulations predict a qualitatively similar response pattern, though the

magnitude of the response differs. (Figures 4-6, 4-7) Similarly, before the exogenously

introduced AHL is overwhelmed by endogenous AHL, our model simulations qualitatively

describe the QS response of wild-type A. fischeri MJ11. It is unsurprising that our

model diverges from these MJ11 lane experiments as our model does not incorporate

bacterial production of AHL, an important feature of the wild-type strain MJ11. The

good qualitative agreement between our model predictions and the A. fischeri lane

experiments (at least at early times and small x in the case of the wild-type strain MJ11)

suggests that even though A. fischeri controls bioluminescence through three entwined

QS circuits (AinS-AinR, LuxS-LuxP/Q and LuxR-LuxI, as described in Section 3.2),

the minimal set of components in our model characterizing the LuxR-LuxI system is

sufficient to describe the bulk of the QS bioluminescence response of A. fischeri.

Our experimental results and model predictions showed two interesting features

that warrant further mention. First, we observed and predicted an extended, coordinated

cell response on the millimeter or centimeter scale. These response distances are

larger than we would expect from a molecule diffusing at a rate of 2 mm2/h. Second,

we observed and predicted a temporal synchronization of response within the bacterial

colony, an important feature of QS systems. Since autoinducer is diffusing away from

64

a point source, the amount of time it takes for the autoinducer concentration to reach a

given value at a given distance x from the source scales as x2. Thus, we may expect

that bacteria slightly farther from the source would respond much later. However, we see

that the qualitative temporal pattern of response remains constant as the distance from

the source x increases, though the intensity decreases (Figures 4-5, 4-6, 4-7).

Our experiments raise an interesting question: In a single bacterial culture, what

is the QS response to multiple autoinducers that have different diffusion coefficients?

S. meliloti makes a case for the relevance of this question. S. meliloti has a complex

QS system involving many different autoinducers. These autoinducers have carbon

chains ranging from 8 carbons to 18 carbons in length.[40] These short- and long-chain

autoinducers have a range of diffusion coefficients, and thus we would expect S. meliloti

to respond to them in different ways. However, the difference in diffusion coefficient

alone may not be sufficient to characterize the difference in response. The strength of

the auto-inductive QS response to a particular autoinducer may vary independently of

the magnitude of diffusion coefficient. Thus, a slower-diffusing autoinducer may induce

a farther-reaching response if the QS circuit is more sensitive to it. This is an area of

current research.

In the following chapter, we formulate a simplified version of the model (4–9)-(4–21)

to further examine patterns in QS gene activation. Specifically, we study the existence of

a traveling wave solution to our simplified model. This line of research could potentially

give insight as to the speed of signal propagation as a compliment to this chapter’s

comments on the distance of signal propagation.

4.6 Methods

This section and the experiments described herein are attributed to Gabriel Dilanji.

4.6.1 Bacterial cultures

Figure 4-2 shows the QS bacterial strains used in this work. The quorum “sensor”

strain is Escherichia coli MT102 harboring plasmid pJBA132, constructed by Andersen

65

et al. [2] and containing the sequence luxR-PluxI-gfp(ASV) (Figure 4-2). The strain

was provided by Dr. Fatma Kaplan. Cultures in exponential phase were prepared by

growing the E. coli to OD600 = 0.3 in Luria-Bertani (LB) medium, approximately pH

7, at 37 ◦C. The culture was prewarmed for 15 s at 50 ◦C Celsius in order to promote

survival in warm agar [5], and then diluted 100× into molten 0.75% LB agar at 50 ◦C.

250µL of the agar mix was then quickly pipetted into each of the four parallel lanes in

the observation device (described below). The lane device was sandwiched between

two glass coverslips as the agar cooled. The upper glass coverslip was then carefully

removed, leaving a very flat and uniform slab of agar within each lane. The device was

incubated at room temperature for 1.5h before measurements began.

Aliivibrio fischeri strain VCW267 is a synthase-deficient (-luxI) mutant produced

from an ES114 wildtype background and was provided to us by Dr. Eric Stabb. A.

fischeri strain MJ11 is a wild type strain that was derived from its symbiotic host fish

Monocentris japonicus and provided to us by Dr. Mark Mandel. Both strains were

grown to OD600 = 0.3 in commercial photobacterium medium (No. 786230, Carolina

Biological), approximately pH 6.9, at room temperature and then prepared as above

for the agar lanes. The photobacterium medium is a rich medium composed of yeast

extract, tryptone, phosphate buffer, and glycerol in artificial seawater.

4.6.2 Well-plate measurements

In order to obtain parameters for our mathematical model for E. coli + pJBA132

growth and its response to the AHL, we measured the optical density and fluorescence

of this strain in the presence of various autoinducer concentrations in a multiwell plate,

using an automated plate reader (Biotek Synergy 2). A dilute culture was loaded

into individual wells containing 0.1% agar in LB medium and AHL (3-oxo-C6-HSL,

N-(3-oxohexanoyl)-L-homoserine lactone, CAS 143537-62-6, Sigma Chemical Co.) at

concentrations ranging from 0 to 500nM. GFP fluorescence and optical density of each

well were measured over a period of ∼ 25h at room temperature. As described in the

66

Supporting Information we then fit the resulting multidimensional dataset (optical density

and GFP fluorescence × [AHL] × time) to the model that is formulated in Section 4.2.

4.6.3 Lane apparatus and imaging

We studied spatiotemporal patterns of QS regulation in bacteria/agar mixtures

that were loaded into rectangular lanes of length 32mm and cross section 3.5mm ×

2mm (width × depth). These agar lanes were prepared by casting the agar mixture

into a black-anodized aluminum “comb” or frame that defined four parallel channels

(Figure 4-2). The frame rested on a glass coverslip that was coated with a thin,

transparent silicone elastomer sealant (Sylgard 184, Dow Corning Inc.). The humidity

of the agar was maintained by covering the lanes with a clear polycarbonate lid during

measurements. GFP fluorescence excitation was provided by blue LED light passing

through an excitation filter (Thorlabs MF469-35) and diffusively scattered toward

the sample by a light dome (Figure 4-2). The light dome was a plastic hemisphere

(15 cm diameter) whose interior was coated with a high-reflectance, non-fluorescing

BaSO4 paint [85]. Multiple scattering of the excitation light within the dome yielded

highly uniform illumination of the agar lanes: The variation in illumination across the

image field was less than 3%. No excitation light was required for the bioluminescence

measurements. Luminescence and OD probe light (see below) from the four parallel

lanes were collected through the same optical path (Figure 4-2) and imaged on a CCD

camera.

The lane/coverslip assembly was seated on a black anodized aluminum baseplate

that contained an array of pinholes (0.7mm diameter), allowing in situ measurements of

the agar optical density (OD): green LED light was directed upward through the pinholes

(from beneath the baseplate) and through the agar to produce a transmitted light image

on the camera. Using a timer circuit to switch between two light sources (blue GFP

fluorescence excitation versus green pinhole light for OD) in alternate exposures, we

67

collected a sequence of OD and fluorescence/bioluminescence images of each lane

over the measurement period.

We introduced exogenous AHL into an agar lane by depositing 1µL of a concentrated

AHL (3-oxo-C6-HSL) solution onto the surface of the agar at one terminus of the lane.

(C∞ in the figure labels refers to the fully diffused, t → ∞, AHL concentration that

resulted from this initial loading.) To generate the simple diffusion pattern shown in

Figure 4-4, we used fluorescein dye (CAS No. 2321-07-5, Sigma Chemical Co.) instead

of autoinducer solution.

Images were recorded on a CCD camera (1300 × 1030 array of 6.7µm pixels

with 12-bit readout, cooled to −10 ◦C, MicroMax, Princeton Instruments) through a

2× achromatic doublet lens (MAP1075150-A, ThorLabs) and a GFP emission filter

(MF525-39, Thorlabs). CCD images were collected with exposure times ∼ 1-10 s

and a repetition rate of 0.004Hz over periods of 20-24 h. The CCD images were

hardware-binned by 5 pixels in the y -direction (transverse to diffusion) and by 2 pixels in

the x-direction (along the direction of diffusion). The image frame captured a 13.8mm

length along each of the four lanes, or nearly half of each 32 mm lane.

68

CHAPTER 5SIGNAL PROPAGATION IN A QUORUM SENSING SYSTEM

In this chapter, we introduce a simple, spatially extended QS model intended to

represent the LuxR-LuxI system present in Aliivibrio fischeri. In this system, the luxR

gene encodes the transcription factor LuxR. The luxICDABEG operon encodes the

LuxI enzyme as well as the components necessary for bioluminescence. LuxI catalyzes

the synthesis of an AHL that transcriptionally activates LuxR. The LuxR/AHL complex

activates the expression of the luxICDABEG operon, thus creating a positive feedback

loop. (See Section 3.2 for a more complete description of the LuxR-LuxI system.)

As in the previous chapter, we model a colony of bacteria (in this case, A. fischeri)

embedded in agar in a long rectangular lane environment. We again assume that the

lane is homogeneous across its width and therefore describe the lane in one spatial

dimension x . However, in this chapter, we assume that the lane has infinite length, that

is, that x ∈ R.

Our model is able to describe a QS shift in gene expression on the colony level,

which appears as a traveling wave. As we mentioned in Section 3.3, Danino et al. and

Ward et al. both studied waves of gene expression in QS systems via experiments

and modeling.[20, 90] However, neither of these teams of authors proves the existence

of a traveling wave solution to their models, as we do. Our mathematical proof of the

existence of a solution that exhibits the characteristics of a colony-level shift in gene

expression gives credence to the model we use to describe the QS system in A.

fischeri. Furthermore, our mathematical rigor guarantees that the solution we describe is

unaffected by any numerical error or artifacts of computational technique.

In the following section, we present our model of the LuxR-LuxI system in A.

fischeri. In Section 5.2, we give parameter conditions under which there exists a

traveling wave solution to our model. In Section 5.3, we give a general theorem and

69

proof of the existence of a traveling wave solution to a class of reaction-diffusion

systems that includes our model.

5.1 Mathematical Model

To model the LuxR-LuxI system found in A. fischeri, we consider only AHL

concentration and LuxI concentration in an infinite one-dimensional domain x . Though

the concentration of LuxR plays a role in the LuxR-LuxI system, we assume that it is not

a limiting factor in our system and do not model it explicitly. We also do not model the

population density of A. fischeri, though the growth phase of bacteria plays a role in the

regulation of protein synthesis.[22, 90] Thus, we expect that our model will only be valid

for short time scales relative to cell growth. The following model is largely inspired by

that in Section 4.2, but has been simplified to facilitate the mathematical exploration of a

traveling wave.

As AHL is freely diffusible through the A. fischeri cell membrane, we describe the

spatial spread of AHL (concentration A(x , t), nM) by the diffusion equation:

∂A

∂t= D

∂2A

∂x2

where D (mm2 /h) is the diffusion constant. [35, 48] Since LuxI catalyzes A. fis-

cheri synthesis of AHL, the production rate of AHL depends explicitly on the LuxI

concentration L (units of nM). We denote the per unit LuxI production rate of AHL

by λ (h−1). Finally, we assume that AHL degrades at a constant per-capita rate γ

(h−1). Though AHL is stable on the scale of hours at neutral pH, AHL degrades at a

non-negligible rate under alkaline conditions and in the presence of quorum-quenching

enzymes. [1, 27, 45] Hence, we describe the AHL concentration A(x , t) by

∂A

∂t= D

∂2A

∂x2+ λL− γA.

The rate of change of the concentration of LuxI per unit volume of agar is explicitly

dependent on the concentration of AHL through the activation of the expression of

70

the luxICDABEG operon by the LuxR/AHL complex, modeled here with a Hill function

(5–1). The Hill function enables a cooperative switch from the synthesis-off state to the

synthesis-on state via the parameters a and m:

f (A) =hAm

am + Am(5–1)

where a (nM) is the half-saturation constant, m (unitless) is the Hill coefficient and h (nM

per h) is the maximum LuxI production rate. We denote the enzymatic per-capita LuxI

decay rate by β (h−1). Thus, the concentration of LuxI is given by

∂L

∂t= f (A)− βL

where f (A) is given in (5–1).

Our complete model is

∂A

∂t= D

∂2A

∂x2+ λL− γA (5–2)

∂L

∂t= f (A)− βL (5–3)

where

f (A) =hAm

am + Am.

Table 5-1 gives a summary of parameter and variable definitions.

We note that in our future experiments, we will not explicitly detect the concentration

of LuxI. Instead, we will assume that the measured luminescence of the system is

proportional to the concentration of LuxI as follows: Since the transcription of lux-

ICDABEG is necessary to synthesize both LuxI and the components necessary

for bioluminescence, we assume that the production rate of LuxI (due to the AHL

concentration A) is proportional to the rate of increase of luminescence of the system.

Assuming that the system begins devoid of both LuxI and luminescent compounds, the

concentration of LuxI is then proportional to the measured luminescence of the system.

71

Table 5-1. A summary of variables and parameters used in the model (5–2),(5–3) .Variable / Definition UnitsParameterA AHL concentration nMD AHL diffusion constant mm2/hλ AHL production rate per unit of LuxI h−1

γ AHL per-capita degradation rate h−1

L LuxI concentration nMf cooperative switch functionh maximum LuxI production rate nM/ha half-activation coefficient nMm Hill coefficient (unitless)β per-capita decay rate of LuxI h−1

5.2 Traveling Wave Solution of (5–2), (5–3)

We will now find parameter ranges under which (5–2), (5–3) admits a traveling wave

solution. Let the domains of x and t be infinite. Mathematically, a traveling wave solution

of (5–2), (5–3) is a solution of the form (A(τ),L(τ)) where τ = x + ct for some real

number c , called the wave speed, and there exist some finite real numbers A0 < A2 and

L0 < L2 such that

limτ→−∞

(A(τ),L(τ)) = (A0,L0)

and limτ→∞

(A(τ),L(τ)) = (A2,L2).

We use the following theorem to show the existence of such a solution.

Theorem 5.1. Suppose the system (5–2), (5–3) satisfies the following three conditions:

(i) f (A) is continuously differentiable for A ∈ [0,∞).

(ii) There exist three points E0 = (0, 0), E1 = (A1,L1) and E2 = (A2,L2) with0 < A1 < A2 and 0 < L1 < L2 such that E0, E1 and E2 are the only zeros of thereaction system

dA

dt= λL− γA (5–4)

dL

dt= f (A)− βL (5–5)

in the order interval [E0,E2].

72

(iii) E0 and E2 are stable and E1 is a saddle.

Then there exists some c ∈ R such that (A(x + ct),L(x + ct)) = (A(τ),L(τ)) is a bistable

monotone increasing traveling wave solution to (5–2), (5–3) with limτ→−∞(A(τ),L(τ)) =

E0 and limτ→∞(A(τ),L(τ)) = E2. Furthermore, the wave speed c has the same sign as

the integral∫ A2

01βf (A)− γ

λAdA.

Remark 1. It is interesting to note that the sign of the wave speed c is determined by

an integral with integrand equal to the difference of the L− and A−nullclines of system

(5–4), (5–5).

We defer the proof of Theorem 5.1 until the following section (§5.3), where we

give a general theorem and proof of the existence of a traveling solution to a class of

reaction-diffusion equations that includes system (5–2), (5–3).

We now examine parameter ranges under which Conditions (i)-(iii) are satisfied.

Recall that λ > 0, h > 0, a > 0,m > 0, β > 0 and γ > 0. Condition (i) requires that f (A)

be continuously differentiable for A ∈ [0,∞). For 0 < m < 1, ddAf (A)

∣∣A=0

= ∞, so f (A) is

not differentiable at A = 0 and Condition (i) does not hold. However, for m ≥ 1, Condition

(i) holds.

Figure 5-1. The A-nullcline (blue solid) and L-nullcline (red dashed) of system (5–4),(5–5) for parameter values λ = 0.2 h−1, h = 1 nM/h, a = 3 nM, m = 2,β = 0.1 h−1 and γ = 0.3 h−1.

73

We assume for the remainder of this section that m ≥ 1. We now examine Condition

(ii). Clearly, (0, 0) is a zero of system (5–4), (5–5). The nullclines of (5–4) and (5–5) are

given by

L =γ

λA and L =

1

βf (A),

respectively. (Figure 5-1) In order to show that there exist two strictly positive steady

states of (5–4), (5–5), it suffices to show that there exist (at least) two positive solutions

A1 and A2 to

γ

λA =

1

βf (A)

γ

λA =

1

β

hAm

am + Am

γβ

λh=

Am−1

am + Am:= g(A). (5–6)

First suppose that m = 1. Then g(A) = (a + A)−1, g(0) = 1a, limA→∞ g(A) = 0, g(A) > 0

for all A > 0 and ddAg(A) < 0 for all A > 0. Thus, (5–6) has exactly one positive solution

and Condition (ii) does not hold.

Suppose now that m > 1. Then g(0) = 0, limA→∞ g(A) = 0, g(A) > 0 for all A > 0

finite, ddAg(A) > 0 for A ∈

(0, (m − 1)

1

m a)

, and ddAg(A) < 0 for A > (m − 1)

1

m a. Let

B := g((m − 1)

1

m a)> 0

denote the value of g evaluated at its unique positive critical point. Then (5–6) has

exactly two positive solutions when 0 < γβλh

< B, exactly one positive solution when

γβλh

= B, and no positive solution when γβλh> B. Thus, Condition (ii) may be satisfied only

if 0 < γβλh< B. Note that 0 < γβ

λhtrivially. Since β,λ and h will all be found experimentally,

let these values be fixed. Then Condition (ii) may be satisfied for m > 1 only if

γ <λh(m − 1)

m−1

m

βam. (5–7)

74

We have shown that if the above condition holds, then the reaction system (5–4), (5–5)

has exactly three fixed points, E0 = (0, 0), E1 = (A1,L1) and E2 = (A2,L2) where E1 and

E2 are strictly positive and 0 < A1 < A2. Condition (ii) also requires that 0 < L1 < L2,

but this fact is immediate after noting that the A−nullcline L = γλA is a strictly increasing

function of A.

Suppose that Condition (ii) holds and consider Condition (iii). We wish to show that

E0 and E2 are stable and that E1 is a saddle. The Jacobian of (5–4), (5–5) is given by

J(A,L) :=

−γ λ

ddAf (A) −β

.

The determinant of J is given by

det(J) = γβ − λd

dAf (A) = βλ

(γ

λ− 1

β

d

dAf (A)

). (5–8)

Note that βλ > 0 and that the rightmost factor of the above equation is the difference of

the slopes of the A−nullcline and the L−nullcline of system (5–4), (5–5). We now claim

that if Condition (ii) holds, then det(J(0, 0)) > 0, det(J(A1,L1)) < 0, and det(J(A2,L2)) >

0.

Indeed, since Condition (ii) holds, m > 1 and ddA

1βf (A)|A=0 = 0 < γ

λ. Then

γ

λA >

1

βf (A)

for small positive A. Note that f (A) is a sigmoid function on the domain A ∈ [0,∞).

Specifically, f (A) is positive, strictly increasing, has exactly one inflection point, and is

bounded. Then since the A− and L−nullclines intersect exactly twice for A > 0 (at A1

75

and A2), it must be true that

γλA > 1

βf (A) for A ∈ (0,A1)

γλA < 1

βf (A) for A ∈ (A1,A2)

γλA > 1

βf (A) for A ∈ (A2,∞)

and so

γ

λ<

1

β

d

dAf (A)

∣∣∣∣A=A1

andγ

λ>

1

β

d

dAf (A)

∣∣∣∣A=A2

.

(See, for example, Figure 5-1.) Then by (5–8), det(J(0, 0)) > 0, det(J(A1,L1)) < 0, and

det(J(A2,L2)) > 0.

First consider E0 = (0, 0). Since the determinant of J(0, 0) is the product of its

eigenvalues, and since this quantity is positive, the eigenvalues of J(0, 0) must either be

real and have the same sign or be a complex conjugate pair. Since the trace of J(0, 0) is

the sum of its eigenvalues and tr(J(0, 0)) = −γ−β < 0, the real parts of the eigenvalues

of J(0, 0) must be negative. Similarly, the eigenvalues of J(A2,L2) have negative real

part. Then E0 = (0, 0) and E2 = (A2,L2) are stable.

Now consider J(A1,L1). Since det(J(A1,L1)) < 0, J(A1,L1) must have one positive

real eigenvalue and one negative real eigenvalue. Then E1 = (A1,L1) is a saddle. Thus

we have shown that if Condition (ii) holds, then Condition (iii) holds.

Since Condition (ii) holds if and only if (5–7) holds, Theorem 5.1 implies that if

λ > 0, h > 0, a > 0,m > 1,β > 0 and

0 < γ <λh(m − 1)

m−1

m

βam,

then there exists a traveling wave solution (A(τ),L(τ)), τ = x + ct, to (5–2), (5–3) where

the wave speed c has the same sign as the integral∫ A2

0

1

βf (A)− γ

λAdA.

76

5.3 The Existence of a Traveling Wave Solution to a Class of Reaction-DiffusionSystems

Here, we show the existence of a traveling wave solution to the system

∂u∂t

= D ∂2u∂x2

+ F (u, v)

∂v∂t

= G(u, v)

(5–9)

where

1. F ,G ∈ C 1(R2+,R) and there exist three points E0 = (0, 0)T , E1 = (a1, b1)

T andE2 = (a2, b2)

T with 0 < a1 < a2 and 0 < b1 < b2 such that E0, E1 and E2 are the onlyzeros of g(u, v) := (F (u, v),G(u, v))T in the order interval [E0,E2].

2. All eigenvalues of the Jacobian matrices Dg(E0) and Dg(E2) have negative realparts, and Dg(E1) has one positive eigenvalue and one negative eigenvalue.

3. Fv(u, v) > 0, Gv(u, v) < 0 and Fu(u, v) ≤ 0 for (u, v) ∈ [0, a2]× [0, b2], Gu(u, v) > 0for (u, v) ∈ R2

+\{(0, 0)} and Gu(u, v)|(0,0) ≥ 0.

4. If Gu(u, v)|(0,0) = 0 and∫ a20F (u,VG(u))du > 0 where VG(u) satisfies G(u,VG(u)) =

0 for all u ∈ [0, a2], then G(u, v) satisfies G(u, v) = ϕ(u, v) − mv where m >0, ϕ(u, v) > 0 for all u > 0, v ∈ R and bounded. If Gu(u, v)|(0,0) = 0 and∫ a20F (u,VG(u))du < 0, then G(u, v) satisfies G(u, v) = −ϕ(a2−u, b2−v)+m(b2−v)

where m > 0, ϕ(u, v) > 0 for all u < a2, v ∈ R and bounded.

Jin and Zhao [47] prove a very similar theorem and many of the lemmas contained

herein are similar to those in Jin and Zhao [47]. If both the statement and the proof of

the lemma are the same as appear in [47], then we note the corresponding lemma in

[47]. We include these proofs for completeness.

5.3.1 Preliminaries

First, as a consequence of Conditions 1 and 3, we prove the following lemma.

Lemma 6. Let F ,G satisfy Conditions 1 and 3. Then there exists a unique continuously

differentiable function VG(u) such that G(u,VG(u)) = 0 for all u ∈ [0, a2], VG(0) = 0,

VG(a1) = b1 and VG(a2) = b2. Similarly, there exists a unique continuously differentiable

77

function VF (u) such that F (u,VF (u)) = 0 for all u ∈ [0, a2], VF (0) = 0, VF (a1) = b1 and

VF (a2) = b2.

Proof. By the Implicit Function Theorem (Theorem 1.3), since G(0, 0) = 0 and Gv < 0

by Conditions 1 and 3, there exists some 0 < ϵ1 and a unique continuously differentiable

function VG(u) such that G(u,VG(u)) = 0 for all u ∈ [0, ϵ1) and VG(0) = 0. If ϵ1 > a2,

then our construction of VG(u) is complete. Otherwise, ϵ1 ≤ a2. We claim that we can

extend VG(u) to the closed interval such that G(u,VG(u)) = 0 for all u ∈ [0, ϵ1]. In order

to make this extension, we need only show that limu→ϵ1 VG(u) exists:

First, note that everywhere VG(u) is defined,

d

duG(u,VG(u)) = Gu(u,VG(u)) + Gv(u,VG(u))

dVG(u)

du. (5–10)

Then for all u ∈ [0, ϵ1), since G(u,VG(u)) = 0, dduG(u,VG(u)) = 0. By Condition 3,

Gu ≥ 0 and Gv < 0, so by (5–10), dVG

du= −Gu

Gv≥ 0. That is, VG(u) is a nondecreasing

function of u for all u ∈ [0, ϵ1). Furthermore, since Gu and Gv are continuous functions,

Gu is bounded on [0, a2] × [0, b2] and there exists some δG > 0 such that Gv < −δG < 0

on [0, a2]×[0, b2]. Then∣∣∣Gu

Gv

∣∣∣ remains bounded on [0, a2]×[0, b2], so dVG

duremains bounded

on [0, a2]× [0, b2].

We claim that VG(u) ∈ [0, b2] whenever u ∈ [0, a2] and VG(u) exists. Suppose

not. Then since VG(u) is a nondecreasing function and VG(0) = 0, there exists some

�u ∈ (0, a2] such that VG(�u) > b2. By the construction of VG(u), G(�u,VG(�u)) = 0.

By Condition 3, Gv < 0 for (u, v) ∈ [0, a2] × [0, b2]. Since Gv is continuous, we may

assume without loss of generality that VG(�u) is close enough to b2 that Gv(�u, v) < 0 for

all v ∈ [0,VG(�u)]. Since VG(�u) > b2, we have that G(�u, b2) > G(�u,VG(�u)) = 0. By

Condition 3, Gu > 0. Then since �u ≤ a2, G(a2, b2) ≥ G(�u, b2) > 0, a contradiction to

that G(a2, b2) = 0 by Condition 1. Then VG(u) ∈ [0, b2] whenever u ∈ [0, a2] and dVG

duis

bounded for u ∈ [0, ϵ1) ⊂ [0, a2]. Thus, VG(u) is bounded on the interval [0, ϵ1).

78

Since VG(u) is nondecreasing and bounded on [0, ϵ1), limu→ϵ1 VG(u) = VG(ϵ1)

exists. Then, by the continuity of G , G(u,VG(u))|u=ϵ1 = 0. Thus, VG(u) is a continuously

differentiable function such that G(u,VG(u)) = 0 for all u ∈ [0, ϵ1) and G(ϵ1,VG(ϵ1)) = 0.

Then the Implicit Function Theorem (Theorem 1.3) can be applied at u = ϵ1 and VG(u)

can be extended to the interval [0, ϵ1 + ϵ2) for some ϵ2 > 0.

Iterating this unique construction and extension, we have that there exists a unique

continuously differentiable function VG(u) such that G(u,VG(u)) = 0 for all u ∈ [0, a2]

and VG(0) = 0. We claim that VG(a2) = b2. Suppose not. Then VG(a2) > b2 (or

VG(a2) < b2). By Condition 3, Gv(u, v) < 0 for (u, v) ∈ [0, a2] × [0, b2]. Since Gv is

continuous, we may assume without loss of generality that VG(a2) is close enough to b2

that Gv(a2, v) < 0 for v ∈ [0,VG(a2)]. We have that G(a2, b2) = 0, so G(a2,VG(a2)) < 0

(respectively, G(a2,VG(a2)) > 0), a contradiction to the definition of VG(u). Similarly,

VG(a1) = b1.

A similar argument shows that there exists a unique continuously differentiable

function VF (u) such that F (u,VF (u)) = 0 for all u ∈ [0, b2], VF (0) = 0, VF (a1) = b1 and

VF (a2) = b2.

Now suppose that Condition 2 holds and consider the Jacobian Dg of the reaction

system g(u, v) as defined in Condition 1:

Dg(u, v) =

Fu Fv

Gu Gv

.

By Condition 2, both eigenvalues of Dg(E0) have negative real part, so det(Dg(E0)) > 0.

Then

FuGv − FvGu|E0 > 0

FuGv |E0 > FvGu|E0Fu

Fv

∣∣∣∣E0

<Gu

Gv

∣∣∣∣E0

(5–11)

79

since Fv > 0 and Gv < 0 by Condition 3. We have that

0 =d

duG(u,VG(u)) = Gu + Gv

dVG

du∀ u ∈ [0, a2]

and

0 =d

duF (u,VF (u)) = Fu + Fv

dVF

du∀ u ∈ [0, a2],

so −Gu

Gv= dVG

duand −Fu

Fv= dVF

dufor all u ∈ [0, a2]. Then by (5–11), dVG

du|E0 < dVF

du|E0. Thus,

close to the origin, VG(u) < VF (u). By Condition 1, VG and VF only intersect at (0, 0),

(a1, b1) and (a2, b2) for (u, v) ∈ [0, a2]× [0, b2]. Therefore,

VG(u) < VF (u) for all u ∈ (0, a1). (5–12)

A similar set of inequalities hold at E2 instead of E0. This set implies that

VG(u) > VF (u) for all u ∈ (a1, a2). (5–13)

By Condition 3, we have that Gv(u, v) < 0 and Fv(u, v) > 0 for (u, v) ∈ [0, a2] ×

[0, b2]. Thus, we can determine the regions of [0, a2]× [0, b2] in which F and G are strictly

positive or strictly negative. These regions are displayed in Figure 5-2.

5.3.2 The Wave Speed c

Now, let us consider the result we wish to prove. We seek a monotone traveling

wave solution of system (5–9) with lower and upper limits at E0 and E2, respectively.

That is, we seek a solution to system (5–9) of the form (u(x , t), v(x , t))T = (U(τ),V (τ))T ,

τ = x + ct for some c ∈ R such that

limτ→−∞

(U(τ),V (τ))T = (0, 0)T and limτ→∞

(U(τ),V (τ))T = (a2, b2)T (5–14)

where (0, 0)T = E0 and (a2, b2)T = E2 are as described in Condition 1.

80

Figure 5-2. VF (u) (blue solid) is the unique continuously differentiable function such thatF (u,VF (u)) = 0 for all u ∈ [0, a2], VF (0) = 0, VF (a1) = b1 and VF (a2) = b2.VG(u) (red dashed) is the unique continuously differentiable function suchthat G(u,VG(u)) = 0 for all u ∈ [0, a2], VG(0) = 0, VG(a1) = b1 andVG(a2) = b2.

Substituting (U(τ),V (τ))T into (5–9), we see that (U(τ),V (τ))T must satisfycU ′ = DU ′′ + F (U,V )

cV ′ = G(U,V )

(5–15)

where the prime (′) denotes ddτ

. Letting W (τ) = U ′(τ), (5–15) is equivalent toU ′ = W

cV ′ = G(U,V )

cW = DW ′ + F (U,V ).

(5–16)

Suppose (U(τ),V (τ),W (τ))T solves (5–16) for some c ∈ R, U ′(τ) > 0 for all τ

finite, V ′(τ) > 0 for all τ finite,

limτ→−∞

(U(τ),V (τ),W (τ))T = (0, 0, 0)T and limτ→∞

(U(τ),V (τ),W (τ))T = (a2, b2, 0)T .

(5–17)

Then (U(τ),V (τ))T satisfies (5–15), (5–14) and is therefore a traveling wave solution of

system (5–9) with lower and upper limits at E0 and E2, respectively.

81

Since U ′(τ) > 0 for all τ finite, we may reparameterize V and W as functions of U.

Let V(U) = V (τ(U)) and W(U) = W (τ(U)) for U ∈ (0, a2). By (5–16) and (5–17), we

have

c

∫ ∞

−∞(W (τ))2dτ = D

∫ ∞

−∞W (τ)W ′(τ)dτ +

∫ ∞

−∞F (U(τ),V (τ))W (τ)dτ

c

∫ ∞

−∞(W (τ))2dτ = D

(W (τ))2

2

∣∣∣∣∞−∞

+

∫ ∞

−∞F (U(τ),V (τ))U ′(τ)dτ

c

∫ ∞

−∞(W (τ))2dτ =

∫ a2

0

F (U,V(U))dU. (5–18)

Then since the sign of the right hand side of (5–18) is determined by c ,∫ a2

0

F (U,V(U))dU > 0 ⇔ c > 0 (5–19)∫ a2

0

F (U,V(U))dU = 0 ⇔ c = 0 (5–20)∫ a2

0

F (U,V(U))dU < 0 ⇔ c < 0 (5–21)

Furthermore, since W (τ) = U ′(τ) > 0 for all τ finite, V ′(τ) > 0 for all τ finite, and

cdVdU

= cV ′

U ′ =G(U,V(U))

W, (5–22)

the sign of G(U,V(U)) is the same as the sign of c for all U ∈ (0, a2).

If c > 0, then by the construction of VG(U),

G(U,V(U)) > 0 = G(U,VG(U))

for all U ∈ (0, a2). By Condition 3, GV < 0, so we have that V(U) < VG(U) for all

U ∈ (0, a2). By Condition 3, FV > 0 for all (U,V ) ∈ [0, a2]× [0, b2], so

F (U,V(U)) < F (U,VG(U)) for all U ∈ (0, a2)∫ a2

0

F (U,V(U))dU <

∫ a2

0

F (U,VG(U))dU

0 <

∫ a2

0

F (U,VG(U))dU by (5–19).

82

That is, if c > 0, then∫ a20F (U,VG(U))dU > 0.

Similarly, if c < 0, then ∫ a2

0

F (U,V(U))dU < 0

and G(U,V(U)) < 0 = G(U,VG(U)) for all U ∈ (0, a2) by (5–21) and (5–22), so by

Condition 3, F (U,VG(U)) < F (U,V(U)) for all U ∈ (0, a2). Thus,∫ a2

0

F (U,VG(U))dU <

∫ a2

0

F (U,V(U))dU < 0.

That is, if c < 0, then∫ a20F (U,VG(U))dU < 0.

If c = 0, then G(U,V(U)) = 0 = G(U,VG(U)) for all U ∈ [0, a2] by (5–22), Condition

1 and the fact that V(0) = 0 = V(a2). Then by the uniqueness of VG(U), V(U) = VG(U)

for all U ∈ [0, a2]. By (5–20),

0 =

∫ a2

0

F (U,V(U))dU =

∫ a2

0

F (U,VG(U))dU.

That is, if c = 0, then∫ a20F (U,VG(U))dU = 0.

We summarize the previous discussion in a lemma:

Lemma 7. Suppose (U(τ),V (τ),W (τ))T solves (5–16) and (5–17) for some c ∈ R,

U ′(τ) > 0 for all τ finite and V ′(τ) > 0 for all τ finite. Then (U(τ),V (τ))T is a monotone

increasing traveling wave solution of system (5–9) with lower and upper limits at E0 and

E2, respectively, and the sign of c is the same as the sign of∫ a20F (U,VG(U))dU where

VG(U) satisfies G(U,VG(U)) = 0 for all U ∈ [0, a2].

5.3.3∫ a20F (U,VG(U))dU = 0

Suppose first that∫ a20F (U,VG(U))dU = 0. Then by Lemma 7, if (U(τ),V (τ),W (τ))T

solves (5–16) and (5–17) for some c ∈ R, U ′(τ) > 0 for all τ finite and V ′(τ) > 0 for all

τ finite, then there exists a monotone increasing stationary wave solution to (5–9) and

83

c = 0. If c = 0, then (5–16) is equivalent toU ′ = W

W ′ = −F (U,VG (U))D

(5–23)

where VG(U) is the unique continuously differentiable function which satisfies G(U,VG(U)) =

0 for all U ∈ [0, a2], VG(0) = 0, VG(a1) = b1 and VG(a2) = b2. Such a function exists by

Lemma 6.

We claim that (0, 0) and (a2, 0) are saddle points of (5–23). Indeed, the Jacobian of

(5–23) at (U,W ) is

J =

0 1

− 1D

ddUF (U,VG(U)) 0

which has zero trace and determinant equal to 1

DddUF (U,VG(U)). Since the trace is

zero, the eigenvalues of J must either be real and of opposite sign or be a complex

conjugate pair with zero real part. In the former case, (0, 0) and (a2, 0) are saddle

points of (5–23). We will exclude the latter possibility. Suppose by way of contradiction

that the eigenvalues of J are a complex conjugate pair with zero real part. Without

loss of generality, we will consider (0, 0). Then the determinant of J is positive, that is,

1D

ddUF (U,VG(U))|U=0 > 0. Since D > 0, this implies that

FU(U,VG(U)) + FV (U,VG(U))d

dUVG(U)

∣∣∣∣U=0

=d

dUF (U,VG(U))

∣∣∣∣U=0

> 0.

By construction, ddUVG(U) = −GU

GV. Then the above inequality and the fact that GV (0, 0) <

0 imply that FU(0, 0)GV (0, 0) − FV (0, 0)GU(0, 0) < 0. On the other hand, by Condition 2,

FU(0, 0)GV (0, 0)− FV (0, 0)GU(0, 0) > 0, a contradiction.

Let x = ( x1x2 ) be an eigenvector of the Jacobian matrix J with eigenvalue λ. Then 0 1

− 1D

ddUF (U,VG(U)) 0

x1x2

= λ

x1x2

.

84

We have a dependent system of equations and may assume without loss of generality

that x1 = 1. Then x = ( 1λ ). If we consider eigenvectors as existing in U,W -space,

then this shows that there exists an eigenvector tangent to the unstable manifold of

(0, 0) pointing northeast from (0, 0) and that there exists an eigenvector tangent to the

stable manifold of (a2, 0) pointing northwest from (a2, 0). Note that since VG(U) is strictly

increasing, the same analysis shows that in U,V ,W -space there exists an eigenvector

tangent to the unstable manifold of (0, 0, 0) pointing into the positive orthant and that

there exists an eigenvector tangent to the stable manifold of (a2, b2, 0) pointing into the

region {(U,V ,W )|0 < U < a2, 0 < V < b2,W > 0}.

Consider the energy of (5–23):

K(U,W ) =1

2W 2(τ) +

∫ U

0

1

DF (s,VG(s))ds. (5–24)

K(U,W ) is a conserved quantity:

d

dτK(U,W ) = WW ′ + U ′ 1

DF (U,VG(U))

= −W 1

DF (U,VG(U)) +W

1

DF (U,VG(U))

= 0

Since K(U,W ) is conserved, any trajectory that starts on a level set of K(U,W ) must

remain on that same level set for all τ . With this in mind, we examine the energy of the

points (0, 0) and (a2, 0):

K(a2, 0) =

∫ a2

0

1

DF (U,VG(U))dU = 0 = K(0, 0). (5–25)

Now, if the level set S0 = {(U,W )|K(U,W ) = 0,U ∈ [0, a2]} is connected, then since

VG(a2) = b2, (5–25) is a sufficient condition to show that a trajectory that starts on the

unstable manifold of (0, 0) will approach (a2, 0) in infinite time. We claim that the set S0

85

is connected. Indeed, by (5–24), (U,W ) ∈ S0 if and only if U ∈ [0, a2] and

W = ±

√−2

∫ U

0

1

DF (s,VG(s))ds. (5–26)

We choose the positive root since we require U ′(τ) = W (τ) > 0 for all τ finite. Note that

by the analysis in Section 5.3.1 (summarized in Figure 5-2),F (U,VG(U)) < 0 for U ∈ (0, a1)

F (U,VG(U)) > 0 for U ∈ (a1, a2)

F (U,VG(U)) = 0 for U ∈ {0, a1, a2}

Then (5–26) gives W (U) as a real, continuous function of U defined on the interval

U ∈ [0, a2]. Since S0 is the graph of W (U) as defined in (5–26) for U ∈ [0, a2], and since

the continuous image of a connected set is connected, S0 is connected.

5.3.4∫ a20F (U,VG(U))dU > 0

Suppose that∫ a20F (U,VG(U))dU > 0. Then by Lemma 7, if (U(τ),V (τ),W (τ))T

solves (5–16) and (5–17) for some c ∈ R, U ′(τ) > 0 for all τ finite and V ′(τ) > 0 for all

τ finite, then there exists a monotone traveling wave solution to (5–9) and c > 0. Then

(5–16) is equivalent to U ′ = W

V ′ = G(U,V )c

W ′ = cW−F (U,V )D

.

(5–27)

First, we will show that (0, 0, 0) is a saddle point of (5–27) with a one-dimensional

unstable manifold, denoted W u(E0), and that W u(E0) intersects the positive orthant. To

do so, we will need three lemmas.

86

Lemma 8. Let E be a linearly stable steady state of the reaction systemdudt

= F (u, v)

dvdt

= G(u, v).

Then for any c > 0, the Jacobian matrix of the vector field of the system (5–27) at (E , 0)

has one real positive eigenvalue and two eigenvalues with negative real parts.

Proof. Recall that the trace of a Jacobian matrix is the sum of its eigenvalues and that

the determinant of a Jacobian matrix is the product of its eigenvalues. Then since E is

linearly stable,

Fu + Gv < 0 and FuGv − GuFv > 0. (5–28)

The characteristic equation of the Jacobian matrix of (5–27) is

det

λ 0 −1

−Gu

cλ− Gv

c0

FuD

FvD

λ− cD

= λ3 −(Gv

c+

c

D

)λ2 +

(Fu + Gv

D

)λ− FuGv − FvGu

cD= 0

which can be rewritten as

p(λ) = λ3 + α2λ2 + α1λ+ α0 = 0

= (λ− λ1)(λ− λ2)(λ− λ3),

where λ1,λ2 and λ3 are the roots of p(λ). By (5–28), Condition 3 and since c > 0,

λ1λ2 + λ1λ3 + λ2λ3 = α1 < 0 (5–29)

λ1λ2λ3 = −α0 > 0. (5–30)

First, observe that there must be a real positive root of p(λ). Indeed, p(0) = α0 < 0 and

limλ→+∞ p(λ) = +∞. Therefore, by the Intermediate Value Theorem, p(λ) has a root

λ1 ∈ (0,+∞). Then λ2 and λ3 must either both be real or be a complex conjugate pair.

87

Since λ1 ∈ (0,+∞), (5–30) implies that λ2λ3 > 0. Then Re(λ2) and Re(λ3) must

have the same sign. Since λ1 ∈ (0,+∞), (5–29) implies that not both of Re(λ2) and

Re(λ3) are positive. Thus, both λ2 and λ3 have negative real parts.

Lemma 9. If L(τ) satisfies L′(τ) + mL(τ) = ϕ(τ) where m > 0, ϕ(τ) > 0 for

all finite τ and bounded, and if L(τ) remains bounded for all τ ∈ R, then L(τ) =

e−mτ∫ τ−∞ emsϕ(s)ds > 0.

Proof. By assumption,

L′(τ) +mL(τ) = ϕ(τ)

emτ (L′(τ) +mL(τ)) = emτϕ(τ)

L(τ)emτ = L(0) +∫ τ

0

emsϕ(s)ds

L(τ) = e−mτ(L(0) +

∫ τ

0

emsϕ(s)ds

)(5–31)

Since limτ→−∞ L(τ) is bounded and limτ→−∞ e−mτ = ∞, we must have that

limτ→−∞

(L(0) +

∫ τ

0

emsϕ(s)ds

)= 0.

That is,

L(0) = −∫ −∞

0

emsϕ(s)ds

L(0) =∫ 0

−∞emsϕ(s)ds.

Then by (5–31),

L(τ) = e−mτ(∫ 0

−∞emsϕ(s)ds +

∫ τ

0

emsϕ(s)ds

)= e−mτ

∫ τ

−∞emsϕ(s)ds.

Since ϕ is positive, L(τ) = e−mτ∫ τ−∞ emsϕ(s)ds > 0 for any τ = −∞.

88

Lemma 10 (Lemma 2.2 in Jin and Zhao [47]). Let (U(τ),V (τ),W (τ))T be a solution of

(5–27) such that

limτ→−∞

(U(τ),V (τ),W (τ))T = (0, 0, 0)T . (5–32)

Then near τ = −∞, (U(τ),V (τ),W (τ))T satisfies U(τ) > 0, V (τ) > 0, W (τ) = U ′(τ) >

0 and V ′(τ) > 0.

Proof. By Lemma 8, (0, 0, 0)T has a one dimensional unstable manifold corresponding

to λ(c). Let x = (x1, x2, x3)T be an eigenvector of the Jacobian corresponding to λ(c).

Then 0 0 1

Gu(0,0)c

Gv (0,0)c

0

−Fu(0,0)D

−Fv (0,0)D

cD

x1

x2

x3

= λ(c)

x1

x2

x3

or equivalently

x3 = λ(c)x1

Gu(0,0)c

x1 +Gv (0,0)

cx2 = λ(c)x2

−Fu(0,0)D

x1 − Fv (0,0)D

x2 +cDx3 = λ(c)x3

Without loss of generality, we can assume that x1 = 1. Then

x =

1

Gu(0,0)cλ(c)−Gv (0,0)

λ(c)

.

Since c > 0, λ(c) > 0 and by Condition 3, x2 ≥ 0 and x3 > 0. Note that the solution

(U(τ),V (τ),W (τ))T lies tangent to the eigenvector x at the origin. Then near τ = −∞,

(U(τ),V (τ),W (τ))T satisfies U(τ) > 0 and W (τ) = U ′(τ) > 0.

If Gu(0, 0) > 0, then x2 > 0 and we have that V (τ) > 0 and V ′(τ) > 0 near τ = −∞.

If Gu(0, 0) = 0, then by Condition 4, V (τ) satisfies V ′(τ) + mcV (τ) = 1

cϕ(U(τ),V (τ)) =

89

1cϕ(τ) where m > 0, ϕ(τ) > 0 for all finite τ and bounded. Then by (5–32), Lemma 9 and

since c > 0,

V (τ) = e−mτc

∫ τ

−∞e

msc

(ϕ(s)

c

)ds > 0

and hence V ′(τ) > 0 near τ = −∞.

By Condition 2 and Lemma 8, E0 is a saddle point of (5–27) with a one-dimensional

unstable manifold W u(E0). By Lemma 10, W u(E0) intersects the positive orthant. LetU(τ)

V (τ)

W (τ)

be the solution of (5–27) with initial condition contained in W u(E0). (U(τ),V (τ),W (τ))T

is defined for all τ ∈ (−∞, τmax) where τmax may be infinite. We now note that a solution

(U(τ),V (τ),W (τ))T to (5–27) depends on the parameter c . We will not explicitly

denote each solution’s dependence on c , but the appropriate association will be clear by

context.

We have shown that we may choose the initial condition of (U(τ),V (τ),W (τ))T

such that it lies both in W u(E0) and in the positive orthant. We may choose the initial

condition such that it lies in the region

:= {(U,V ,W )|0 < U ≤ a2, 0 < V ≤ b2,W > 0}. (5–33)

(See Figure 5-3.) To show that (U(τ),V (τ),W (τ))T is a traveling wave solution

to (5–27), we need to show that limτ→∞(U(τ),V (τ),W (τ))T = (a2, b2, 0)T and

(U(τ),V (τ),W (τ))T ∈ for all τ ∈ (−∞,∞). To this end, we examine the possible

paths of (U(τ),V (τ),W (τ))T .

First, we show that a solution may not exit through the face where V = 0 or the

face where U = 0. Note that since c > 0, GU > 0 on (0, a2] × (0, b2] and G(0, 0) = 0

90

(by Conditions 1 and 3), V ′ > 0 in the region {(U,V ,W )|0 < U < a2,V = 0,W > 0}.

Additionally, since U ′ = W , U ′ > 0 in the region {(U,V ,W )|U = 0, 0 ≤ V ≤ b2,W > 0}.

Thus, (U(τ),V (τ),W (τ))T may intersect ∂ only in the faces {(U,V ,W )|U = a2, 0 ≤

V ≤ b2,W ≥ 0}, {(U,V ,W )|0 ≤ U ≤ a2,V = b2,W ≥ 0} or {(U,V ,W )|0 ≤ U ≤ a2, 0 ≤

V ≤ b2,W = 0}. Our strategy is to classify each c ∈ (0,∞) based on which region of ∂

the trajectory (U(τ),V (τ),W (τ))T intersects.

Figure 5-3. The region .

5.3.4.1 The Sets P1 and P2

We classify each c > 0 into three sets: P1, P2 and (0,∞)\(P1 ∪ P2). P1 will be the

set of parameter values c for which the solution (U(τ),V (τ),W (τ))T exits through

{(U,V ,W )|U = a2, 0 ≤ V ≤ b2,W > 0} or {(U,V ,W )|0 ≤ U ≤ a2,V = b2,W >

0}. P2 will be the set of parameter values c for which the solution exits through

{(U,V ,W )|0 < U < a2, 0 < V < b2,W = 0}. (We will formally define P1 and P2 shortly.)

We say that parameter values c ∈ P1 overshoot (a2, b2, 0) and that parameter values

c ∈ P2 undershoot (a2, b2, 0). We will show that P1 and P2 satisfy

1. P1,P2 = ∅,

2. P1,P2 open,

3. and P1 ∩ P2 = ∅.

Then, since (0,∞) is connected, that is, since (0,∞) cannot be written as the union of

two disjoint, nonempty, open sets, there must exist some wave speed c∗ ∈ (0,∞)\(P1 ∪

91

P2). We will then show that this speed c∗ admits a heteroclinic orbit of (5–27) connecting

(0, 0, 0)T and (a2, b2, 0)T .

Define

P1 = {c > 0| ∃ τ1 <∞ such that U(τ1) > a2 or V (τ1) > b2 (5–34)

and W (τ) > 0 ∀ τ ∈ (−∞, τ1]}

P2 = {c > 0| ∃ τ2 <∞ such that W (τ2) < 0,U(τ) < a2 ∀ τ ∈ (−∞, τ2] (5–35)

and V (τ) < b2 ∀ τ ∈ (−∞, τ2]}

5.3.4.2 P1 = ∅

First, we show that P1 = ∅. To do so, we will need five lemmas:

Lemma 11 (Lemma 2.3 in Jin and Zhao [47]). Let (U(τ),V (τ),W (τ))T be a solution

of (5–27) such that (5–32) holds. Then for any τ0 such that U ′(τ) = W (τ) > 0 for all

τ ∈ (−∞, τ0), V ′(τ) > 0 for all τ ∈ (−∞, τ0).

Proof. By Lemma 10, W (τ) > 0 for τ close to −∞. Since we assume that U ′ = W > 0

for all τ ∈ (−∞, τ0), we may reparameterize V and W as functions of U. Let V(U) =

V (τ(U)) and W(U) = W (τ(U)) for U ∈ (0,U(τ0)). Then V and W satisfy the following

equations:

V ′ =dVdU

=G(U,V)cW

(5–36a)

W ′ =dWdU

=cW − F (U,V)

DW(5–36b)

for U ∈ (0,U(τ0)) with the initial conditions

V(0) = 0, W(0) = 0. (5–37)

We claim that V ′(U) > 0 for all U ∈ (0,U(τ0)). Indeed, suppose not. Then by Lemma 10

there exists some �u ∈ (0,U(τ0)) such that V ′(�u) = 0 and V ′(U) > 0 for all U ∈ (0, �u).

92

Since W > 0 for U ∈ (0,U(τ0)) and c > 0, we have that G(U,V(U)) > 0 for all U ∈ (0, �u)

and G(�u,V(�u)) = 0. Then it must be true that ddUG(U,V(U))|U=�u ≤ 0. On the other

hand, we have that

d

dUG(U,V(U))

∣∣∣∣U=�u

= GU(U,V(U)) + GV (U,V(U))dV(U)

dU

∣∣∣∣U=�u

= GU(U,V(U))|U=�u > 0

by Condition 3 since �u > 0, a contradiction.

Lemma 12. Let (U(τ),V (τ),W (τ))T be a solution of (5–27) such that (5–32) holds.

Suppose that 0 < U(τ) < a2, 0 < V (τ) < b2 and W > 0 for all τ ∈ (−∞, τ0) where

τ0 ≤ τmax . Then W (τ) is bounded for all τ ∈ (−∞, τ0). Furthermore, if τ0 = τmax , then

τmax = ∞ and W (τ) is bounded for all τ ∈ (−∞,∞).

Proof. Since U ′ > 0 on τ ∈ (−∞, τ0), we can express V and W as functions of U for

U ∈ (0,U0) where U0 = limτ→τ0 U(τ). Then V(U) and W(U) satisfy (5–36) and (5–37) for

U ∈ (0,U0).

Suppose by way of contradiction that W(U) is unbounded for U ∈ (0,U0). Then

limU→U0W(U) = ∞. Since 0 < U(τ) < a2 and 0 < V (τ) < b2 for all τ ∈ (−∞, τ0),

0 < U0 ≤ a2 and 0 < V(U) < b2 for U ∈ (0,U0). Consider max[0,a2]×[0,b2] F (U,V ). Since

F (a1, b1) = 0 and FV > 0 for (U,V ) ∈ [0, a2]× [0, b2], we have that

~M := max[0,a2]×[0,b2]

|F (U,V )|

is positive. Then for all U ∈ (0,U0),

W ′ =c

D− F (U,V)

DW≤ c

D+

~M

DW

Since limU→U0W(U) = ∞, there exists some �U such that for all U ∈ (�U,U0), W(U) > 1

and

|W ′| <

∣∣∣∣∣c + ~M

D

∣∣∣∣∣ ,

93

and hence

W(U) <W(�U) +

∣∣∣∣∣c + ~M

D

∣∣∣∣∣ (U − �U)<∞ for all U ∈ (�U,U0).

This contradicts the assumption that limU→U0W(U) = ∞. Thus, W (τ) is bounded for all

τ ∈ (−∞, τ0).

Suppose that τ0 = τmax . Then since U(τ), V (τ) and W (τ) are bounded, Lemma 3

shows that the maximum interval of existence of (U(τ),V (τ),W (τ))T is τ ∈ (−∞,∞).

Lemma 13 (Lemma 2.4 in Jin and Zhao [47]). Let (U(τ),V (τ),W (τ))T be a solution

of (5–27) such that (5–32) holds. Suppose that there exists some τ0 such that U(τ) ∈

(0, a2) and W (τ) > 0 for all τ ∈ (−∞, τ0). Then V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0).

Proof. By Lemma 10, there exists some �τ ≤ τ0 such that V (τ) ∈ (0, b2) for all

τ ∈ (−∞, �τ). Let �τ be the largest such τ . If �τ ≥ τ0, then our proof is complete. Suppose

by way of contradiction that �τ < τ0. Then since V ′(τ) > 0 for all τ ∈ (−∞, τ0) ⊃ (−∞, �τ)

by Lemma 11 and the maximality of �τ , V (�τ) = b2. Then

G(U(�τ),V (�τ)) = G(U(�τ), b2) < G(a2, b2) = 0

since U(�τ) ∈ (0, a2) and GU > 0 by Condition 3. Then

V ′(τ)|τ=�τ =G(U(τ),V (τ))

c

∣∣∣∣τ=�τ

< 0

since c > 0. This contradicts that V ′(τ) > 0 for all τ ∈ (−∞, τ0) ⊃ (−∞, �τ).

Lemma 14 (Step 1 in the proof of Theorem 2.1 in Jin and Zhao [47]). Let (U(τ),V (τ),W (τ))T

be a solution of (5–27) such that (5–32) holds. Let �τ be such that 0 < U(τ) < a1 for all

τ ∈ (−∞, �τ). Then W (τ) > cDU(τ) > 0 and W ′(τ) > c

DU ′(τ) > 0 for all τ ∈ (−∞, �τ).

Proof. First, we show that W (τ) > 0 for all τ ∈ (−∞, �τ). Suppose by way of

contradiction that there exists some τ0 < �τ such that W (τ0) = 0. Without loss of

94

generality, assume that τ0 is the first such τ . Then W (τ) > 0 for all τ ∈ (−∞, τ0). Note

that by Lemma 13, V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0).

Since U ′ > 0 for all τ ∈ (−∞, τ0), we may reparameterize the solution and consider

the system (5–36), (5–37) for U ∈ (0,U0) where limτ→τ0 U(τ) = U0 ≤ a1. By Lemma 11,

0 < V ′ = G(U,V(U))cW for U ∈ (0,U0). Since W > 0 and c > 0,

G(U,V(U)) > 0 = G(U,VG(U)) for U ∈ (0,U0). (5–38)

By Condition 3, GV < 0. This combined with (5–38) shows that V(U) < VG(U) for

U ∈ (0,U0). Then by (5–12), V(U) < VF (U). By Condition 3, FV > 0, so F (U,V(U)) <

F (U,VF (U)) = 0. Then

W ′ =cW − F (U,V)

DW>

c

D> 0 (5–39)

for all U ∈ (0,U0). Since W(0) = 0, this implies that W(U0) >W(0) = 0, a contradiction.

To show that W (τ) > cDU(τ) > 0 for all τ ∈ (−∞, �τ), we consider the above

proof. Note that (5–39) holds for all τ ∈ (−∞, �τ) since U ′ > 0 for all τ ∈ (−∞, �τ). Then

W(U) > cDU for all U ∈ (0,U(�τ)) and hence, W (τ) > c

DU(τ) for all τ ∈ (−∞, �τ).

Similarly, by (5–39), W ′(τ) > cDU ′(τ) > 0 for all τ ∈ (−∞, �τ).

Lemma 15 (Step 2 in the proof of Theorem 2.1 in Jin and Zhao [47]). Let (U(τ),V (τ),W (τ))T

be a solution of (5–27) such that (5–32) holds. Let

c > �c :=

√2MD

a1(5–40)

where

0 < M := max[0,a2]×[0,b2]

F (U,V ). (5–41)

Let τa1 and τ0 be such that U(τ) ∈ (0, a1) for all τ ∈ (−∞, τa1) and U(τ) ∈ [a1, a2) for all

τ ∈ [τa1, τ0). Then W (τ) > ca14D

> 0 for all τ ∈ [τa1, τ0).

95

Proof. First, note that since F (a1, b1) = 0 and FV > 0 for (U,V ) ∈ [0, a2] × [0, b2], M is

positive. Now we will prove the claim of the lemma.

By the continuity of U(τ), since U(τ) < a1 for all τ < τa1 and since U(τ) ≥ a1 for

all τa1 ≤ τ < τ0, U(τa1) = a1. Then by Lemma 14, W (τa1) ≥ ca1D

> 0. Suppose by

way of contradiction that there exists some �τ ∈ [τa1, τ0) such that W (�τ) = ca14D

. Without

loss of generality, suppose that �τ is the first such τ ∈ [τa1, τ0). Then W (τ) > ca14D

for all

τ ∈ [τa1, �τ).

By Lemma 14, U ′(τ) > 0 for all τ ∈ (−∞, τa1), and U ′(τ) > ca14D

> 0 for all τ ∈ [τa1, �τ)

by the above assumption, so we may reparameterize the solution and consider the

system (5–36), (5–37) for U ∈ (0,U(�τ)). Since W(U(�τ)) = ca14D

, there must exist some

u ∈ (a1,U(�τ)) such that W(u) = ca12D

and W(U) > ca12D

for all U ∈ [a1, u). Then for all

U ∈ [a1, u),

dWdU

=cW − F (U,V(U))

DW

≥ cW −M

DW=

c

D− M

DW

>c

D− M

D ca12D

=c2a1 − 2MD

Dca1(5–42)

since V(U) ∈ (0, b2) by Lemma 13 and where M defined in (5–41) is positive. By (5–40)

and (5–42), ddU

W(U) > 0,U ∈ [a1, u). Therefore W(u) >W(a1), a contradiction.

Lemma 16. P1 as defined in (5–34) is nonempty.

Proof. We claim that for all c > �c as defined in (5–40) and (5–41), c ∈ P1. Suppose by

way of contradiction that there exists some ~c > �c such that ~c /∈ P1. Then either

(i)U(τ) ≤ a2 and V (τ) ≤ b2 for all τ ∈ (−∞, τmax)

or

(ii)there exists some τ1 < τmax such that U(τ1) > a2 or V (τ1) > b2 and there existssome �τ ≤ τ1 such that W (�τ) = 0.

96

Case 1: Suppose that (i) holds. We will show that then there must exist some τ < τmax

such that U(τ) > a2, yielding a contradiction.

We claim that there exists some τa1 < τmax such that U(τa1) = a1. If not, then

U(τ) < a1 for all τ ∈ (−∞, τmax). By Lemma 14, W (τ) > 0 for all τ ∈ (−∞, τmax) so by

Lemma 13, V (τ) ∈ (0, b2) for all τ ∈ (−∞, τmax). Then, by Lemma 12, τmax = ∞ and we

have that

U(τ) =

∫ τ

−∞W (s)ds

>

∫ τ

0

W (s)ds since W (τ) > 0 ∀ τ ∈ (−∞,∞)

>

∫ τ

0

W (0)ds since W ′(τ) > 0 ∀ τ ∈ (−∞,∞) by Lemma 14

= τW (0).

Since τmax = ∞, we can find some τ large enough such that τW (0) > a1, a

contradiction. Thus, there exists some τa1 < τmax such that U(τa1) = a1.

Now, we claim that there exists some τa2 < τmax such that U(τa2) = a2. If not, then

U(τ) < a2 for all τ ∈ (−∞, τmax). By Lemma 14, W (τ) > 0 for all τ ∈ (−∞, τa1). We

claim that W (τ) > 0 for all τ ∈ [τa1, τmax). Suppose not. Then there exists some least

τ0 ∈ [τa1, τmax) such that W (τ0) = 0 and W (τ) > 0 for all τ ∈ [τa1, τ0). By Lemma

15, W (τ) > ~ca14D

for all τ ∈ [τa1, τ0), so by the continuity of W (τ) and since τ0 < τmax ,

W (τ0) > 0, a contradiction. Thus, W (τ) > 0 for all τ ∈ [τa1, τmax). Now, by Lemma 13,

V (τ) ∈ (0, b2) for all τ ∈ (−∞, τmax). Then, by Lemma 12, τmax = ∞. Then we have that

U(τ) = a1 +

∫ τ

τa1

W (s)ds

> a1 +

∫ τ

τa1

~ca14D

ds by Lemma 15

= a1 +~ca14D

(τ − τa1) .

97

Since τmax = ∞, we can find some τ large enough such that U(τ) > a2, a contradiction.

Thus, there exists some τa2 < τmax such that U(τa2) = a2.

Furthermore, since W (τ) > ~ca14D

whenever U(τ) ∈ [a1, a2) by Lemma 15 and since

τa2 < τmax , we have that W (τa2) > 0. Then there exists some ϵ > 0 small enough such

that τa2 + ϵ < τmax and U(τa2 + ϵ) > a2. This contradicts the assumption that (i) holds.

Case 2: Suppose that (ii) holds. Then there exists some τ1 < τmax such that

U(τ1) > a2 or V (τ1) > b2 and there exists some finite �τ ≤ τ1 such that W (�τ) = 0.

Without loss of generality, let �τ be the first such τ . Then W (τ) > 0 for all τ ∈ (−∞, �τ).

Note that U(τ) is increasing for all τ ∈ (−∞, �τ). Since �τ is finite, U(�τ) = 0. If

0 < U(�τ) < a1, then by Lemma 14, W (�τ) > 0, a contradiction to the definition of �τ . If

a1 ≤ U(�τ) < a2, then by Lemma 15, W (�τ) > 0, a contradiction to the definition of �τ . If

U(�τ) = a2, then by Lemma 15, W (τ) > ~ca14D

for all τ ∈ [τa1, �τ). By the continuity of W (τ)

and since �τ < τmax , W (�τ) > 0, a contradiction to the definition of �τ . If U(�τ) > a2, then

~c ∈ P1, a contradiction to the assumption that ~c /∈ P1. Hence, (ii) does not hold.

Since neither (i) nor (ii) hold, we must have that (�c ,∞) ⊆ P1. Thus, P1 = ∅.

5.3.4.3 P2 = ∅

To show that P2 is nonempty, we will need another lemma:

Lemma 17. Let (U(τ),V (τ),W (τ))T be a solution of (5–27) such that (5–32) holds. If

there exists some τ0 < τmax such that W (τ0) = 0, W (τ) > 0 for all τ ∈ (−∞, τ0) and

U(τ) ∈ (0, a2) for all τ ∈ (−∞, τ0], then c ∈ P2.

Proof. Note that since W (τ) > 0 for all τ ∈ (−∞, τ0) and U(τ) ∈ (0, a2) for all

τ ∈ (−∞, τ0], V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0) by Lemma 13. We claim that

V (τ0) ∈ (0, b2). Indeed, since V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0), V (τ0) ∈ [0, b2]. By

Lemma 11, V ′(τ) > 0 for all τ ∈ (−∞, τ0), so V (τ0) ∈ (0, b2]. Since V ′(τ) > 0 for

all τ ∈ (−∞, τ0), V ′(τ0) ≥ 0. Then, by the equation for V ′ in (5–27) and since c > 0,

98

G(U(τ0),V (τ0)) ≥ 0. If V (τ0) = b2, then

G(U(τ0), b2) = G(U(τ0),V (τ0)) ≥ 0 = G(a2, b2). (5–43)

By Condition 3, GU > 0 on (0, a2]× (0, b2], so by (5–43), U(τ0) ≥ a2. This contradicts the

assumption that U(τ0) ∈ (0, a2). Hence, V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0].

Since W (τ) > 0 for all τ ∈ (−∞, τ0) and W (τ0) = 0, W ′(τ0) ≤ 0. If W ′(τ0) < 0,

then there exists some ϵ > 0 small enough such that τ0 + ϵ < τmax , W (τ0 + ϵ) < 0,

U(τ) ∈ (0, a2) for all τ ∈ (−∞, τ0 + ϵ] and V (τ) ∈ (0, b2) for all τ ∈ (−∞, τ0 + ϵ]. Then

c ∈ P2.

We will show that if W ′(τ0) = 0, then W ′′(τ0) < 0. Since W (τ0) = 0, this implies

that there exists some �τ < τ0 such that W (�τ) < 0, a contradiction to the assumption that

W (τ) > 0 for all τ ∈ (−∞, τ0). Consider W ′′(τ):

W ′′(τ) =cW ′(τ)− FU(U(τ),V (τ))U ′(τ)− FV (U(τ),V (τ))V ′(τ)

D

W ′′(τ)|τ=τ0 = −FV (U(τ),V (τ))V ′(τ)

D

∣∣∣∣τ=τ0

since W ′(τ0) = 0 and U ′(τ0) = W (τ0) = 0. Since U(τ0) ∈ (0, a2) and V (τ0) ∈ (0, b2),

FV (U(τ0),V (τ0)) > 0 by Condition 3. Thus, it suffices to show that V ′(τ0) > 0. First, we

claim that U(τ0) > a1. If not, then since U(τ) is monotonically increasing, U(τ) ∈ (0, a1)

for all τ ∈ (−∞, τ0). Then by Lemma 14, W (τ) > cDU(τ) > 0 for all τ ∈ (−∞, τ0). This

implies that W (τ0) > 0, a contradiction to that W (τ0) = 0. Hence, U(τ0) > a1.

Now, by the equation for W ′ in (5–27) and since W (τ0) = 0, we have that

F (U(τ0),V (τ0)) = 0 = F (U(τ0),VF (U(τ0))). (5–44)

By Condition 3, FV (U,V ) > 0 for all (U,V ) ∈ [0, a2] × [0, b2], so V (τ0) = VF (U(τ0)) by

(5–44). Since U(τ0) ∈ (a1, a2) and by (5–13),

VG(U(τ0)) > VF (U(τ0)) = V (τ0).

99

By Condition 3, GV < 0 on [0, a2]× [0, b2], so

0 = G(U(τ0),VG(U(τ0))) < G(U(τ0),V (τ0)).

Hence, V ′(τ0) =G(U(τ0),V (τ0))

c> 0.

Lemma 18. P2 as defined in (5–35) is nonempty.

Proof. We claim that there exists some c > 0 such that (0, c) ⊆ P2. Suppose not. Then

there exists some sequence {ci}∞i=1 such that ci → 0 as i → ∞ and ci /∈ P2 for all i . In

Step 1, we show that for all ci ,

(I) there exists some least τ0 (which may be infinite) such that limτ→τ0 U(τ) = a2and W (τ) > 0 for all τ ∈ (−∞, τ0).

In Step 2, we will use this fact to obtain a contradiction. For each ci , let (Ui ,Vi ,Wi)T

be a solution of (5–27) such that (5–32) holds. We will suppress this subscript i for the

duration of Step 1, though it will be used explicitly in Step 2.

Step 1 We claim that (I) holds for all ci /∈ P2. For each ci , either W (τ) > 0 for all

τ < τmax or there exists some τ < τmax such that W (τ) ≤ 0.

If W (τ) > 0 for all τ < τmax and there exists some τ0 < τmax such that U(τ0) = a2,

then (I) holds. If W (τ) > 0 for all τ < τmax and U(τ) < a2 for all τ < τmax then by

Lemma 10, U(τ) ∈ (0, a2) for all τ ∈ (−∞, τmax). By Lemma 13, V (τ) ∈ (0, b2) for all

τ < τmax . By Lemma 12, τmax = ∞ and W (τ) is bounded for all τ ∈ (−∞,∞). Since

U(τ) is monotonically increasing and bounded for all τ ∈ (−∞,∞), limτ→∞U(τ) exists.

By Lemma 11, V ′(τ) > 0 for all τ ∈ (−∞,∞). Since V (τ) is monotonically increasing

and bounded for all τ ∈ (−∞,∞), limτ→∞ V (τ) exists. It is easy to see that W ′ (defined

in (5–27)) is bounded, so by Barbalat’s Lemma (Lemma 1), limτ→∞W (τ) = 0.

Since U,V and W converge, Lemma 4 implies that the trajectory must approach a

steady state. The only steady states contained in the closure of as defined in (5–33)

100

are (0, 0, 0)T , (a1, b1, 0)T and (a2, b2, 0)T . We will show that limτ→∞ U(τ) > a1. This

implies that limτ→∞ U(τ) = a2.

We claim that there exists some τa1 < ∞ such that U(τa1) = a1. If not, then

U(τ) < a1 for all τ ∈ (−∞,∞) and

U(τ) =

∫ τ

−∞W (s)ds

>

∫ τ

0

W (s)ds since W (τ) > 0 ∀ τ ∈ (−∞,∞)

>

∫ τ

0

W (0)ds since W ′(τ) > 0 ∀ τ ∈ (−∞,∞) by Lemma 14

= τW (0).

Since τmax = ∞, we can find some τ large enough such that τW (0) > a1, a

contradiction. Thus, there exists some τa1 < τmax such that U(τa1) = a1.

Since U(τ) is strictly increasing, it must be true that limτ→∞ U(τ) > a1. Since

limτ→∞ U(τ) ∈ {0, a1, a2}, limτ→∞ U(τ) = a2. Then (I) holds.

If there exists some τ < τmax such that W (τ) ≤ 0, then there exists some τ0 ≤ τ

such that W (τ0) = 0 and W (τ) > 0 for all τ ∈ (−∞, τ0). If U(τ0) < a2, then U(τ) ∈ (0, a2)

for all τ ∈ (−∞, τ0] and c ∈ P2 by Lemma 17, a contradiction. If U(τ0) = a2, then (I)

holds. If U(τ0) > a2, then there exists some �τ < τ0 such that U(�τ) = a2, U(τ) ∈ (0, a2) for

all τ ∈ (−∞, �τ) and W (τ) > 0 for all τ ∈ (−∞, �τ). Then (I) holds.

Step 2 (Step 3 in the proof of Theorem 2.1 in Jin and Zhao [47]) We have shown

that (I) holds for all ci /∈ P2. That is, for all ci /∈ P2, there exists some least τ i0 (which

may be infinite) such that limτ→τ i0

Ui(τ) = a2 and Wi(τ) > 0 for all τ ∈ (−∞, τ i0). Since

U ′i (τ) > 0 for all τ ∈ (−∞, τ i0), we may reparameterize the solution and consider the

system (5–36), (5–37) for U ∈ (0, a2). Then for each i ∈ N and for all U ∈ (0, a2),

W ′i (U) =

ci

D− F (U,Vi(U))

DWi(U)∫ U

0

Wi(s)W ′i (s)ds =

∫ U

0

ci

DWi(s)−

1

DF (s,Vi(s))ds

101

(Wi(U))2

2=

∫ U

0

ci

DWi(s)−

1

DF (s,Vi(s))ds. (5–45)

Define

Ai := supU∈[0,a2]

Wi(U) for each i and m := min(U,V )∈[0,a2]×[0,b2]

F (U,V ).

We claim that Ai is finite for each i . Since Wi is continuous and since Wi(0) = 0

by (5–32), it suffices to show that limA→A(τ i0)Wi(A) < ∞. If τ i0 < τmax , then Wi(a2) =

Wi(τi0) < ∞. If τ i0 = τmax , then limτ→τ i

0

Wi(τ) = 0 by the argument in the second

paragraph of Step 1. Thus, Ai is finite for each i . Furthermore, since Wi(τ) > 0 for all

τ ∈ (−∞, τ i0) by Lemma 10, Ai > 0 for all i ∈ N. Also note that since F (a1, b1) = 0 and

FV > 0 for (U,V ) ∈ [0, a2]× [0, b2], m is negative.

Step 2.1 We claim that Wi(U) is uniformly bounded on [0, a2] for large i . That is,

there exists some N1 > 0 and some A∗ > 0 such that Wi(U) ≤ A∗ for all i ≥ N1 and

all U ∈ [0, a2]. To prove this, we will show that Ai is bounded for large i . That is, we will

show that there exists some N1 > 0 and some A∗ > 0 such that 0 < Ai ≤ A∗ for all

i ≥ N1. Indeed, by (5–45),

A2i

2≤∫ a2

0

ci

DAi −

m

Dds

A2i

2≤ cia2

DAi −

ma2

DAi

2+ma2

DAi

≤ cia2

D

DA2i + 2ma22a2Ai

≤ ci (5–46)

for all i since Ai > 0. Note that

h(A) :=DA2 + 2ma2

2a2A

102

is an increasing function of A. Since m < 0, limA→0 h(A) = −∞. Then since

limA→∞ h(A) = ∞, there exists some unique A∗ such that h(A∗) = c1 and

h(A) ≤ c1 if and only if A ≤ A∗. (5–47)

Since limi→∞ ci = 0, there exists some N1 > 0 such that ci ≤ c1 for all i ≥ N1. Then by

(5–46) and (5–47), Ai ≤ A∗ for all i ≥ N1. By the definition of Ai , 0 < Wi(U) ≤ A∗ for all

i ≥ N1 and all U ∈ [0, a2].

Step 2.2 We claim that G(U,Vi(U)) → G(U,VG(U)) pointwise for each U ∈ [0, a2]

as i → ∞. That is, for all ϵ > 0 and each U ∈ [0, a2], there exists some N2 > 0 (which

depends on U) such that |G(U,Vi(U))− G(U,VG(U))| < ϵ for all all i ≥ N2.

For ease of notation, let

Mi(U) := G(U,Vi(U)), U ∈ (0, a2).

Then

M ′i (U) = GU(U,Vi(U)) + GV (U,Vi(U))V ′

i (U)

= GU(U,Vi(U)) + GV (U,Vi(U))Mi(U)

ciWi(U)

M ′i (U)− GV (U,Vi(U))

Mi(U)

ciWi(U)= GU(U,Vi(U)).

Multiplying through by an integrating factor, we have that

d

dU

[Mi(U) exp

(−∫ U

a1

GV (s,Vi(s))ciWi(s)

ds

)]= GU(U,Vi(U)) exp

(−∫ U

a1

GV (s,Vi(s))ciWi(s)

ds

)for all U ∈ [0, a2]. Then by integrating we see that

Mi(U) exp

(−∫ U

a1

GV (s,Vi(s))ciWi(s)

ds

)=

∫ U

0

GU(y ,Vi(y)) exp(−∫ y

a1

GV (s,Vi(s))ciWi(s)

ds

)dy

since Vi(0) = 0 by (5–32) and G(0, 0) = 0 by Condition 1. Then

Mi(U) =

∫ U

0

GU(y ,Vi(y)) exp(∫ U

y

GV (s,Vi(s))ciWi(s)

ds

)dy (5–48)

103

for all U ∈ [0, a2].

First, consider∫ Uy

GV (s,Vi (s))ciWi (s)

ds for fixed U, y ∈ [0, a2], y < U. By Condition 3,

GV < 0 on [0, a2] × [0, b2], so there exists some δ > 0 such that GV (U,V ) ≤ −δ for all

(U,V ) ∈ [0, a2] × [0, b2]. By Lemma 13, Vi(U) ∈ (0, b2) for all U ∈ (0, a2). Then, by

continuity, Vi(U) ∈ [0, b2] for all U ∈ [0, a2] for all i . Hence, GV (U,Vi(U)) ≤ −δ for all

U ∈ [0, a2]. Then ∫ U

y

GV (s,Vi(s))ciWi(s)

ds ≤∫ U

y

−δciWi(s)

ds.

By Step 2.1, 0 <Wi(U) ≤ A∗ for all i ≥ N1 and all U ∈ [0, a2]. Then for all i ≥ N1,∫ U

y

−δciWi(s)

ds ≤∫ U

y

−δciA∗ds = −δ(U − y)

ciA∗

and

limi→∞

∫ U

y

GV (s,Vi(s))ciWi(s)

ds ≤ limi→∞

−δ(U − y)

ciA∗ (5–49)

for all U, y ∈ [0, a2], y < U. Since δ, U − y , and A∗ are all fixed positive values, the right

hand side of (5–49) approaches negative infinity as ci goes to zero. Then

limi→∞

∫ U

y

GV (s,Vi(s))ciWi(s)

ds = −∞

for all y ,U ∈ [0, a2], y < U and hence

limi→∞

exp

(∫ U

y

GV (s,Vi(s))ciWi(s)

ds

)= 0 (5–50)

for all y ,U ∈ [0, a2], y < U.

By Condition 3, GU > 0 on [0, a2] × [0, b2], so there exists some γ > 0 such that

0 ≤ GU(U,V ) ≤ γ for all (U,V ) ∈ [0, a2] × [0, b2]. By Lemma 13, Vi(U) ∈ (0, b2) for

all U ∈ (0, a2). Then, by continuity, Vi(U) ∈ [0, b2] for all U ∈ [0, a2] for all i . Hence,

0 ≤ GU(U,Vi(U)) ≤ γ for all U ∈ [0, a2].

104

By (5–48),

|Mi(U)| =∣∣∣∣∫ U

0

GU(y ,Vi(y)) exp(∫ U

y

GV (s,Vi(s))ciWi(s)

ds

)dy

∣∣∣∣≤ γ

∫ U

0

exp

(∫ U

y

GV (s,Vi(s))ciWi(s)

ds

)dy

for all i ∈ N and for each U ∈ [0, a2]. Then

limi→∞

|Mi(U)| ≤ limi→∞

γ

∫ U

0

exp

(∫ U

y

GV (s,Vi(s))ciWi(s)

ds

)dy = 0 (5–51)

for each U ∈ [0, a2] by (5–50) and the Dominated Convergence Theorem (Theorem 1.4).

Then, since G(U,VG(U)) = 0 for all U ∈ [0, a2] by definition, and by the definition of

Mi(U),

|G(U,Vi(U))− G(U,VG(U))| = |G(U,Vi(U))| = |Mi(U)|

Then by (5–51),

limi→∞

|G(U,Vi(U))− G(U,VG(U))| = limi→∞

|Mi(U)| = 0

for each U ∈ [0, a2]. Then for all ϵ > 0 and each U ∈ [0, a2], there exists some N2 > 0

(which depends on U) such that |G(U,Vi(U))− G(U,VG(U))| < ϵ for all i ≥ N2.

Step 2.3 We claim that F (U,Vi(U)) → F (U,VG(U)) pointwise for each U ∈ [0, a2]

as i → ∞.

Recall from Step 2.2 that by Condition 3 and Lemma 13, there exists some δ > 0

such that GV (U,V ) ≤ −δ for all (U,V ) ∈ [0, a2] × [0, b2]. Then for all i ∈ N and for each

U ∈ [0, a2],

δ|Vi(U)− VG(U)| ≤∣∣∣∣(Vi(U)− VG(U))

∫ 1

0

GV (U,Vi(U) + s[VG(U)− Vi(U)])ds

∣∣∣∣= |G(U,VG(U))− G(U,Vi(U))| (5–52)

105

By Step 2.2, for all ϵ > 0 and each U ∈ [0, a2], there exists some N2 > 0 (which depends

on U) such that |G(U,Vi(U)) − G(U,VG(U))| < ϵ for all i ≥ N2. Then by (5–52),

|Vi(U)− VG(U)| < ϵδ

for all i ≥ N2.

Since F (U,V ) is continuous on R2+, F (U,V ) is uniformly continuous for (U,V ) ∈

[0, a2]× [0, b2]. (Recall that Vi(U) ∈ [0, b2] for all U ∈ [0, a2] by Lemma 13 and continuity.)

Let µ > 0. Then there exists some η > 0 such that |F (U,Vi(U)) − F (U,VG(U))| < µ

whenever |Vi(U) − VG(U)| < η. Choose ϵ < ηδ. Then |Vi(U) − VG(U)| < ϵδ< η for all

i ≥ N2. Hence, |F (U,Vi(U))− F (U,VG(U))| < µ for all i ≥ N2. Thus the claim is proven.

Step 2.4 Now, by (5–45),

D(Wi(a2))2

2=

∫ a2

0

ciWi(s)− F (U,Vi(s))ds

limi→∞

D(Wi(a2))2

2= lim

i→∞

∫ a2

0

ciWi(s)ds − limi→∞

∫ a2

0

F (U,Vi(s))ds

limi→∞

D(Wi(a2))2

2= −

∫ a2

0

F (U,VG(s))ds (5–53)

by Steps 2.1 and 2.3 and the Dominated Convergence Theorem (Theorem 1.4). Since

(Wi(a2))2 ≥ 0 for all i ∈ N, limi→∞

D(Wi (a2))2

2≥ 0. Then by (5–53),

∫ a20F (U,VG(s))ds ≤ 0,

a contradiction to the assumption that∫ a20F (U,VG(s))ds > 0.

5.3.4.4 P1 and P2 are Open and Disjoint

Next, we show that P1 and P2 are open by the continuity of solutions with respect to

parameters, time and initial conditions (Lemma 5) and by the continuity of the unstable

manifold with respect to parameters (Appendix C).

Lemma 19. P1 and P2 as defined in (5–34) and (5–35) are open.

Proof. The proofs of openness of P1 and P2 are similar. Here, we will give only the proof

that P1 is open. Recall that

P1 = {c > 0| ∃ τ1 <∞ such that U(τ1) > a2 or V (τ1) > b2 and W (τ) > 0 ∀ τ ∈ (−∞, τ1]}.

106

Let c1 ∈ P1. We will show that there exists some ϵ > 0 such that c ∈ P1 whenever

|c1 − c | < ϵ.

Let (U0c1,V 0

c1,W 0

c1)T ∈ W u(0, c1) denote the initial condition (τ = 0) chosen on

the unstable manifold of the origin and in the open positive orthant. (Recall that such

a choice is possible by Lemmas 8 and 10.) Then, since c1 ∈ P1, there exists some

τ1 < ∞ such that U(τ1) > a2 or V (τ1) > b2 and W (τ) > 0 for all τ ∈ (−∞, τ1].

Without loss of generality, suppose that U(τ1) > a2. For each τ ∈ [0, τ1], choose δτ such

that Bδτ((U(τ , c1),V (τ , c1),W (τ , c1))

T)

is contained in the open positive orthant and

such that Bδτ1((U(τ1, c1),V (τ1, c1),W (τ1, c1))

T)

is additionally contained in the region

{U > a2}. Note that this precludes the possibility that any steady states are contained in

Bδτ((U(τ , c1),V (τ , c1),W (τ , c1))

T). By the continuity of solutions with respect to initial

conditions, time, and parameters (Lemma 5), for each τ ∈ [0, τ1], there exist λτ , γτ , and

ρτ such that

(U(s, c),V (s, c),W (s, c))T ∈ Bδτ((U(τ , c1),V (τ , c1),W (τ , c1))

T)

whenever∥∥(U0

c1,V 0

c1,W 0

c1)T − (U0

c ,V0c ,W

0c )

T∥∥ < λτ , |c1 − c | < γτ and |τ − s| < ρτ . This

gives us an infinite cover {(τ − ρτ , τ + ρτ)|τ ∈ [0, τ1]} of the compact interval [0, τ1]. Then

there exists a finite subcover {(τi − ρi , τi + ρi)}Ni=1 of [0, τ1]. Let

λ = min1≤i≤N

{λi} and γ = min1≤i≤N

{γi}.

Then given an initial condition and a parameter c such that∥∥(U0

c1,V 0

c1,W 0

c1)T − (U0

c ,V0c ,W

0c )

T∥∥ <

λ and |c1−c | < γ, for any s ∈ [0, τ1] there exists some τi , 1 ≤ i ≤ N, such that |τi−s| < ρi

and

(U(s, c),V (s, c),W (s, c))T ∈ Bδi((U(τi , c1),V (τi , c1),W (τi , c1))

T). (5–54)

By the continuity of the unstable manifold with respect to parameters (Appendix C),

there exists some �γ > 0 such that Bλ((U0

c1,V 0

c1,W 0

c1)T)∩ W u(0, c) = ∅ whenever

107

|c1 − c | < �γ. Let ϵ = min{γ, �γ}. Then for any |c1 − c | < ϵ, we may choose

(U0c ,V

0c ,W

0c )

T ∈ W u(0, c) such that∥∥(U0

c1,V 0

c1,W 0

c1)T − (U0

c ,V0c ,W

0c )

T∥∥ < λ. Then

by (5–54), (U(s, c),V (s, c),W (s, c))T is contained in the open positive orthant for

s ∈ [0, τ1] and (U(τ1, c),V (τ1, c),W (τ1, c))T is additionally contained in the set

{U > a2}. Then by Lemma 10, c ∈ P1. Thus, P1 is open.

Lastly, P1 ∩ P2 = ∅ by definition. Thus, P1 and P2 are nonempty, disjoint, and open.

Now, as we argued in Section 5.3.4.1, there exists some c∗ ∈ (0,∞)\(P1 ∪ P2).

5.3.4.5 The Existence of a Heteroclinic Connection

Lemma 20. Let c∗ ∈ (0,∞)\(P1 ∪ P2). Then for all τ ∈ (−∞, τmax), U(τ) < a2, V (τ) < b2

and 0 <W (τ).

Remark 2. This lemma together with Lemma 10 and the discussion following the

definition of in (5–33) implies that for c∗ ∈ (0,∞)\(P1 ∪ P2), (U(τ),V (τ),W (τ))T

remains in the interior of for all τ ∈ (−∞, τmax). That is, for all τ ∈ (−∞, τmax),

0 < U(τ) < a2, 0 < V (τ) < b2 and 0 <W (τ).

Proof of Lemma 20. Suppose by way of contradiction that Lemma 20 does not hold.

Then either

∃ τa < τmax such that U(τa) ≥ a2

or ∃ τb < τmax such that V (τb) ≥ b2

or ∃ τ0 < τmax such that W (τ0) ≤ 0.

We assume without loss of generality that τa, τb and τ0 are the first time at which each of

their respective conditions hold. That is, τa, τb and τ0 are the least values of τ such that

U(τa) = a2, V (τb) = b2 and W (τ0) = 0. If U(τ) < a2 for all τ < τmax , we define τa = τmax .

Similarly, we allow for τb = τmax and τ0 = τmax . Allowing for the possibility that (no more

than two of) τa, τb and τ0 may equal τmax , there are seven cases under which we may

contradict the lemma.

108

Case 1: τa < τb, τ0. Since τa < τ0, U ′(τa) = W (τa) > 0. Then there exists some ϵ > 0

small enough such that U(τa + ϵ) > a2 and W (τ) > 0 for all τ ∈ (−∞, τa + ϵ]. That is,

c∗ ∈ P1, a contradiction.

Case 2: τb < τa, τ0. We have that U(τb) < a2, V (τb) = b2 and since V (τ) < b2

for all τ < τb, V ′(τb) ≥ 0. By Condition 3, GU > 0 on (0, a2] × (0, b2] and hence

G(U(τb),V (τb)) = G(U(τb), b2) < 0 = G(a2, b2). Then since c∗ > 0, V ′(τb) =

G(U(τb),V (τb))c∗

< 0, a contradiction.

Case 3: τ0 < τa, τb. In this case, we have that W (τ0) = 0 and W (τ) > 0 for all

τ < τ0. Then it must be true that W ′(τ0) ≤ 0 and since W ′ = c∗W−F (U,V )D

, we have

that F (U(τ0),V (τ0)) ≥ 0. If F (U(τ0),V (τ0)) > 0, then W ′(τ0) = c∗W (τ0)−F (U(τ0),V (τ0))D

=

−F (U(τ0),V (τ0))D

< 0. Then there exists some ϵ > 0 such that for all τ ∈ (τ0, τ0 + ϵ),

W (τ) <W (τ0) = 0. Thus, c∗ ∈ P2.

If F (U(τ0),V (τ0)) = 0, then W ′(τ0) = −F (U(τ0),V (τ0))D

= 0 and W (τ0) = U ′(τ0) = 0.

Since W ′ does not provide us with a contradiction, we examine W ′′:

W ′′(τ0) =c∗

DW ′(τ0)−

1

DFU(U(τ0),V (τ0))U

′(τ0)−1

DFV (U(τ0),V (τ0))V

′(τ0)

= − 1

DFV (U(τ0),V (τ0))V

′(τ0).

By Condition 3 and since τ0 < τa, τb, FV (U(τ0),V (τ0)) > 0. By Lemma 11, V ′(τ0) ≥ 0.

If V ′(τ0) > 0, then W ′′(τ0) < 0, in which case there exists some τ < τ0 such that

W (τ) < 0, a contradiction to the definition of τ0. If V ′(τ0) = 0, then since V ′ = G(U,V )c

,

G(U(τ0),V (τ0)) = 0. Recall that F (U(τ0),V (τ0)) = 0. Then (U(τ0),V (τ0),W (τ0))T is a

steady state of system (5–27). This contradicts the uniqueness of solutions.

Case 4: τa = τb < τ0. Define

LI ={(U,V ,W )T |U = a2,V = b2,W > 0

}.

109

(See Figure 5-3.) Then (U(τa),V (τa),W (τa))T ∈ LI . Consider

V ′′(τa) =1

c∗(GU(U(τa),V (τa))U

′(τa) + GV (U(τa),V (τa))V′(τa)) .

By Condition 3, GU > 0 on (0, a2] × (0, b2]. By Condition 1, G(a2, b2) = 0, so

V ′(τa) = 0. By assumption, U ′(τa) = W (τa) > 0 and c∗ > 0. Hence, V ′′(τa) > 0. Then,

since V (τa) = b2, V ′(τa) = 0 and V ′′(τa) > 0, there exists some �τ < τa = τb such that

V (�τ) = b2, a contradiction to the definition of τb.

Case 5: τa = τ0 < τb. Define

LII ={(U,V ,W )T |U = a2, 0 < V < b2,W = 0

}.

(See Figure 5-3.) Then (U(τa),V (τa),W (τa))T ∈ LII . Since τa is the first value of τ at

which W (τ) = 0, we have that W ′(τa) ≤ 0. Since W ′ = c∗W−F (U,V )D

and W (τa) = 0, we

see that −F (U(τa),V (τa)) = −F (a2,V (τa)) ≤ 0. Then F (a2,V (τa)) ≥ 0 = F (a2, b2). By

Condition 3, FV > 0 for (U,V ) ∈ [0, a2] × [0, b2], so by the continuity of FV , V (τa) ≥ b2.

Then τb ≤ τa, a contradiction.

Case 6: τb = τ0 < τa. Define

LIII ={(U,V ,W )T |0 < U < a2,V = b2,W = 0

}.

(See Figure 5-3.) Then (U(τb),V (τb),W (τb))T ∈ LIII . Since τb is the first value of

τ at which V (τ) = b2, we have that V ′(τb) ≥ 0. Since c∗ > 0, G(U(τb),V (τb)) =

G(U(τb), b2) ≥ 0 = G(a2, b2). By Condition 3, GU > 0 for (U,V ) ∈ R2+\{(0, 0)}, so we

must have that U(τb) ≥ a2. Then τa ≤ τb, a contradiction.

Case 7: τa = τb = τ0. Here, τa, τb and τ0 must all be less than τmax and therefore

finite. Then (U(τa),V (τa),W (τa))T = (a2, b2, 0)

T , an equilibrium point of (5–27). This

contradicts the uniqueness of solutions.

110

We now have all of the lemmas necessary to show that c∗ ∈ R+\(P1 ∪ P2) admits a

heteroclinic orbit of (5–27) connecting (0, 0, 0)T and (a2, b2, 0)T .

Lemma 21. Let (U(τ),V (τ),W (τ))T be a solution of (5–27) such that (5–32) holds.

Suppose that for all τ ∈ (−∞, τmax), 0 < U(τ) < a2, 0 < V (τ) < b2 and 0 < W (τ). Then

τmax = ∞ and limτ→∞(U(τ),V (τ),W (τ))T = (a2, b2, 0)T .

Proof. By Lemma 12, the W component of (U(τ),V (τ),W (τ))T is bounded and

τmax = ∞. By Lemma 11 and since U ′ > 0 for all τ , the U and V components of the

trajectory are monotonically increasing for all τ . Thus, U(τ) and V (τ) have limits as

τ → ∞. By Barbalat’s Lemma (Lemma 1), if U ′ = W is uniformly continuous, then

limτ→∞W (τ) = 0. Indeed, since U,V , and W are bounded, F is continuous and

W ′ = cW−F (U,V )D

, we have that W ′ is bounded and hence W is uniformly continuous.

Since U,V and W converge, Lemma 4 implies that the trajectory must approach a

steady state. The only steady states contained in the closure of as defined in (5–33)

are (0, 0, 0)T , (a1, b1, 0)T and (a2, b2, 0)T . We claim that (U(τ),V (τ),W (τ))T cannot

converge to (0, 0, 0)T or to (a1, b1, 0)T .

Indeed, since U and V are monotonically increasing for all τ ∈ (−∞,∞) by

Lemmas 11 and 20, (U(τ),V (τ),W (τ))T cannot converge to (0, 0, 0)T .

Suppose by way of contradiction that (U(τ),V (τ),W (τ))T converges to (a1, b1, 0)T .

Then U(τ) < a1 for all τ ∈ (−∞,∞) and limτ→∞ U(τ) = a1. By Lemma 14, W ′(τ) > 0 for

all τ ∈ (−∞,∞). Then limτ→∞W (τ) > 0. Since W does not approach zero as τ → ∞,

(U(τ),V (τ),W (τ))T cannot converge to (a1, b1, 0)T .

Since (U(τ),V (τ),W (τ))T cannot converge to (0, 0, 0)T or to (a1, b1, 0)T , the

solution must converge to (a2, b2, 0)T .

5.3.4.6 The Existence of a Traveling Wave

By Remark 2 (after Lemma 20) and Lemmas 20, 21 and 11, (U(τ),V (τ),W (τ))T

solves (5–16) and (5–17) for c∗ ∈ R+\(P1 ∪ P2), U ′(τ) > 0 for all τ finite and V ′(τ) > 0

111

for all τ finite. Then (U(τ),V (τ))T satisfies (5–15), (5–14) and is therefore a traveling

wave solution of system (5–9) with lower and upper limits at E0 and E2, respectively.

5.3.5∫ a20F (U,VG(U))dU < 0

(Step 4 in the proof of Theorem 2.1 in Jin and Zhao [47]) Suppose that∫ a20F (U,VG(U))dU <

0. By the change of variables

�u = a2 − u and �v = b2 − v ,

(5–9) becomes ∂�u∂t

= D ∂2�u∂x2

+ �F (�u, �v)

∂�v∂t

= �G(�u, �v)

(5–55)

where

�F (�u, �v) = −F (a2 − �u, b2 − �v) and �G(�u, �v) = −G(a2 − �u, b2 − �v).

Define

�vG(�u) := b2 − VG(a2 − �u).

Then

�G(�u, �vG(�u)) = −G(u,VG(u)) = 0 for all u ∈ [0, a2]

by the construction of VG(u) in Section 5.3.1.

By the definitions of �F and �G , �g(�u, �v) := (�F (�u, �v), �G(�u, �v))T has only three zeros

E0 = (0, 0), E1 = (a2 − a1, b2 − b1) and E2 = (a2, b2) in the order interval [E0,E2]. Since F

and G satisfy Conditions 1-4 for (5–9), it is easy to see that∫ a2

0

�F (�u, �vG(�u))d�u = −∫ a2

0

F (U,VG(U))dU > 0

112

and that �F and �G satisfy Conditions 1-4 for (5–55). Then by the series of proofs in

Section 5.3.4, there exists some c∗ ∈ R+ such that (�U(x + c∗t), �V (x + c∗t))T is a

monotone increasing traveling wave solution of (5–55) such that

limτ→−∞

(�U(τ), �V (τ))T = (0, 0)T and limτ→∞

(�U(τ), �V (τ))T = (a2, b2)T

where τ = x + c∗t.

Define

U(ξ) = a2 − �U(−ξ) and V (ξ) = b2 − �V (−ξ) for all ξ ∈ R.

Then

limξ→−∞

(U(ξ),V (ξ))T = (0, 0)T and limξ→∞

(U(ξ),V (ξ))T = (a2, b2)T .

It follows that (U(ξ),V (ξ))T is a monotone increasing traveling wave solution of system

(5–9) with lower and upper limits at E0 and E2, respectively. By Lemma 7, this wave has

negative speed, that is, c < 0.

5.3.6 Statement of Existence Theorem

We summarize the results of this chapter in a theorem:

Theorem 5.2. Suppose (5–9) satisfies Conditions 1-4. Then there exists some c ∈ R

such that (U(x + ct),V (x + ct)) = (U(τ),V (τ)) is a bistable monotone increasing travel-

ing wave solution to (5–9) with limτ→−∞(U(τ),V (τ)) = E0 and limτ→∞(U(τ),V (τ)) = E2.

Furthermore, the wave speed c has the same sign as the integral∫ a20F (U,VG(U))dU

where VG(U) satisfies G(U,VG(U)) = 0 for all U ∈ [0, a2].

113

CHAPTER 6FUTURE WORK

There are several directions in which we may extend this work. One obvious

extension is determining conditions under which the traveling wave solution found

in Section 5.3 is unique up to translation. We expect that such a result would follow

from monotonicity properties of the functions F and G similar to, but possibly stronger

than, those listed in Condition 3. A sufficient proof would show that the sets P1 and P2

(defined in (5–34) and (5–35)) are connected and that the lower bound of P1 is equal to

the upper bound of P2, or possibly that as c > 0 increases, the value of U at which the

trajectory (U(τ),V (τ),W (τ))T intersects the U,V -plane increases monotonically.

Another extension to our existence proof is an analysis of the stability of our

traveling wave solution. In a series of articles, Evans (1972, 1975) proves that for a

certain class of reaction-diffusion systems (that includes our model), a traveling wave

solution is stable if and only if the corresponding linearization about this solution is

stable. [28–31] We expect that these results could be readily applied to our traveling

wave solution.

Finally, we plan to experimentally parameterize our model of QS up-regulation

in Aliivibrio fischeri, described in Section 5.1. The parameterization will require

incorporating cell growth dependence in the rate of production of LuxI as a function

of AHL concentration. This dependence will then need to be appropriately removed,

perhaps by approximating the time-dependent cell growth rate by the instantaneous

cell growth rate at an appropriate time. Another complication in our parameterization

lies in that we will not be able to directly measure the AHL concentration A. Without

a time series representation of A(t), we would not be able to determine λ, the AHL

production rate per unit of LuxI. To deduce the AHL concentration A in a culture of

A. fischeri at time t, we extract a sample of the culture and remove the supernatant.

We then introduce the sensor strain E. coli + pJBA132, whose response to AHL

114

is well-characterized, and measure the fluorescence response. This allows us to

approximate the AHL concentration A(t). Once we have parameterized the model in

Section 5.1, we will be able to determine parameter ranges under which a traveling

wave solution exists and to numerically approximate the speed of signal propagation in a

spatially extended colony of A. fischeri.

115

APPENDIX APROOF OF THEOREM 2.2

In this Appendix we shall prove Theorem 2.2. First we set Ii =∫ l0n(x)dx and

Io =∫ 1

ln(x)dx . Using the definition of n(x) in (2–11) and (2–14), we have:

Ii =

∫ l

0

n(x)dx

=

∫ l

0

c cosh(αix) +R

µidx

=c

αisinh(αi l) +

Rl

µi

= −R

�

(1

µi− 1

µo

)DoαoDiαi

1

αisinh(αi l) sinh(αo(1− l)) +

Rl

µi

where � is given by (2–13), and similarly for Io :

Io =

∫ 1

l

n(x)dx

=

∫ 1

l

d cosh(αo(1− x)) +R

µodx

=d

αosinh(αo(1− l)) +

R(1− l)

µo

=R

�

(1

µi− 1

µo

)1

αosinh(αi l) sinh(αo(1− l)) +

R(1− l)

µo

Recalling the definitions of αi and αo in (2–12), and using (2–20), it follows that:

Ii = −(R

(1

µi− 1

µo

) õo

µi

)√D

β

sinh(√

µi l1√D

)sinh

(√µo(1− l)

√βD

)�

+Rl

µi

Io =

(R

(1

µi− 1

µo

)1

õo

)√D

β

sinh(√

µi l1√D

)sinh

(√µo(1− l)

√βD

)�

+R(1− l)

µo

and

� =

√µoµi

1√βcosh

(õi l

1√D

)sinh

(√µo(1− l)

√β

D

)

116

+ sinh

(õi l

1√D

)cosh

(√µo(1− l)

√β

D

)

Defining

ηi = R

(1

µi− 1

µo

) õo

µi, ηo = R

(1

µi− 1

µo

)1

õo

, σi =√µi l σo =

√µo(1− l),

we find that:

Ii = −ηi

√D

β

sinh(σi

1√D

)sinh

(σo

√βD

)�

+Rl

µi(A–1)

Io = ηo

√D

β

sinh(σi

1√D

)sinh

(σo

√βD

)�

+R(1− l)

µo(A–2)

and

� =

√µoµi

1√βcosh

(σi

1√D

)sinh

(σo

√β

D

)+ sinh

(σi

1√D

)cosh

(σo

√β

D

)(A–3)

The Yield Y , Total Abundance A and Log Ratio L as defined in (2–17), (2–18) and

(2–19) respectively, can be written more compactly in terms of Ii and Io :

Y = (µo − µi)Io ,

A = Ii + Io ,

and

L = ln

[(1− l)Ii

lIo

].

We start by examining the signs of the derivatives dIidD

and dIodD

, and will then use this

information to determine the signs of dYdD

, dAdD

and dLdD

.

Fact:

dIidD

≤ 0 and dIodD

≥ 0.

117

Before proving this, we introduce more notation. We set ~σo = σo√β, γ =

√µoµiβ

,

~ηi = ηi/√β, and we let y = 1√

D. Then we can rewrite � in (A–3) as follows:

� = γ sinh(~σoy) cosh(σiy) + sinh(σiy) cosh(~σoy), (A–4)

and (A–1) becomes:

Ii = −~ηisinh(~σoy) sinh(σiy)

�y+Rl

µi,

and then

dIi

dD=

dy

dD

dIi

dy

= −~ηiy 3

−2

d

dy

[sinh(~σoy) sinh(σiy)

�y

]= ~ηi

y 3

2

1

(�y)2

[(~σo cosh(~σoy) sinh(σiy) + σi sinh(~σoy) cosh(σiy)) (�y)

− sinh(~σoy) sinh(σiy)

(yd�

dy+ �

)]= ~ηi

y

2�2f (y),

where we used (A–4) in the last step and introduced

f (y) := sinh2(σiy) [~σoy − sinh(~σoy) cosh(~σoy)]

+ γ sinh2(~σoy) [σiy − sinh(σiy) cosh(σiy)] . (A–5)

Thus, the sign of dIidD

is equal to the sign of f (y). To determine this, we examine the

function a − sinh(a) cosh(a) = a − 12sinh(2a) for a ≥ 0. Note that when a = 0,

12sinh(2a) = 0 and that d

da(a) = 1 ≤ cosh(2a) = d

da

(12sinh(2a)

)for a ≥ 0. Thus,

a ≤ 12sinh(2a) for all a ≥ 0 and consequently (A–5) is nonpositive for y > 0. This shows

that dIidD

≤ 0.

We use a similar calculation for Io . Setting ~ηo = ηo/√β, we rewrite Io as

Io = ~ηosinh(~σoy) sinh(σiy)

�y+R(1− l)

µo.

118

Then

dIo

dD=

dy

dD

dIo

dy

= ~ηoy 3

−2

d

dy

[sinh(~σoy) sinh(σiy)

�y

]= −~ηo

y

2�2f (y)

≥ 0 for all y > 0.

Proof of Theorem 2.2.

1. dIodD

≥ 0.This has already been shown.

2. dYdD

≥ 0.Indeed, since Y = (µo − µi)Io and dIo

dD≥ 0, it follows that

dY

dD= 2(µo − µi)

dIo

dD≥ 0.

3. dAdD

≤ 0.Since A = Ii + Io , it follows that

dA

dD=

(dIi

dD+

dIo

dD

)=

(~ηi

1

2√D�2

f(1/

√D)− ~ηo

1

2√D�2

f(1/

√D))

=1

2√D�2

(ηi − ηo)f(√

D)

Recall from (A–5) that f (y) ≤ 0 if y > 0. Moreover,

~ηi − ~ηo =R√β

(1

µi− 1

µo

)(µo − µiµi√µo

)> 0,

and therefore dAdD

≤ 0.

4. dLdD

≤ 0.Notice that

L = ln

[(1− l)Ii

lIo

]= ln

[1− l

l

]+ ln

[Ii

Io

].

119

Moreover,Ii

Io=

∫ l0n(x)dx∫ 1

ln(x)dx

=

∫ 1

0n(x)dx∫ 1

ln(x)dx

− 1 =A

Y /(µ0 − µi)− 1

Since A is nonincreasing and Y is nondecreasing with D, it follows that Ii/Io andhence L is nonincreasing with D.

120

APPENDIX BSKETCH OF THE PROOF OF THEOREM 2.4

Here we only sketch part of the proof of Theorem 2.4, as it is similar to that of

Theorem 2.2.

Since Ii =∫ l0n(x)dx , and n(x) is given by (2–25), with (2–26), there holds that

Ii = − DoαoDiα2

i �sinh(αo(1− l)) sinh(αi l)R(z) +

Rl

µi

where R(z) is given by (2–27), and � by (2–28). By (2–20), recalling (2–12) and setting

~σo =√µo(1− l)

√β and σi =

õi l , it follows that

Ii = −√µo

µi√β

sinh(~σoy) sinh(σiy)

�yR(z) +

Rl

µi

where as before, we have set y = 1/√D, and where:

� = γ sinh(~σoy) cosh(σiy) +1− z

1 + z

√β cosh(~σoy) sinh(σiy)

with γ =√µo/(βµi). Therefore,

dIi

dD=

dy

dD

dIi

dy= +

õo

µi√β

y 3

2R(z)

d

dy

[sinh(~σoy) sinh(σiy)

�y

]Proceeding similarly as in the proof of Theorem 2.2, but being cautious because now the

factor (1− z)/(1 + z) appears in the second term of � here, it follows that:

dIi

dD= +

õo

µi√β

y

2�2R(z)

[γ sinh2(~σoy)g(σiy) +

1− z

1 + z

√β sinh2(σiy)g(~σoy)

],

where

g(x) = x − sinh(x) cosh(x).

Since g(x) ≤ 0 if x ≥ 0, it follows that the sign of dIi/dD is opposite to the sign of R(z)

(R(z)is defined in (2–27)). As the latter is negative if z < z∗, zero if z = z∗, and positive

121

if z > z∗, it follows that

dIi

dD

< 0, if z > z∗(weak bias)

= 0, if z = z∗(critical bias)

> 0, if z < z∗(strong bias)

In particular, we notice that the sign of dIi/dD is reversed when z moves from weak bias

values to strong ones. We omit the rest of the proof of the Theorem 2.4 as it is similar to

that of Theorem 2.2.

122

APPENDIX CCONTINUITY OF THE STABLE MANIFOLD WITH RESPECT TO PARAMETERS

In this appendix, we prove that the stable manifold is locally continuous with respect

to parameters. This proof is an extension of that found in Coddington and Levinson’s

classic text, Theory of Ordinary Differential Equations. [17] Section C.5 closely follows

the proof found in Coddington and Levinson, but differs in that Coddington and Levinson

use a Picard Iteration to show the existence of the stable manifold while we use the

Contraction Mapping Theorem.

C.1 Assumptions

Consider the system

x ′ = A(c)x + f (t, x , c) (C–1)

where x ∈ Rn, t ∈ R, the prime(′ = d

dt

)denotes the time derivative, c ∈ R is a

parameter and A(c) is a real-valued matrix whose entries depend continuously on c , but

are independent of t. Suppose that f (t, x , c) depends continuously on (t, x , c) for x ∈ �,

t ≥ 0, and c ∈ � where � ⊂ Rn, 0 ∈ int(�) and � ⊂ R. Suppose that

f (t, 0, c) = 0

for all t ≥ 0, c ∈ �. Moreover, suppose that given any ϵ > 0, there exist some δ > 0 and

some T > 0 such that for any t ≥ T , ∥x∥ ≤ δ, ∥~x∥ ≤ δ and all c ∈ �,

∥f (t, ~x , c)− f (t, x , c)∥ ≤ ϵ∥~x − x∥. (C–2)

This condition is stronger than the usual assumption that f is locally Lipschitz.

C.2 Outline

Fix c ∈ � and suppose that A(c) has k characteristic roots with negative real parts

and n − k characteristic roots with positive real parts. We will show that there exists a

k−dimensional manifold ~S(c) ∈ Rn, some k curvilinear coordinates y1, y2, ... yk and

123

some n − k real continuous functions ψj(y1, y2, ... , yk , c), k + 1 ≤ j ≤ n, such that

y1

...

yk

ψk+1(y1, y2, ... , yk , c)

...

ψn(y1, y2, ... , yk , c)

defines ~S(c) for sufficiently small |yi |, 1 ≤ i ≤ k . We will also show that there exists

a real nonsingular matrix P(c) whose entries depend continuously on c such that

y = P(c)x and

x = P(c)−1

y1

...

yk

ψk+1(y1, y2, ... , yk , c)

...

ψn(y1, y2, ... , yk , c)

defines the sought-after stable manifold S(c) in terms of the k curvilinear coordinates

y1, ... , yk .

Lastly, we will show that if we fix c1, then whenever c is sufficiently close to c1, S(c)

is close to S(c1). More precisely, we show that there exists some compact set × �

(which will be defined later) containing (0, c1) such that given any ϵ > 0 there exists

124

some γ > 0 such that∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

y1

...

yk

ψk+1(y1, y2, ... , yk , c)

...

ψn(y1, y2, ... , yk , c)

−

�y1...

�yk

�ψk+1( �y1, �y2, ... , �yk , c1)

...

�ψn( �y1, �y2, ... , �yk , c1)

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

< ϵ

whenever c ∈ �, |c−c1| < γ, P(c)−1(y1, y2, ... yk , 0, ... , 0)T ∈ and P(c1)

−1( �y1, �y2, ... �yk , 0, ... , 0)T ∈

.

C.3 Fix the Dimension of Stable Manifold

Fix c1. Suppose that A(c1) has k characteristic roots with negative real parts and

n − k characteristic roots with positive real parts. We will show that for c close enough

to c1, A(c) also has k characteristic roots with negative real parts, n − k characteristic

roots with positive real parts, and in particular, no characteristic roots with zero real part.

To prove this, we will show that the roots of the characteristic polynomial of A(c) depend

continuously on c for c close enough to c1. We will use the following lemma from Sontag

(1998).[83]

Lemma 22. Let p(z) = zn + a1zn−1 + ... an be a polynomial of degree n and complex

coefficients having distinct roots λ1, ... ,λq, with multiplicities m1 + · · · + mq = n,

respectively. Given any small enough ϵ > 0, there exists a δ > 0 so that if

r(z) = zn + b1zn−1 + · · ·+ bn, |ai − bi | < δ for 1 ≤ i ≤ n,

then r has precisely mi roots in the open disk Bϵ(λi) for each 1 ≤ i ≤ q.

Let p(z) be the characteristic polynomial of A(c1). Suppose that p(z) has distinct

roots λ1, ... ,λq, with multiplicities m1 + · · ·+mq = n, respectively. For each λi (1 ≤ i ≤ q),

choose ϵi such that the open disk Bϵi (λi) does not intersect the real axis of the complex

125

plane and such that Bϵi (λi) ∩ Bϵj (λj) = ∅ for all i = j . Let ϵ = min1≤i≤q ϵi . Then by Lemma

22, there exists some δ > 0 such that if

r(z) = zn + b1zn−1 + · · ·+ bn, |ai − bi | < δ for 1 ≤ i ≤ n,

then r has precisely mi roots in the open disk Bϵ(λi) for each 1 ≤ i ≤ q.

Let r(z) = zn + b1zn−1 + · · ·+ bn be the characteristic polynomial of A(c). Since the

entries of A(c) depend continuously on c and each bi (1 ≤ i ≤ n) depends continuously

on the entries of A(c), there exists some γ1 > 0 such that |ai − bi | < δ for 1 ≤ i ≤ n

whenever |c1 − c | < γ1. That is, A(c) has k characteristic roots with negative real parts

and n − k characteristic roots with positive real parts whenever |c1 − c | < γ1.

C.4 P(c) Varies Continuously with Respect to c

Let c ∈ � such that |c1 − c | < γ1 where γ1 is the same as was determined in Section

C.3. Suppose that A(c) has qc distinct roots λci with multiplicities mci . Let Pi(c) be the

Riesz projection associated with λci . That is, let

Pi(c) = − 1

2π i

∫�ci

R(ξ, c)dξ

where

R(ξ, c) := (A(c)− ξI )−1

is the resolvent of A(c) and where �ci is a closed rectifiable curve in the complex

plane that encloses λci but no other eigenvalue of A(c) and that does not intersect the

spectrum of A(c). Then for each 1 ≤ i ≤ qc , Pi(c) projects the domain X = Rn onto ~E ci ,

the generalized eigenspace corresponding to λci :

Pi(c)X = ~E ci := N

((A(c)− λci I )

mci

)where N(·) denotes the null space.

126

Since A(c) varies continuously with respect to c , and therefore R(ξ, c) does, each

Pi(c) varies continuously with respect to c so long as the conditions under which we

constructed Pi(c) still hold: Fix c1 and let c be close to c1. Then Pi(c) varies from Pi(c1)

continuously with respect to c so long as mc1i characteristic roots of A(c) (including

multiplicity) are enclosed by �c1i and �c1i does not intersect the spectrum of A(c).

Let each �c1i be the circle of radius ϵ centered at λc1i , where ϵ is chosen so that �c1i

does not intersect the real axis of the complex plane and no two �c1i intersect. Then

by Lemma 22 and the above discussion, Pi(c) varies continuously with respect to c

whenever |c1 − c | < γ1, where γ1 is the same as was determined in Section C.3.

Recall that since |c1 − c | < γ1, A(c) has no eigenvalues with zero real part.

Suppose that A(c) has q−c distinct eigenvalues with negative real part (λci , 1 ≤ i ≤ q−

c ),

q+c = qc − q−c distinct eigenvalues with positive real part (λci , q

−c + 1 ≤ i ≤ qc ). Define

P±(c) := − 1

2π i

∫�±c

R(ξ, c)dξ

where �−c (�+c ) is a simple closed contour in the complex plane that encloses all of the

characteristic roots of A(c) with negative (positive) real part, each �ci , 1 ≤ i ≤ q−c

(q−c + 1 ≤ i ≤ qc ), and that does not intersect the spectrum of A(c). Then by Cauchy’s

Residue Theorem (Theorem 1.2), P−(c) (P+(c)) is the sum of the projections onto all

generalized eigenspaces corresponding to eigenvalues enclosed by �−c (�+c ):

P−(c) = − 1

2π i

∫�−c

R(ξ, c)dξ

= −q−c∑i=1

Resξ=λciR(ξ, c)

= −q−c∑i=1

1

2π i

∫�ci

R(ξ, c)dξ

=

q−c∑i=1

Pi(c).

127

Similarly,

P+(c) =

qc∑i=q−c +1

Pi(c).

Furthermore, by the Primary Decomposition Theorem (Theorem 1.1) and since each

Pi(c) is a linear operator (and therefore P−(c) and P+(c) are),

E−(c) := P−(c)X = ⊕q−ci=1

~E ci and E+(c) := P+(c)X = ⊕qc

i=q−c +1~E ci .

In particular, P−(c) maps the domain X = Rn onto the generalized stable eigenspace

E−(c) and P+(c) maps the domain X = Rn onto the generalized unstable eigenspace

E+(c). Since each Pi(c) varies continuously with respect to c whenever |c1 − c | < γ1,

P±(c) varies continuously with respect to c whenever |c1 − c | < γ1.

Suppose that A(c1) has k eigenvalues with negative real part and n − k eigenvalues

with positive real part. Then dim(E−(c1)) = k and dim(E+(c1)) = n−k . Let {v1, v2, ... , vk}

be a basis for E−(c1) and {vk+1, ... , vn} be a basis for E+(c1). Then there exists

some γ2 > 0 such that {P−(c)v1,P−(c)v2, ... ,P−(c)vk} is a basis for E−(c) and

{P+(c)vk+1, ... ,P+(c)vn} is a basis for E+(c) whenever |c1 − c | < γ2.

Let γ = min{γ1, γ2} and define

P(c) := [P−(c)v1,P−(c)v2, ... ,P−(c)vk ,P+(c)vk+1, ... ,P+(c)vn] (C–3)

for |c1 − c | < γ. Then since P−(c) and P+(c) vary continuously with respect to c , so

does P(c).

Recall that for |c1 − c | < γ1 ≤ γ, A(c) has k eigenvalues with negative real part and

n − k eigenvalues with positive real part. By the above construction of P(c), we have

that

P(c)−1A(c)P(c) =

B1(c) 0

0 B2(c)

= B(c) (C–4)

128

for |c1 − c | < γ, where B1(c) is a matrix of k rows and columns having all its

characteristic roots with negative real parts and B2(c) is a matrix of n − k rows and

columns having all its characteristic roots with positive real parts. Additionally, since

P(c) and A(c) vary continuously with respect to c , B1(c) and B2(c) vary continuously

with respect to c .

C.5 Construction of the Stable Manifold

C.5.1 Preliminaries

Fix some c ∈ � such that |c1 − c | < γ where γ is as was determined in Section C.4.

Then P(c), as defined in (C–3), is a real nonsingular constant matrix. Letting y = xP(c),

(C–1) becomes

y ′ = B(c)y + g(t, y , c) (C–5)

where B(c) is defined in (C–4), g = f (t, yP(c)−1, c)P(c) and

g(t, 0, c) = f (t, 0, c)P(c) = 0

for all t ≥ 0, c ∈ �. From (C–2), it follows that given any ϵ > 0, there exist δ > 0, T > 0,

not necessarily equal to those from (C–2), such that

|g(t, ~y , c)− g(t, y , c)| ≤ ϵ|~y − y | (C–6)

for |~y | ≤ δ, |y | ≤ δ, t ≥ T and all c ∈ �. Let

U1(t, c) =

etB1(c) 0

0 0

and

U2(t, c) =

0 0

0 etB2(c)

.

129

Then etB(c) = U1(t, c) + U2(t, c) and

∂

∂tUj(c) = B(c)Uj(c) for j = 1, 2.

Furthermore, as a direct consequence of Theorem 1.5, we obtain the following bounds

on U1(t, c) and U2(t, c):

Lemma 23. Let 0 < αc ∈ R be chosen so that the real parts of the characteristic roots of

B1(c) are less than −αc . Then there exist positive constants Kc and σc such that

∥U1(t, c)∥ ≤ Kce−(αc+σc)t for all t ≥ 0

∥U2(t, c)∥ ≤ Kceσc t for all t ≤ 0.

C.5.2 Existence of the Stable Manifold

Let ϵ > 0 be chosen so that

2ϵKc

σc<

1

2(C–7)

and let δ,T be as in (C–6) for the chosen ϵ. Fix t0 ≥ T and let t ≥ t0. Consider the

integral equation

θ(t, a, c) = U1(t − t0, c)a +

∫ t

t0

U1(t − s, c)g(s, θ(s, a, c), c)ds

−∫ ∞

t

U2(t − s, c)g(s, θ(s, a, c), c)ds (C–8)

where a ∈ Rn is a constant vector. We will show that for a fixed c , there exists a unique

continuous solution θ(t, a, c) to (C–8) on the space [T ,∞) × B δ2Kc

× {c}, where B δ2Kc

is

the open n−dimensional ball of radius δ2Kc

centered at the origin and B δ2Kc

denotes its

closure.

Let us denote by C(Y ) the set of continuous functions on a metric space Y .

Consider the complete metric space of functions

Cb

([T ,∞)× B δ

2Kc

× {c})={~θ(t, a, c) ∈C

([T ,∞)× B δ

2Kc

× {c})

such that (C–9)

130

~θ(t, 0, c) = 0,∥∥∥~θ(t, a, c)∥∥∥

∞< δ}

where the norm is the sup norm, defined by

∥~θ∥∞ = sup{|~θ(y)|, y ∈ Y

}.

Note that Cb = ∅ since 0 ∈ Cb.

Define the map

ϕ : Cb

([T ,∞)× B δ

2Kc

× {c})→ C

([T ,∞)× B δ

2Kc

× {c})

by

ϕ(θ)(t, a, c) = U1(t − t0, c)a +

∫ t

t0

U1(t − s, c)g(s, θ(s, a, c), c)ds

−∫ ∞

t

U2(t − s, c)g(s, θ(s, a, c), c)ds (C–10)

and note that a solution of (C–8) must satisfy ϕ(θ) = θ. We claim that the map ϕ is

well-defined for (t, a, c) ∈ [T ,∞) × B δ2Kc

× {c}, t0 ≥ T . Indeed, for any t ∈ [T ,∞) and

any a ∈ B δ2Kc

, the first term on the right hand side of (C–10) is clearly finite. Since θ ∈ Cb,

and by (C–6), ∥g(s, θ(s, a, c), c)∥ is uniformly bounded for all s ∈ [T ,∞). By Lemma 23,

U1 and U2 are bounded by exponential functions that are integrable on [T ,∞). Then all

terms of (C–10) are finite.

C.5.2.1 ϕ : Cb → C

First, we will show that if θ ∈ Cb, then ϕ(θ) ∈ C . Fix some (t1, a1, c) ∈ [T ,∞) ×

B δ2Kc

× {c}. Let ν > 0. We will show that there exists some η > 0 such that

∥ϕ(θ)(t2, a2, c)− ϕ(θ)(t1, a1, c)∥ < ν

whenever (t2, a2, c) ∈ [T ,∞)× B δ2Kc

× {c} and ∥(t2, a2, c)− (t1, a1, c)∥ < η. By (C–10),

∥ϕ(θ)(t2, a2, c)− ϕ(θ)(t1, a1, c)∥

131

≤ ∥U1(t2 − t0, c)a2 − U1(t1 − t0, c)a1∥ (C–11)

+

∥∥∥∥∥∫ t2

t0

U1(t2 − s, c)g(s, θ(s, a2, c), c)ds −∫ t1

t0

U1(t1 − s, c)g(s, θ(s, a1, c), c)ds

∥∥∥∥∥(C–12)

+

∥∥∥∥∥∫ ∞

t2

U2(t2 − s, c)g(s, θ(s, a2, c), c)ds −∫ ∞

t1

U2(t1 − s, c)g(s, θ(s, a1, c), c)ds

∥∥∥∥∥(C–13)

First, we consider (C–11). By definition,

∥U1(t2 − t0, c)a2 − U1(t1 − t0, c)a1∥

≤ ∥U1(t2 − t0, c)∥ ∥a2 − a1∥+ ∥a1∥ ∥U1(t2 − t0, c)− U1(t1 − t0, c)∥

=∥∥e(t2−t0)B1(c)

∥∥ ∥a2 − a1∥+ ∥a1∥∥∥e(t2−t0)B1(c) − e(t1−t0)B1(c)

∥∥ .Since the matrix exponential is continuous, and since c is fixed, there exists some η1 > 0

such that

∥∥e(t2−t0)B1(c) − e(t1−t0)B1(c)∥∥ < Kcν

3δ

whenever |t2 − t1| < η1. Then since ∥a1∥ ≤ δ2Kc

,

∥U1(t2 − t0, c)a2 − U1(t1 − t0, c)a1∥ ≤∥∥e(t2−t0)B1(c)

∥∥ ∥a2 − a1∥+ ∥a1∥∥∥e(t2−t0)B1(c) − e(t1−t0)B1(c)

∥∥<∥∥e(t2−t0)B1(c)

∥∥ ∥a2 − a1∥+ ∥a1∥Kcν

3δ

<ν

6+ν

6=ν

3(C–14)

for (t2, a2, c) such that ∥(t2, a2, c)− (t1, a1, c)∥ < min{η1,

ν6N

}where

N = max|t1−t|≤η1

∥∥e(t−t0)B1(c)∥∥

and N is finite since the matrix exponential is bounded on the compact interval |t1 − t| ≤

η1.

132

Next, consider (C–12). First note that by (C–6) and since θ is continuous, we may

consider g(t, θ(t, a, c), c) as a function that is continuous in (t, a). Let

G(t, a) := g(t, θ(t, a, c), c). (C–15)

Then G(t, a) is continuous in (t, a). Suppose without loss of generality that t2 > t1. Then

by the definition of U1 and rewriting g as in (C–15),∥∥∥∥∥∫ t2

t0

U1(t2 − s, c)G(s, a2)ds −∫ t1

t0

U1(t1 − s, c)G(s, a1)ds

∥∥∥∥∥≤

∥∥∥∥∥∫ t1

t0

U1(t2 − s, c)G(s, a2)− U1(t1 − s, c)G(s, a1)ds

∥∥∥∥∥+

∥∥∥∥∫ t2

t1

U1(t2 − s, c)G(s, a2)ds

∥∥∥∥=

∥∥∥∥∥∫ t1

t0

e(t2−s)B1(c)G(s, a2)− e(t1−s)B1(c)G(s, a1)ds

∥∥∥∥∥ (C–16)

+

∥∥∥∥∫ t2

t1

e(t2−s)B1(c)G(s, a2)ds

∥∥∥∥ (C–17)

Consider (C–16) alone. Since G is continuous and by the continuity of the matrix

exponential,∥∥∥∥∥∫ t1

t0

e(t2−s)B1(c)G(s, a2)− e(t1−s)B1(c)G(s, a1)ds

∥∥∥∥∥≤∫ t1

t0

∥∥e(t2−s)B1(c) − e(t1−s)B1(c)∥∥ ∥G(s, a2)∥ ds

+

∫ t1

t0

∥∥e(t1−s)B1(c)∥∥ ∥G(s, a1)− G(s, a2)∥ ds

≤∫ t1

t0

∥∥e(t2−s)B1(c) − e(t1−s)B1(c)∥∥M1ds +

∫ t1

t0

M2 ∥G(s, a1)− G(s, a2)∥ ds

≤∥∥e(t2−t1)B1(c) − 1

∥∥∫ t1

t0

∥∥e(t1−s)B1(c)∥∥M1ds +

∫ t1

t0

M2 ∥G(s, a1)− G(s, a2)∥ ds

≤∥∥e(t2−t1)B1(c) − 1

∥∥M3 +

∫ t1

t0

M2 ∥G(s, a1)− G(s, a2)∥ ds (C–18)

133

where

M1 = max(s,a)∈[T ,t1]×B δ

2Kc

∥G(s, a)∥

M2 = maxs∈[T ,t1]

∥∥e(t1−s)B1(c)∥∥

and M3 = M2(t1 − T ). Note that all three of M1, M2 and M3 are nonnegative and finite.

By the continuity of the matrix exponential, there exists some η2 > 0 such that

∥∥e(t2−t1)B1(c) − 1∥∥M3 <

ν

12

whenever |t2 − t1| < η2. Since G is continuous in (t, a), G is uniformly continuous for

(t, a) ∈ [t0, t1]× B δ2Kc

. Then there exists some η3 > 0 such that

∥G(s, a1)− G(s, a2)∥ <ν

12M2(t1 − T )

whenever ∥a2 − a1∥ < η3 for all s ∈ [T , t1]. Then by (C–18),∥∥∥∥∥∫ t1

t0

e(t2−s)B1(c)G(s, a2)− e(t1−s)B1(c)G(s, a1)ds

∥∥∥∥∥ < ν

12+

ν

12=ν

6(C–19)

whenever ∥(t2, a2, c)− (t1, a1, c)∥ < min{η2, η3}.

Now consider (C–17). Since G and the matrix exponential are continuous, they are

uniformly continuous on the compact sets [T , 2t1]× B δ2Kc

and [T , 2t1], respectively. Let

M4 = max(s,a)∈[T ,2t1]×B δ

2Kc

∥G(s, a)∥

and

M5 = maxs∈[T ,2t1]

∥∥e(t2−s)B1(c)∥∥.

134

Then for t2 ≤ 2t1, ∥∥∥∥∫ t2

t1

e(t2−s)B1(c)G(s, a2)ds

∥∥∥∥ ≤ |t2 − t1|M4M5.

Then there exists some 0 < η4 < t1 such that∥∥∥∥∫ t2

t1

e(t2−s)B1(c)G(s, a2)ds

∥∥∥∥ < ν

6

whenever |t2 − t1| < η4. Thus, by the above and by (C–19),∥∥∥∥∥∫ t2

t0

U1(t2 − s, c)G(s, a2)ds −∫ t1

t0

U1(t1 − s, c)G(s, a1)ds

∥∥∥∥∥ < ν

6+ν

6=ν

3(C–20)

whenever ∥(t2, a2, c)− (t1, a1, c)∥ < min{η2, η3, η4}.

Lastly, consider (C–13). Since the integrals in (C–13) are well-defined, there exists

some T1 > 0 such that∥∥∥∥∥∫ ∞

T1

U2(t2 − s, c)g(s, θ(s, a2, c), c)ds −∫ ∞

T1

U2(t1 − s, c)g(s, θ(s, a1, c), c)ds

∥∥∥∥∥ < ν

6.

Then by an analysis similar to that for (C–12), there exists some η5 such that∥∥∥∥∥∫ T1

t2

U2(t2 − s, c)g(s, θ(s, a2, c), c)ds −∫ T1

t1

U2(t1 − s, c)g(s, θ(s, a1, c), c)ds

∥∥∥∥∥ < ν

6

whenever ∥(t2, a2, c)− (t1, a1, c)∥ < η5. Now, by the above two equations, (C–14) and

(C–20),

∥ϕ(θ)(t1, a1, c)− ϕ(θ)(t2, a2, c)∥ < ν

whenever ∥(t2, a2, c)− (t1, a1, c)∥ < η = min{

ν6N

, η1, η2, η3, η4, η5}

. Thus we have shown

that ϕ(θ) ∈ C for θ ∈ Cb.

C.5.2.2 ϕ : Cb → Cb

Now, we claim that ϕ maps Cb onto itself. It is obvious that ϕ(θ)(t, 0, c) = 0 since

g(t, 0, c) = 0 for all t ≥ 0. We show that ∥ϕ(θ)(t, a, c)∥∞ < δ.

135

By (C–10), Lemma 23 and by (C–6) since ∥θ(t, a, c)∥∞ < δ,

∥ϕ(θ)(t, a, c)∥∞ ≤ Kce−(αc+σc)(t−t0) ∥a∥+

∫ t

t0

Kce−(αc+σc)(t−s)ϵδds +

∫ ∞

t

Kceσc(t−s)ϵδds

< Kce−(αc+σc)(t−t0) δ

2Kc

+Kcϵδ

αc + σc

[1− e−(αc+σc)(t−t0)

]+Kcϵδ

σc

<δ

2+Kcϵδ

σc+Kcϵδ

σc

<δ

2+δ

4+δ

4= δ

since t > t0, αc > 0, ∥a∥ ≤ δ2Kc

and by (C–7). Thus the claim is proven.

C.5.2.3 ϕ is a Contraction

Next, we claim that ϕ is a contraction. Let θ1, θ2 ∈ Cb. Then by Lemma 23, (C–6),

(C–7), and (C–10),

∥ϕ(θ1)(t, a, c)− ϕ(θ2)(t, a, c)∥∞ ≤∫ t

t0

Kce−(αc+σc)(t−s)ϵ ∥θ1(s, a, c)− θ2(s, a, c)∥∞ ds

+

∫ ∞

t

Kceσc(t−s)ϵ ∥θ1(s, a, c)− θ2(s, a, c)∥∞ ds

≤ ∥θ1(s, a, c)− θ2(s, a, c)∥∞ ·[Kcϵ

αc + σc

(1− e−(αc+σc)(t−t0)

)+Kcϵ

σc

(1− eσc t

)]≤ ∥θ1(s, a, c)− θ2(s, a, c)∥∞

2Kcϵ

σc

< ∥θ1(s, a, c)− θ2(s, a, c)∥∞1

2

Thus the claim is proven.

C.5.2.4 Existence of the Stable Manifold

By the Contraction Mapping Theorem (Theorem 1.6), there exists a unique fixed

point θ of ϕ in Cb that solves (C–8). From (C–8) it is clear that the last n − k components

of the vector a do not enter into the solution and may be taken as zero. That θ(t, a, c) is

a solution of (C–5) is immediate for a ∈ B δ2Kc

since by Lemma 23 the integrals in (C–8)

converge.

136

From (C–8) it follows that the first k components of θj(t0, a, c) are

θj(t0, a, c) = aj 1 ≤ j ≤ k

and the later components are given by

θj(t0, a, c) = −(∫ ∞

t0

U2(t0 − s, c)g(s, θ(s, a, c), c)ds

)j

k + 1 ≤ j ≤ n

where ( )j denotes the j th component. If the functions ψj are defined by

ψj(a1, ... , ak , c) = −(∫ ∞

t0

U2(t0 − s, c)g(s, θ(s, a, c), c)ds

)j

for k + 1 ≤ j ≤ n, then clearly the initial values yj = θj(t0, a, c) satisfy the equations

yj = ψj(y1, ... , yk , c) k + 1 ≤ j ≤ n

in y space, which define a manifold ~S(c) in y space.

We claim that no solution p of (C–5) with p(t0) ∈ B δ2Kc

and p(t0) not on ~S can satisfy

∥p(t)∥ ≤ δ for all t ≥ t0, where δ is the same as was defined in (C–7). Indeed, suppose

by way of contradiction that there exists some solution p of (C–5) with p(t0) ∈ B δ2Kc

such

that ∥p(t)∥ ≤ δ for all t ≥ t0. Then by the variation of parameters formula,

p(t) = e(t−t0)B(c)p(t0) +

∫ t

t0

e(t−s)B(c)g(s, p(s), c)ds.

By the definitions of U1 and U2, we have that

p(t) = (U1(t − t0) + U2(t − t0))p(t0) +

∫ t

t0

(U1(t − s) + U2(t − s))g(s, p(s), c)ds

p(t) = U1(t − t0)p(t0) + U2(t − t0)b +

∫ t

t0

U1(t − s)g(s, p(s), c)ds

−∫ ∞

t

U2(t − s)g(s, p(s), c)ds (C–21)

where

b =

∫ ∞

t0

U2(t0 − s)g(s, p(s), c)ds + p(t0)

137

is a finite constant vector by Lemma 23 and by (C–6). Since the left hand side of (C–21)

is bounded as t → ∞ by assumption, the right hand side must be also. Each term on

the right hand side of (C–21), except U2(t − t0)b, is bounded as t → ∞ by Lemma 23

and by (C–6). Then it must be true that U2(t − t0)b is bounded as t → ∞. By Lemma 2

and since all eigenvalues of B2(c) have positive real parts, each nonzero component of

U2(t− t0) grows unbounded as t → ∞. Then it must be true that bj = 0 for k+1 ≤ j ≤ n.

Thus, by (C–21),

p(t) = U1(t − t0)p(t0) +

∫ t

t0

U1(t − s)g(s, p(s), c)ds −∫ ∞

t

U2(t − s)g(s, p(s), c)ds.

Then p(t) solves (C–8), so p(t) is on ~S , a contradiction.

We claim that

x = P(c)−1

y1

...

yk

ψk+1(y1, ... , yk , c)

...

ψn(y1, ... , yk , c)

defines the stable manifold S(c) in terms of k curvilinear coordinates y1, ... , yk . To prove

this claim, we need only show that for a fixed a ∈ B δ2Kc

(and a fixed c), limt→∞ θ(t, a, c) =

0. Indeed, note that since θ solves (C–8), ϕ(θ) = θ. Recall that ϕ is a contraction with

contraction coefficient 12. Then for any fixed t ∈ [t0,∞), a ∈ B δ

2Kc

and c ∈ �,

∥ϕ(θ)(t, a, c)− ϕ(0)∥ ≤ 1

2∥θ(t, a, c)∥

∥θ(t, a, c)− U1(t − t0, a)a∥ ≤ 1

2∥θ(t, a, c)∥

∥θ(t, a, c)∥ ≤ 2 ∥a∥ ∥U1(t − t0, a)∥

∥θ(t, a, c)∥ ≤ 2 ∥a∥Kce−(αc+σc)(t−t0) (C–22)

138

by (C–10) and by Lemma 23. Since ∥a∥ ≤ δ2Kc

, the right hand side of (C–22) goes to

zero as t approaches infinity. Then limt→∞ ∥θ(t, a, c)∥ = 0 and limt→∞ θ(t, a, c) = 0.

Thus the claim is proven.

C.6 θ(t, a, c) is Continuous with Respect to c

We now show that θ(t, a, c) is continuous with respect to c . First, we prove a lemma

and establish the domain of existence of θ(t, a, c) for a range of c values.

Lemma 24. For a fixed c1, there exist some α,σ,K and γ1 all positive such that

∥U1(t, c)∥ ≤ Ke−(α+σ)t for all t ≥ 0

∥U2(t, c)∥ ≤ Keσt for all t ≤ 0

for all |c1 − c | < γ1.

Proof. We will show that there exists some �K > 0 and some γ1 > 0 such that

∥∥etB1(c)w∥∥ ≤ �Ke−αt ∥w∥ (C–23)

for all w ∈ Rk , t ≥ 0, and c ∈ [c1 − γ1, c1 + γ1] where α is chosen such that the real parts

of all eigenvalues of B1(c1) are less than −2α. This gives us an analog to Theorem 1.5

that holds uniformly for all c ∈ [c1 − γ1, c1 + γ1]. Then by a similar proof and result for

−B2(c), Lemma 24 holds.

Let λi , 1 ≤ i ≤ k be the eigenvalues of B1(c1). Recall that all eigenvalues of B1(c1)

have negative real part. Choose α > 0 such that Re(λi) < −2α for all 1 ≤ i ≤ k . Then by

Theorem 1.5, there exists some �K > 0 such that

∥∥etB1(c1)w∥∥1≤ �Ke−2αt ∥w∥1 (C–24)

where ∥·∥1 denotes the L1 norm, defined as follows. Let w ∈ Rk , w = (w1,w2, ... ,wk)T .

Then

∥w∥1 :=k∑i=1

|wi |.

139

Let A be a k × k constant matrix with entries aij . Then the induced norm on A

∥A∥1 := max1≤j≤k

k∑i=1

|aij |

is the maximum of the absolute column sums of A. For the remainder of this proof, let

∥·∥ = ∥·∥1.

We claim that for any constant matrix A such that �K ∥A∥ < α,

∥∥e(B1(c1)+A)t∥∥ ≤ �Ke−αt . (C–25)

Indeed, consider the initial value problem

dx

dt= B1(c1)x + Ax , x(0) = x0.

By the variation of parameters formula, and by (C–24), we have that

x(t) = eB1(c1)tx0 +

∫ t

0

eB1(c1)(t−s)Ax(s)ds

∥x(t)∥ ≤∥∥eB1(c1)tx0

∥∥+ ∫ t

0

∥∥eB1(c1)(t−s)∥∥ ∥A∥ ∥x(s)∥ ds

≤ �Ke−2αt ∥x0∥+∫ t

0

�Ke−2α(t−s) ∥A∥ ∥x(s)∥ ds

e2αt ∥x(t)∥ ≤ �K ∥x0∥+∫ t

0

�K ∥A∥ e2αs ∥x(s)∥ ds.

Let

z(t) := e2αt ∥x(t)∥ .

Then the above inequality becomes

z(t) ≤ �Kz(0) +

∫ t

0

�K ∥A∥ z(s)ds.

By Gronwall’s Inequality (Theorem 1.7),

z(t) ≤ �Kz(0)e�K∥A∥t

140

e2αt ∥x(t)∥ ≤ �K ∥x0∥ e�K∥A∥t

∥x(t)∥ ≤ �K ∥x0∥ e(−2α+�K∥A∥)t . (C–26)

Now, since e(B1(c1)+A)t is the principal fundamental matrix solution of

dx

dt= (B1(c1) + A)x ,

column i of e(B1(c1)+A)t is the solution to the initial value problem

dx

dt= (B1(c1) + A)x , x(0) = ei

where ei denotes the i th standard basis vector. Let xi(t) be this solution. Then by

(C–26),

∥xi(t)∥ ≤ �Ke(−2α+�K∥A∥)t .

Then the norm of each column of e(B1(c1)+A)t is bounded by the same function. Thus,

∥∥e(B1(c1)+A)t∥∥ ≤ �Ke(−2α+�K∥A∥)t .

Choose A such that �K ∥A∥ < α. Then the claim (C–25) holds.

Let c2 ∈ � and let A = B1(c2) − B1(c1). Then since the entries of B1(c) vary

continuously with respect to c for |c1 − c | < γ (where γ is as in Section C.4), and

since the norm is a continuous function, there exists some γ1 < γ such that �K ∥A∥ =

�K ∥B1(c2)− B1(c1)∥ < α whenever |c2 − c1| < γ1. Then by (C–25),

∥∥eB1(c2)t∥∥ =

∥∥e(B1(c1)+B1(c2)−B1(c1))t∥∥ ≤ �Ke−αt .

Then (C–23) holds for this choice of γ1.

141

Let � ⊆ � ⊂ R be a compact set such that � ⊂ (c1 − γ1, c1 + γ1) and c1 lies in the

interior of �. Let ϵ1 > 0 be chosen such that

2ϵ1K

σ<

1

2(C–27)

where K and σ are as in Lemma 24. By (C–6), there exist some δ > 0, T > 0 such that

∥g(t, ~y , c)− g(t, y , c)∥ < ϵ1 ∥~y − y∥ (C–28)

for ∥y∥ < δ, ∥~y∥ < δ, t ≥ T and c ∈ � ⊆ �. Then by the proof of the existence of the

stable manifold (Section C.5), for each c ∈ �, there exists a unique

θ(t, a, c) ∈ Cb

([T ,∞)× B δ

2K× {c}

),

where Cb is defined in (C–9), that solves (C–8). Note that

∥θ(t, a, c)∥ ≤ δe−(α+σ)(t−t0) (C–29)

for all (t, a, c) ∈ [T ,∞) × B δ2K

× �, t ≥ t0 ≥ T by (C–22), where the estimate (C–29)

holds for all c ∈ � since Lemma 24 holds for all c ∈ � and since ϵ1 is chosen (and δ and

T are found) uniformly for all c ∈ �.

We now show that for any t0 ≥ T , θ(t, a, c) varies continuously with respect to c for

(t, a, c) ∈ [t0,∞) × B δ2K

× �. Let ϵ > 0 and fix some t0 ≥ T , t ≥ t0, a ∈ B δ2K

. Recall that

c1 is fixed. We show that there exists some γ > 0 such that

∥θ(t, a, c1)− θ(t, a, c2)∥ < ϵ

for all t ∈ [t0,∞) and a ∈ B δ2K

whenever |c1 − c2| < γ.

Let c2 ∈ �. By (C–29), there exists some T ≥ t0 such that

∥θ(t, a, c1)− θ(t, a, c2)∥ ≤ 2δe−(α+σ)(t−t0) < ϵ

142

for all t ≥ T . If T = t0, our proof is complete. Otherwise, let t ∈ [t0, T ]. By (C–8),

∥θ(t, a, c1)− θ(t, a, c2)∥ ≤∥U1(t − t0, c1)− U1(t − t0, c2)∥ ∥a∥ (C–30)

+

∫ t

t0

∥U1(t − s, c1)− U1(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds

(C–31)

+

∫ t

t0

∥U1(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)− g(s, θ(s, a, c2), c1)∥ ds

(C–32)

+

∫ t

t0

∥U1(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds

(C–33)

+

∫ ∞

t

∥U2(t − s, c1)− U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds

(C–34)

+

∫ ∞

t

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)− g(s, θ(s, a, c2), c1)∥ ds

(C–35)

+

∫ ∞

t

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds

(C–36)

First consider (C–30). Note that ∥a∥ ≤ δ2K

. Since U1(t, c) is continuous with respect

to (t, c) for (t, c) ∈ R × �, U1(t, c) is uniformly continuous with respect to (t, c) for

(t, c) ∈ [T , T ]× �. Then there exists some γ1 > 0 such that

∥U1(t − t0, c1)− U1(t − t0, c2)∥ < ϵK

5δ

for each t ∈ [T , T ] whenever |c1 − c2| < γ1. Then

∥U1(t − t0, c1)− U1(t − t0, c2)∥ ∥a∥ <ϵ

10(C–37)

whenever |c1 − c2| < γ1.

143

Consider (C–31). Since θ ∈ Cb, ∥θ(t, a, c)∥ < δ for all (t, a, c) ∈ [T ,∞)× B δ2K

× � by

(C–29). Recall that g(t, 0, c) = 0 for all t ≥ 0 and c ∈ �. Then by (C–28),

∥g(t, θ(s, a, c), ~c)∥ < ϵ1δ (C–38)

for all t, s ≥ T , a ∈ B δ2K

, c , ~c ∈ �.

Since U1(t−s, c) is uniformly continuous with respect to (s, c) for (s, c) ∈ [T , T ]×�,

there exists some γ2 > 0 such that

∥U1(t − s, c1)− U1(t − s, c2)∥ <ϵ

10T ϵ1δ

whenever |c1 − c2| < γ2. Then∫ t

t0

∥U1(t − s, c1)− U1(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds <ϵ

10(C–39)

whenever |c1 − c2| < γ2.

Consider (C–33). Since g(s, x , c) is uniformly continuous for (s, x , c) ∈ [T , T ] ×

[−δ, δ]× � and ∥θ∥ < δ, there exists some γ3 > 0 such that

∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ <ϵ(α+ σ)

10K

whenever |c1 − c2| < γ3. Then by the above and by Lemma 24,∫ t

t0

∥U1(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds <ϵ

10(C–40)

whenever |c1 − c2| < γ3.

Consider (C–34). By (C–38) and Lemma 24, the integral (C–34) is finite. Since the

integrand is nonnegative, there exists some T1 ≥ t0 such that∫ ∞

T1

∥U2(t − s, c1)− U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds <ϵ

20.

Note that since g is bounded uniformly and since Lemma 24 gives a uniform bound for

all c ∈ �, T1 is independent of c1, c2 ∈ �. If t ≥ T1, this bound is enough. Otherwise,

144

consider ∫ T1

t

∥U2(t − s, c1)− U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds.

As we showed for equation (C–31), there exists some γ4 > 0 such that∫ T1

t

∥U2(t − s, c1)− U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds <ϵ

20

whenever |c1 − c2| < γ4. Then∫ ∞

t

∥U2(t − s, c1)− U2(t − s, c2)∥ ∥g(s, θ(s, a, c1), c1)∥ ds <ϵ

10(C–41)

whenever |c1 − c2| < γ4.

Consider (C–36). By (C–38) and Lemma 24, the integral (C–36) is finite. Since the

integrand is nonnegative, there exists some T2 ≥ t0 such that∫ ∞

T2

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds <ϵ

20.

Similarly as for T1, T2 is independent of c1, c2 ∈ �. If t ≥ T2, this bound is enough.

Otherwise, consider∫ T2

t

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds.

As we showed for equation (C–33), there exists some γ5 > 0 such that∫ T2

t

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds <ϵ

20

whenever |c1 − c2| < γ5. Then∫ ∞

t

∥U2(t − s, c2)∥ ∥g(s, θ(s, a, c2), c1)− g(s, θ(s, a, c2), c2)∥ ds <ϵ

10(C–42)

whenever |c1 − c2| < γ5.

145

Combining the estimates (C–37), (C–39), (C–40), (C–41) and (C–42) and by

applying (C–28) to (C–32) and (C–35), we have that

∥θ(t, a, c1)− θ(t, a, c2)∥ <ϵ

2+ ϵ1

∫ t

t0

∥U1(t − s, c2)∥ ∥θ(s, a, c1)− θ(s, a, c2)∥ ds

+ ϵ1

∫ ∞

t

∥U2(t − s, c2)∥ ∥θ(s, a, c1)− θ(s, a, c2)∥ ds (C–43)

whenever |c1 − c2| < min{γ1, γ2, γ3, γ4, γ5}.

Let

H := supt∈[T ,∞),a∈B δ

2K

,c2∈�∥θ(t, a, c1)− θ(t, a, c2)∥

Then H ≤ 2δ is finite. By (C–43),

H ≤ ϵ

2+ ϵ1

∫ t

t0

∥U1(t − s, c2)∥Hds + ϵ1

∫ ∞

t

∥U2(t − s, c2)∥Hds

Then by Lemma 24,

H ≤ ϵ

2+ Hϵ1

(K

α+ σ+K

σ

)<ϵ

2+ H

2ϵ1K

σ

<ϵ

2+ H

1

2

by (C–27). Then

H < 2ϵ

2= ϵ

and

∥θ(t, a, c1)− θ(t, a, c2)∥ < ϵ

for all t ∈ [T ,∞), a ∈ B δ2K, c2 ∈ � whenever |c1 − c2| < γ := min{γ1, γ2, γ3, γ4, γ5}.

146

C.7 The Stable Manifold is Continuous with Respect to c

Since P(c) is continuous with respect to c and invertible, and since θ(t, a, c) is

continuous with respect to c , the composition P−1(c)θ(t, a, c) is continuous with respect

to c . Since this defines the stable manifold, we have shown that the stable manifold is

continuous with respect to c . That is, given c1, ϵ > 0, and a ∈ B δ2K

, there exists some

γ > 0, T > 0 such that

supt∈[T ,∞)

∥∥P−1(c1)θ(t, a, c1)− P−1(c2)θ(t, a, c2)∥∥ < ϵ (C–44)

whenever |c1 − c2| < γ.

Remark 3. A similar statement holds for the unstable manifold. The proof follows by

making the substitution t = −t in (C–1).

Remark 4. Note that if (C–1) is an autonomous system, then we may choose T

arbitrarily in (C–44).

147

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BIOGRAPHICAL SKETCH

Jessica graduated from the University of Florida in 2008 with a B.S. in mathematics

with minors in physics and statistics. Jessica continued studying mathematics, in

particular mathematical biology, at the University of Florida and received her Ph.D. in

mathematics in August 2013.

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