Mathematical methods in biology Master of Mathematics ...perthame/MathMethodBiol.pdfMathematical methods in biology Master of Mathematics, second year Beno^ t Perthame Sorbonne Universit

Mathematical methods in biology

Master of Mathematics, second year

Benôıt PerthameSorbonne Université,

UMR 7598, Laboratoire J.-L. Lions, BC187

4, place Jussieu, F-75252 Paris cedex 5

October 26, 2018

Contents

1 Lotka-Volterra equations 1

1.1 Movement and growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Nonnegativity principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Operator monotonicity for competition or cooperative systems . . . . . . . . . . . . . . . 3

1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Reaction kinetics and entropy 7

2.1 Reaction rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The law of mass action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Elementary examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Small numbers of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Slow-fast dynamics and enzymatic reactions 13

3.1 Slow-fast dynamics (monostable) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Slow-fast dynamics (general) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Enzymatic reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Belousov-Zhabotinskii reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Some material for parabolic equations 23

4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 The spectral decomposition of Laplace operators (Dirichlet) . . . . . . . . . . . . . . . . . 24

4.3 The spectral decomposition of Laplace operators (Neumann) . . . . . . . . . . . . . . . . 26

4.4 The Poincaré and Poincaré-Wirtinger inequalities . . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Rectangles: explicit solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Brownian motion and the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Relaxation, perturbation and entropy methods 31

5.1 Asymptotic stability by perturbation methods (Dirichlet) . . . . . . . . . . . . . . . . . . 31

5.2 Asymptotic stability by perturbation methods (Neuman) . . . . . . . . . . . . . . . . . . 33

5.3 Entropy and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Entropy: chemostat and SI system of epidemiology . . . . . . . . . . . . . . . . . . . . . . 36

5.5 The Lotka-Volterra prey-predator system with diffusion (Problem) . . . . . . . . . . . . . 37

3

CONTENTS 1

5.6 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Blow-up and extinction of solutions 41

6.1 Semilinear equations; the method of the eigenfunction . . . . . . . . . . . . . . . . . . . . 42

6.2 Semilinear equations; the energy method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.3 Keller-Segel system; the method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.4 Non-extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Linear instability, Turing instability and pattern formation 51

7.1 Turing instability in linear reaction-diffusion systems . . . . . . . . . . . . . . . . . . . . . 52

7.2 Spots on the body and stripes on the tail . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.3 The simplest nonlinear example: the non-local Fisher/KPP equation . . . . . . . . . . . . 56

7.4 Phase transition: what is NOT Turing instability . . . . . . . . . . . . . . . . . . . . . . . 58

7.5 Gallery of parabolic systems giving Turing patterns . . . . . . . . . . . . . . . . . . . . . . 59

7.5.1 A cell polarity system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.5.2 The CIMA reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.5.3 The diffusive Fisher/KPP system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.5.4 The Brusselator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.5.5 The Gray-Scott system (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.5.6 Schnakenberg system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.5.7 The FitzHugh-Nagumo system with diffusion . . . . . . . . . . . . . . . . . . . . . 68

7.5.8 The Gierer-Meinhardt system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.6 Models from ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.6.1 Competing species and Turing instability . . . . . . . . . . . . . . . . . . . . . . . 69

7.6.2 Prey-predator system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.6.3 Prey-predator system with Turing instability (problem) . . . . . . . . . . . . . . . 70

7.7 Keller-Segel with growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8 Transport and advection equations 73

8.1 Transport equation; method of caracteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.2 Advection and volumes transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.3 Advection (Dirac masses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.4 Examples and related equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.4.1 Long time behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.4.2 Nonlinear advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.4.3 The renewal equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.4.4 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.4.5 Other stochastic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2 CONTENTS

Chapter 1

Lotka-Volterra equations

As first examples, we present two general classes of equations that are used in biology: Lotka-Volterra

systems and chemical reactions. These are reaction-diffusion equations, or in a mathematical classifica-

tion, semilinear equations. Our goal is to explain what mathematical properties follow from the set-up

of the model: nonnegativity properties, monotonicity and entropy inequalities. These are very general

examples, for more material and more biologically oriented textbooks, the reader can also consult for

instance [33, 32, 38, 22, 9, 1].

1.1 Movement and growth

Class I are used in the area of population biology, and ecological interactions, and are characterized by

birth and death. For 1 ≤ i ≤ I, we denote by ni(t, x) the population densities of I interacting speciesat the location x ∈ Rd (d = 2 for example). We assume that these species move randomly accordingto Brownian motion with a bias according to a velocity Ui(t, x) ∈ Rd and that they have growth ratesRi(t, x) (meaning birth minus death). Then, the system describing the dynamic of these population

densities is

∂

∂tni − Di∆ni︸︷︷︸

random motion

+ div(Uini)︸︷︷︸oriented drift

= ni Ri ,︸︷︷︸growth and death

t ≥ 0, x ∈ Rd, i = 1, ..., I. (1.1)

In the simplest cases, the diffusion coefficients Di > 0 are constants depending on the species (they

represent active motion of individuals) and the bulk velocity Ui vanishes. Here we insist on the birth and

death rates Ri; they depend strongly on the interactions between species and we express this fact as a

nonlinearity

Ri(t, x) = Ri(n1(t, x), n2(t, x), ..., nI(t, x)

). (1.2)

A standard family of such nonlinearities results in quadratic interactions and is written

Ri(n1, n2, ..., nI) = ri +I∑i=1

cijnj ,

2 Relaxation and the energy method

with ri the intrinsic growth rate of the species i (it can be positive or negative) and cij the interaction

effect of species j on species i. The coefficients cij are usually neither symmetric nor nonnegative. One

can distinguish

• cij < 0, cji > 0: species i is a prey for j, species j is a predator for i. When several species eat no otherfood, these are called trophic chains/interactions,

• cij > 0, cji > 0: a mutualistic interaction (both species help the other and benefit from it) to bedistinguished from symbiosis (association of two species whose survival depends on each other),

• cij < 0, cji < 0: a direct competition (both species compete for example for the same food),• cii < 0 is usually assumed to represent intra-specific competition.

The quadratic aspect is related to the necessary binary encounters for the interaction to occur. Better

models include saturation effects: for instance the effect of too numerous predators is to decrease their

individual efficiency. This leads ecologists rather to use (with cii = −cii > 0)

Ri(n1, n2, ..., nI) = ri − ciini +I∑

j=1, j 6=i

cijnj

1 + nj,

the terminology Holing II is used in the case where j refers to a prey, i to a predator. Ecological networks

are described by such systems: diversity refers to the number of species I itself, connectance refers to the

proportion of interacting species, i.e. such that cij 6= 0. Connectance is low for trophic chain and higherfor a food web (species use several foods).

The original prey-predator system of Lotka and Volterra has two species, I = 2 (i = 1 the prey, i = 2

the predator). The prey (small fishe) can feed on abundant zooplankton and thus r1 > 0, while predators

(sharks) will die out without small fishe to eat (r2 < 0). The sharks eat small fishs in proportion to the

number of sharks (c12 < 0), while the shark population grows proportionally to the small fishe they can

consume c21 > 0). Therefore, we find the rule

r1 > 0, r2 < 0, c11 = c22 = 0, c12 < 0, c21 > 0.

1.2 Nonnegativity principle

In all generality, solutions of the Lotka-Volterra system (1.1) satisfy very few qualitative properties; there

are neither conservation laws (because they contain birth and death), nor entropy properties, a concept

which does not seem to be relevant for ecological systems. As we shall see the quadratic aspect may lead

to blow-up (solutions exist only for a finite time, see Chapter 6). Let us only mention here, that the

model is consistent with the property that population density be nonnegative.

Lemma 1.1 (Nonnegativity principle) Assume that the initial data n0i are nonnegative functions of

L2(Rd), that Ui ≡ 0 and that there is a locally bounded function Γ(t) such that |Ri(t, x)| ≤ Γ(t). Then,the weak solutions in C

(R+;L2(Rd)

)to the Lotka-Volterra system (1.1) satisfy ni(t, x) ≥ 0.

The definition and usual properties of weak solutions are given in Chapter 4, but we do not require

1.3. OPERATOR MONOTONICITY FOR COMPETITION OR COOPERATIVE SYSTEMS 3

that theoretical background to show the formal manipulations leading to this result.

Proof. (Formal) Here, we use the method of Stampacchia. Set pi = −ni, then we have

∂

∂tpi −Di∆pi = pi Ri.

Multiply by (pi)+ := max(0, pi) and integrate by parts. We find

1

2

d

dt

∫Rd

(pi(t, x)

)2+dx+Di

∫Rd

∣∣∇(pi)+∣∣2 = ∫Rd

(pi(t, x)

)2+Ri (1.3)

and thus1

2

d

dt

∫Rd

(pi(t, x)

)2+dx ≤ Γ(t)

∫Rd

(pi(t, x)

)2+.

Therefore, we have ∫Rd

(pi(t, x)

)2+dx ≤ e2

∫ t0

Γ(s)ds

∫Rd

(p0i (x)

)2+dx.

But, the assumption n0i ≥ 0 implies that∫Rd(p0i (x)

)2+dx = 0, and thus, for all times we have

∫Rd(pi(t)

)2+dx =

0, which means ni(t, x) ≥ 0.

(Rigorous) The difficulty here stems from the ‘linear’ definition of weak solutions of (1.1), which does not

allow for nonlinear manipulations such as multiplication by (pi)+ because this is not a smmoth function.

Section ?? explains how to resolve this difficulty.

Exercise. Perform the same formal proof with the drift terms Ui. Which assumption is needed on Ui?

Exercise. In the formal context of the Lemma 1.1, show that

d

dt

∫Rdni(t, x)dx =

∫RdRi(t, x)ni(t, x)dx,

∫Rdni(t, x)

2dx ≤∫Rd

(n0i (x)2dx e2

∫ t0

Γ(s)ds .

1.3 Operator monotonicity for competition or cooperative sys-

tems

The positivity property is general but a rather weak property. Comparison principle and monotonicity

principle are much stronger properties but require either competition or cooperative systems.

For simplicity we consider only two cooperative species, I = 2 in equation (1.1)–(1.2), and we make

the cooperative assumption

∂

∂n2R1(n1, n2) ≥ 0,

∂

∂n1R2(n1, n2) ≥ 0. (1.4)


Lemma 1.2 (Monotonicity principle for cooperative systems) Consider a cooperative system, that

is, (1.4) and two solutions (n1, n2), (m1,m2) such that for some constants 0 ≤ ni ≤ Γ1, 0 ≤ mi ≤ Γ1

and |Ri| ≤ Γ2, | ∂∂niRj | ≤ Γ3. Then, the Lotka-Volterra system (1.1)–(1.2) is ordered for all times

n0i ≤ m0i , i = 1, 2 =⇒ ni(t) ≤ mi(t), ∀t ≥ 0, i = 1, 2.

Proof. Subtracting the two equations for n1 and m1, we find successively, by adding and subtracting

convenient terms,

∂

∂t(n1 −m1)−D1∆(n1 −m1) = (n1 −m1)R1(n1, n2) +m1

(R1(n1, n2)−R1(m1,m2)

),

∂

∂t

(n1 −m1)2+2

− D1(n1 −m1)+∆(n1 −m1) = (n1 −m1)2+R1(n1, n2)

+m1(n1 −m1)+(R1(n1, n2)−R1(m1, n2)

)+m1(n1 −m1)+

(R1(m1, n2)−R1(m1,m2)

).

The first and second lines are controlled by bounds on R

d

dt

∫Rd

(n1 −m1)2+2

+D1

∫Rd|∇(n1 −m1)+|2 ≤ (Γ2 + Γ3)

∫Rd

(n1 −m1)2+

+

∫Rdm1(n1 −m1)+

(R1(m1, n2)−R1(m1,m2)

),

We now use 0 ≤ m1 ≤ γ1, ∂∂n2R1(n1, n2) ≥ 0 and a Lipschitz constant of R1(n1, n2), to conclude

d

dt

∫Rd

(n1 −m1)2+2

≤ (Γ2 + Γ3)∫Rd

(n1 −m1)2+ + Γ1Γ3∫Rd

(n1 −m1)+(n2 −m2)+.

The same argument on n2 −m2 leads to the similar inequality

d

dt

∫Rd

(n2 −m2)2+2

≤ (Γ2 + Γ3)∫Rd

(n2 −m2)2+ + Γ1Γ3∫Rd

(n1 −m1)+(n2 −m2)+,

and adding these two inequalities, for u(t) =∫Rd [(n1−m1)

2+ + (n2−m2)2+] and 2ab ≤ a2 + b2, we obtain

for some Γ the inequalitydu(t)

dt≤ Γu(t).

Since u(0) = 0 from our initial order assumption, we conclude that u(t) = 0 for all t ≥ 0. That is theconclusion of the lemma.

Exercise. Show that the result of Lemma 1.2 holds true for the system

∂

∂tni − Di∆ni = Ri

(n1(t, x), n2(t, x), ..., nI(t, x)

).

1.4. CHALLENGES 5

Exercise. For a cooperative system, assume that we are given an initial data such that (n01, n02) is a

smooth subsolution of the steady state equation. Write the equation for ∂ni(t)∂t and prove that

∂ni(t)

∂t≥ 0.

Exercise. Write the proof that order is preserved for a general k × k cooperative system.

Exercise. In the case of 2 by 2 competitive systems, that means,

∂

∂n2R1(n1, n2) ≤ 0,

∂

∂n1R2(n1, n2) ≤ 0. (1.5)

Show that the order 0 ≤ n1 ≤ m1, 0 ≤ m2 ≤ n2 is preserved. There is no such general statement forlarger systems.

1.4 Challenges

Variability. There is always a large variability in living populations. For that reason, solutions of mod-

els with fixed parameters, for movement or growth, usually fit poorly with experiments or observations.

However, the models are useful for explaining qualitatively these observations but not for predicting in-

dividual behavior, a major problem when dealing with medical applications for example. A usual point

of view is that these parameters are distributed; one can use a statistical representation (see the software

Monolix at http://software.monolix.org/) in pharmacology. One can also use so-called structured popula-

tion models, see [32, 34, 38]. When modeling Darwinian evolution (see adaptive evolution in Section ??),

the parameters are part of the model solution and mutations are seen as changes in the model coefficients

selected by adaptation to the environment.

Small numbers. In physics and chemistry, 1023 is the normal order of magnitude for a number of

molecules. Populations that are studied with Lotka-Volterra equations are much smaller and 106 is al-

ready a large number. This means that exponentially decaying tails are very quickly meaningless because

they represent a number of individuals, less than one. The interpretation of such tails should be ques-

tioned carefully and several types of corrections can be included; demographic stochasticity is used in

Monte-Carlo simulations, which correspond to the survival threshold in PDEs1.

1Gauduchon, M. and Perthame, B. Survival thresholds and mortality rates in adaptive dynamic: conciliating determin-

istic and stochastic simulations. Mathematical Medicine and Biology 27 (2010), no. 3, 195–210

Chapter 2

Reaction kinetics and entropy

When large numbers of molecules are involved in a chemical reaction, the kinetics are well described

by reaction rate equations. These are nonlinear equations describing the population densities (concen-

trations) of the reacting molecules and the specific form of these nonlinearities is usually prescribed by

the law of mass action. This section deals with this aspect of reaction kinetics. The derivation of these

equations is postponed to Section ?? where we introduce the chemical master equation, which better

describes a small number of reacting molecules, an important topic in cell biology.

2.1 Reaction rate equations

The general form of the equations leads to a particular structure for the right hand side of semi-linear

parabolic equations. They are written

∂

∂tni − Di∆ni︸︷︷︸

molecular diffusion

+ ni Li = Gi︸︷︷︸reaction

, t ≥ 0, x ∈ Rd, i = 1, 2, ..., I. (2.1)

The quantities ni ≥ 0 are molecular concentrations, the loss terms Li ≥ 0 depend on all the moleculesnj (with which the molecule i can react) and the gain terms Gi ≥ 0, which also depends on the nj ’s,denote the rates of production of ni from the other reacting molecules. The molecular diffusion rate of

these molecules is Di and can be computed according to the Einstein rule from the molecular size.

For a single reaction, the nonlinearities Li and Gi take the form

niLi =∑

reactions p

apikp

I∏j=1

napjj , Gi =

∑reactions p

bpi kp

I∏j=1

napjj . (2.2)

The powers apj , bpj ∈ N represent the number of molecules j necessary for the reaction p; this is the lawof mass action.

Nonnegativity property. We have factored the term ni in front of Li to ensure that this term van-

ishes at ni = 0. The loss term Li is not singular here because the product contains api ≥ 1 when

7

8 CHAPTER 2. REACTION KINETICS AND ENTROPY

kpi > 0 (the reactant i really reacts, otherwise the corresponding Li vanishes). For that reason, as for

the Lotka-Volterra systems, the nonnegativity property holds true.

Lemma 2.1 A weak solution of (2.1) with Gi ≥ 0 and nonnegative initial data satisfies ni(t, x) ≥ 0,∀i = 1, 2, ..., I.

Proof. Adapt the proof of Lemma 1.1.

Even though we do not know that Gi ≥ 0, this lemma is useful because one may argue as follows.Solutions are first built using the positive part of Gi, then the lemma tells us that the ni’s are positive.

From formula (2.2), the positivity of the n′js implies that Gi is positive and thus that we have built the

solution to the correct problem.

Conservation of atoms. The second main property of these models comes from the conservation

of atoms, which asserts that some quantities should be constant in time. For each atom k = 1, ...,K, one

defines the number Nki of atoms k in the molecule i. Then all reactions should preserve the number of

atoms and, for all k ∈ {1, ...,K}, the coefficients α, β, k and k′ should be such thatI∑i=1

Nki [ni Li −Gi] = 0, ∀(ni) ∈ RI . (2.3)

This implies that

∂

∂t

I∑i=1

Nkini = ∆

I∑i=1

NkiDini, (2.4)

d

dt

∫Rd

I∑i=1

Nkini(t, x)dx = 0, 1 ≤ k ≤ K,

and thus, a priori, the weighted L1 bound holds∫Rd

I∑i=1

Nkini(t, x)dx =

∫Rd

I∑i=1

Nkin0i (x)dx, 1 ≤ k ≤ K. (2.5)

Except when all the diffusion rates Di’s are equal (then its main part is the heat equation), it is not easy

to extract conclusions from equation (2.4) except the conservation law (2.5). Very few general tools, such

as M. Pierre’s duality estimate, are available, see the survey1.

There is a third property, the entropy dissipation that we shall discuss later.

2.2 The law of mass action

Irreversible reaction. To begin with, consider molecular species Si undergoing the single irreversible

reaction

a1S1 + ...+ aISIk+−→ b1S1 + ...+ bISI ,

1Pierre, M. Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78 (2) (2010),

417–455

2.2. THE LAW OF MASS ACTION 9

where ai and bi ∈ N, (ai)i=1,...,I 6= (bi)i=1,...,I , denote the number of molecules involved as reactants andproducts. Then, using the law of mass action, the equation for the population number densities ni of

species i is written

∂

∂tni −Di∆ni + aik+

I∏j=1

najj︸︷︷︸

loss of molecules by reaction

= bik+

I∏j=1

najj︸︷︷︸

gain of reaction product

, i = 1, 2, ..., I.

From the conservation of atoms, it is impossible that for all i we have bi ≥ ai (resp. ai ≤ bi). Therefore,at least one reactant (ai < bi) should disappear and one product (bi > ai) be produced.

Reversible reaction. More interesting is when several reactions occur. Then they are represented by

a sum of such terms over each chemical reaction. For example, still with ai 6= bi ∈ N, the reversiblereaction

a1S1 + ...+ aISIk+−⇀↽−k−

b1S1 + ...+ bISI ,

leads to reaction rate equations

∂∂tni −Di∆ni

loss of forward reacting molecules︷︸︸︷+ aik+

I∏j=1

najj +

loss of backward reacting molecules︷︸︸︷bik−

I∏j=1

nbjj

= bik+

I∏j=1

najj︸︷︷︸

gain of forward reaction product

+ aik−

I∏j=1

nbjj .︸︷︷︸

gain of backward reaction product

A more convenient form is

∂

∂tni −Di∆ni = (bi − ai)

k+ I∏j=1

najj − k−

I∏j=1

nbjj

, i = 1, 2, ..., I. (2.6)With this form, one can check the fundamental entropy property for reversible reactions. Define

S(t, x) :=

I∑i=1

ni[

ln(ni) + σi − 1], with

I∑i=1

σi(ai − bi) = ln k+ − ln k−. (2.7)

There are different possible choices for the constant σi, which can be more or less convenient depending

on the case. In all cases, one can check the

Lemma 2.2 (Entropy property for reversible reactions) The entropy dissipation equality holds

d

dt

∫S(t, x)dx = −

I∑i=1

Di

∫|∇ni|2

nidx−D(t, x) ≤ 0,

D(t, x) :=

∫ [ln(k+

I∏j=1

najj )− ln(k−

I∏j=1

nbjj )] [k+

I∏j=1

najj − k−

I∏j=1

nbjj

]dx ≥ 0.


The entropy dissipation property is very important because it dictates the long term behavior of the

reaction. Mathematically it is also useful to prove a priori estimates for the quantity |∇ni|2

nibut also for

integrability properties of powers of ni, which are necessary to define solutions of the reaction-diffusion

equation. This has been used recently in a series of papers by L. Desvillettes and K. Fellner2 for estimates

and relaxation properties, and by T. Goudon and A. Vasseur3 for regularity properties.

2.3 Elementary examples

Isomerization. The reversible isomerization reaction is the simplest chemical reaction. The atoms

within the molecule are not changed but only their spatial arrangement. The reaction is represented as

S1k+−⇀↽−k−

S2

and corresponds, in (2.6), to a1 = 1, b1 = 0, a2 = 0, b2 = 1, which meads to∂∂tn1 −D1∆n1 + k+n1 = k−n2,∂∂tn2 −D2∆n2 + k−n2 = k+n1.

(2.8)

The conserved quantity is simply

d

dt

∫[n1(t, x) + n2(t, x)]dx = 0,

∫[n1(t, x) + n2(t, x)]dx =

∫[n01(x) + n

02(x)]dx.

The formula (2.7) for the entropy (with a1 = b2 = 1, b1 = a2 = 0) gives

S(t, x) = n1[

ln(k+n1)− 1]

+ n2[

ln(k−n2)− 1], (2.9)

and one can check the entropy dissipation relation

ddt

∫Rd S(t, x)dx = −

∫Rd[D1|∇n1|2n1

+D2|∇n2|2n2

]dx

−∫Rd[

ln(k+ n2)− ln(k− n1)][k−n2 − k+n1

]dx ≤ 0.

Dioxygen dissociation. The standard degradation reaction of dioxygen to monoxygen is usually asso-

ciated with hyperbolic models for fluid flows rather than diffusion. This is because it is a very energetic

reaction occurring at very high temperature (for atmosphere re-entry vehicles for example) and with

reaction rates depending critically on this temperature, see [17]. But for our purpose here, we forget

these limitations and consider the dissociation rate k+ > 0 of n1 = [O2] in n2 = [O], and conversely its

recombination with rate k− > 0

O2k+−⇀↽−k−

O +O.

2L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a-priori bounds. Rev.

Mat. Iberoamericana, 24 (2) (2008), 407–431.3T. Goudon and A. Vasseur, regularity analysis for systems of reaction-diffusion equations, Annales de l ’ENS, 43 (1)

(2010), 117–141.

2.3. ELEMENTARY EXAMPLES 11

This leads to a1 = 1, b1 = 0, a2 = 0, b2 = 2, and thus we obtain∂∂tn1 −D1∆n1 + k+n1 = k−(n2)

2,

∂∂tn2 −D2∆n2 + 2k−(n2)

2 = 2k+n1,(2.10)

with initial data n01 ≥ 0, n02 ≥ 0. According to the law of mass action, the term (n2)2 arises because theencounter of two atoms of monoxygen is required for the reaction.

We derive the conservation law (number of atoms is constant) by a combination of the equations

∂

∂t[2n1 + n2]−∆[2D1n1 +D2n2] = 0,

which implies that for all t ≥ 0∫Rd

[2n1(t, x) + n2(t, x)

]dx = M :=

∫Rd

[2n01(x) + n

02(x)

]dx. (2.11)

For the simple case of the reaction (2.10), the formula (2.7) for the entropy (with a1 = 1, b2 = 2,

b1 = a2 = 0) gives

S(t, x) = n1[

ln(k+n1)− 1]

+ n2[

ln(k1/2− n2)− 1

]. (2.12)

One can readily check that

Lemma 2.3 (Entropy inequality)

ddt

∫Rd S(t, x)dx = −

∫Rd[D1|∇n1|2n1

+D2|∇n2|2n2

]dx

−∫Rd[

ln(k− (n2)2)− ln(k+ n1)

][k−(n2)

2 − k+n1]dx ≤ 0.

Exercise. Deduce from (2.11) and n1 ≥ 0, n2 ≥ 0 the a priori bound

2k−

∫ T0

∫Rd

(n2)2 dxdt ≤M(1 + 2k+T ).

Hint. In (2.10), integrate the equation for n1.

Hemoglobin oxidation. Hemoglobin Hb can bind to dioxygen O2 to form HbO2 according to the

reaction

Hb+O2k+−⇀↽−k−

HbO2.

This reaction is important in brain imaging because the magnetic properties of Hb and HbO2 are dif-

ferent; the consumption of oxygen by neurons generates desoxyhemoglobin that can be detected by MRI

(magnetic resonance imaging) and thus, indirectly, indicates the location of neural activity.

The resulting system of PDEs, for n1 = [Hb], n2 = [O2] and n3 = [HbO2], is∂∂tn1 −D1∆n1 + k+n1n2 = k−n3,∂∂tn2 −D2∆n2 + k+n1n2 = k−n3,∂∂tn3 −D3∆n3 + k−n3 = k+n1n2.


From this system, we can derive two conservation laws for the total number of molecules Hb and O2,

∂

∂t[n1 + n3]−∆[D1n1 +D3n3] = 0,

∂

∂t[n2 + n3]−∆[D2n2 +D3n3] = 0.

these imply an L1 control

M :=

∫[n1(t) + n2(t) + n3(t) + n4(t)] =

∫[n01 + n

02 + n

03 + n

04].

As in the case of dioxygene dissociation, integrating the third equation, one concludes the quadratic

estimate

k+

∫ ∞0

∫Rdn1n2dx dt ≤M(1 + k−T ).

From (2.7) (with a1 = 1, b2 = 2, b1 = a2 = 0), this also comes with an entropy

S(t, x) = n1[ln(k1/2+ n1 − 1] + n2[ln(k

1/2+ n2 − 1] + n3[ln(k−n3 − 1].

Mathematical references and studies on this system can be found in B. Andreianov and H. Labani4.

Exercise. Another simple and generic example is the reversible reaction A+Bk+−⇀↽−k−

C +D

∂∂tn1 −D1∆n1 + k+n1n2 = k−n3n4,∂∂tn2 −D2∆n2 + k+n1n2 = k−n3n4,∂∂tn3 −D3∆n3 + k−n3n4 = k+n1n2,∂∂tn4 −D4∆n4 + k−n3n4 = k+n1n2.

(2.13)

with the ni ≥ 0 and the single atom conservation law∫Rd

[n1(t, x) + n2(t, x) + n3(t, x) + n4(t, x)

]dx = M :=

∫Rd

[n01(x) + n

02(x) + n

03(x) + n

04(x)

]dx.

Choose ki so that S =

4∑i=1

ni ln(kini) is a convex entropy and write the entropy inequality for the chemical

reaction (2.13).

2.4 Small numbers of molecules

It may happen that in biological systems, e.g. intracellular bio-molecular interactions, the number of

molecules is not large so that the reaction rate equations are not applicable. The alternative is to use the

so-called chemical master equations that describe the dynamic of individual molecules.

These models are close to the jump processes described in Chapter ?? and with a brief presentation in

Section ??. They allow for a number of asymptotic limits (from discrete to continuous, from continuous

jumps to drift-diffusion). For these reasons, there is a large literature on this subject, on both the

theoretical and numerical aspects5,6,7.4B. Andreianov and H. Labani, Preconditioning operators and L∞ attractor for a class of reaction-diffusion systems.

Comm. Pure Appl. Anal. Vol. 11, No. 6 (2012) 2179–21995A. De Masi, E. Presutti, Mathematical methods for hydrodynamic limits. Springer-Verlag, Berlin, 19916D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25) 2340–2361 (1977)7J.-L. Doob, Markoff chains- Denumerable case. Trans. Amer. Math. Soc. 58(3), 455–473 (1945)

Chapter 3

Slow-fast dynamics and enzymatic

reactions

In many situations, one can encounter very different time scales for different reactions, and more generally

phenomena. When a time scale is very short compared to the other, one wishes to consider that the fast

phenomenon is at equilibrium. This chapter gives circumstances where this is legitimate. A typical

example is the case of enzymatic reactions, where a low level of enzymes regulates a reaction. Other

examples arise in the spiking dynamic of neurons.

Throughout this chapter we ignore diffusion and work with ordinary differential equations. For a

parameter ε > 0, we study the system of two equations with a fast and a slow time scale εduε(t)dt = f

(uε(t), vε(t)

), uε(t = 0) = u

0,

dvε(t)dt = g

(uε(t), vε(t)

), vε(t = 0) = v

0.(3.1)

Formally, the limiting system should be written as 0 = f(ū(t), v̄(t)

),

dv̄(t)dt = g

(ū(t), v̄(t)

), v̄(t = 0) = v0.

(3.2)

There are however several objections to such a general aymptotic result. Firstly, to pass to the limit in

nonlinear expressions as f(uε(t), vε(t)

), one needs compactness and for uε(t) the time derivative is not

under control. Secondly, the loss of an initial condition for ū(t) should be explained. Thirdly, the relation

0 = f(ū(t), v̄(t)

)does not necessarily define ū(t) as a function of v̄(t) so as to insert it in the differential

equation for v̄(t). This last observation leads us to distinguish two cases

• ∂∂uf(u, v) < 0, then we can hope to find a smooth mapping F (v) such that f(F (v), v) = 0. We call itthe ‘monostable’ case.

• Otherwise, some organizing principle should be found to decide which branch of solution of f(u, v) = 0is selected and we present some general results in section 3.2.

14 Slow-fast dynamics and enzymatic reactions

The sign of ∂∂uf(u, v) also appears when one studies the stability of a steady state. The linearized

matrix is 1ε ∂∂uf(u, v) 1ε ∂∂vf(u, v)∂∂u (gu, v)

∂∂v g(u, v)

= 1ε

∂∂uf(u, v) ∂∂vf(u, v)0 0

+O(1).Again stability is related to the property ∂∂uf(u, v) < 0.

3.1 Slow-fast dynamics (monostable)

We begin with the simplest situation where a fast scale can be considered at equilibrium, meaning that

we can invert the relation

0 = f(u, v)⇐⇒ u = F (v),

with F a smooth function from R to R. Then, we end up with the single equation

dv̄(t)

dt= g(F (v̄(t)), v̄(t)

), v̄(t = 0) = v0.

In particular the initial data for ū is lost, a phenomena called initial layer.

We are going to assume

f, g are Lipschitzian, (3.3)

∂f

∂u(u, v) ≤ −α < 0 for some constant α > 0. (3.4)

The second assumption implies that the equation on uε is monostable, as one can see it in the

Exercise. With the assumption (3.4), for the differential equation with v fixed

du(t)

dt= f

(u(t), v

), u(t = 0) = u0,

1. prove that |du(t)dt | ≤ |f(u0, v)|e−αt,

2. show that there is a limit u(t)→ U as t→∞.Hint. Write the equation on z(t) = du(t)dt .

Theorem 3.1 We assume (3.3)–(3.4) and that for some T > 0 there is a constant M such that |uε(t)|+|vε(t)| ≤M . Then,(i) vε(t) is Lipschitzian, uniformly in ε, for t ∈ [0, T ],(ii) uε(t) is Lipschitzian, uniformly in ε, for t ∈ [τ,∞), for all τ ∈]0, T ],(iii) there are ū(t), v̄(t) ∈ C([0, T ]) such that, as ε→ 0, uε(t)→ ū(t) uniformly in [τ, T ] and vε(t)→ v̄(t),uniformly in [0, T ] and (3.2) holds.

Proof. The statement (i) is obvious since g(uε(t), vε(t)) is bounded. For statement (ii), we define

zε :=duε(t)

dt.

Differentiating the equation on uε, we compute using the chain rule

dzε(t)

dt= zε

∂f

∂u(uε, vε) +

∂f

∂v(uε, vε)

dvε(t)

dt,

3.1. SLOW-FAST DYNAMICS (MONOSTABLE) 15

1

2

dzε(t)2

dt= z2ε

∂f

∂u(uε, vε) + zε

∂f

∂v(uε, vε) g(uε, vε).

Thanks to assumption (3.3) and (3.4), we find

dzε(t)2

dt≤ −2αz2ε + αz2ε +

1

α[∂f

∂v(uε, vε) g(uε, vε)]

2 ≤ −αz2ε +K

and thus

zε(t)2 ≤ z20e−αt/ε +

K

α=f(u0, v0)2

ε2e−αt/ε +

K

α.

This proves (ii).

For (iii) , we use the Ascoli-Arzela theorem and conclude that for a subsequence, we have uniform

convergence. Then we may pass to the limit in the integral form. The full family converges because of

uniqueness of solutions of (3.2).

Exercise. if the intital data is well-prepared, that means f(u0, v0) = 0, prove that uε(t) is uniformly

lipschitzian on [0,∞).

Exercise. Consider the system, with Lipschitz continous f and g, εduε(t)dt = f

(uε(t), vε(t)

), uε(t = 0) = u

0 ∈ (0, 1),dvε(t)dt = g

(uε(t), vε(t)

), vε(t = 0) = v

0,

assuming that f(0, v) = f(1, v) = 0, f(u, v) ≥ 0 for u ∈ (0, 1).1. Show that 0 < uε(t) < 1 for all times.

2. Show that∫∞

0|duε(t)dt |dt ≤ 1 and conclude that, after extracting a subsequence, uε(t)→ ū(t) as ε→ 0,

almost everywhere and in all Lp(0,∞), 1 ≤ p 0, there is a constant M(T ) such that

|vε(t)| ≤M(T ), 0 ≤ t ≤ T,

Proof. We can write,

εduε(t)

dt= f

(0, vε(t)

)+

∫ uε(t),0

∂f(σ, vε(t)

)∂u

dσ,

ε2d uε(t)

2

dt ≤ uε(t)f(0, vε(t)

)− αuε(t)2

≤ α2 uε(t)2 + C[1 + vε(t)

2]− αuε(t)2

because f(·, ·) is Lipschitz continuous. In the same way, we find

ε2d vε(t)

2

dt ≤ C|vε(t)|[1 + |uε(t)|+ |vε(t)|]

≤ α2 uε(t)2 + C[1 + vε(t)

2]


Adding up these two inequalities rises

1

2

d

dt[vε(t)

2 + ε uε(t)2] ≤ C[1 + vε(t)2] ≤ C[1 + vε(t)2 + ε uε(t)2]

and the Gronwall inequality gives us that vε(t)2 + ε uε(t)

2 is locally bounded.

To bound uε(t)2 itself, we may now write the inequation for uε(t)

2 as

ε

2

d uε(t)2

dt≤ −α

2uε(t)

2 + C[1 +M(T )]

which proves the result because max((u0)2. 2C[1+M(T )α

)is a solution with a larger initial data.

3.2 Slow-fast dynamics (general)

The assumption (3.4) is very strong and one can extend the result to more general systems but with

weaker conclusions. We keep the system εduε(t)dt = f

(uε(t), vε(t)

), uε(t = 0) = u

0,

dvε(t)dt = g

(uε(t), vε(t)

), vε(t = 0) = v

0,(3.5)

f, g are Lipschitz continuous and C1. (3.6)

Theorem 3.3 With the assumption (3.6) and assuming that uε(t), vε(t) are bounded on [0, T ], we have∫ T0

|f(uε(t), vε(t))|2dt ≤ C(T,K)ε.

If, for all v, f(u, v) does not vanish on an interval in u, then after extaction of a subsequence

uε(n) → ū(t) a.e. vε(n) → v̄(t) uniformly on [0, T ]

and the relations holdsdv̄(t)

dt= g(ū(t), v̄(t)

), f

(ū(t), v̄(t)

)= 0.

Proof. 1. We define

F (u, v) =

∫ u0

f(σ, v)dσ

and write ∫ T0

|f(uε(t), vε(t))|2dt = ε∫ T

0

f(uε(t), vε(t))duε(t)

dtdt.

SincedF (u(t), v(t))

dt= f(u(t), v(t))

du(t)

dt+∂F (u, v)

∂v

dv(t)

dt,

we conclude that∫ T0

|f(uε(t), vε(t))|2dt = ε∫ T

0

[dF (uε(t), vε(t))

dt− ∂F (uε, vε)

∂vg(uε, vε)

]dt

= ε[F (uε(T ), vε(T ))− F (u0, v0) + TK

].

3.2. SLOW-FAST DYNAMICS (GENERAL) 17

Because uε(t), vε(t) are locally uniformly bounded, we conclude.

2. We define

G(u, v) =

∫ u0

f(σ, v)2dσ.

We have

d

dtG(uε(t), vε(t)) = f(uε(t), vε(t))

2 duε(t)

dt+ 2g(uε(t), vε(t))

∫ uε(t)0

f(σ, vε(t)) ∂vf(σ, vε(t))dσ

=f(uε(t), vε(t))

2

εf(uε(t), vε(t)) + 2g(uε(t), vε(t))

∫ uε(t)0

f(σ, vε(t)) ∂vf(σ, vε(t))dσ

and thus G(uε(t), vε(t)) is locally BV. As a consequence for a subsequence, there is Ḡ(t) such that

G(uε(n)(t), vε(n)(t))→ Ḡ(t) a.e. as n→∞.

But because vε is uniformly Lipschitzian, by the Ascoli theorem, we also have vε(n)(t)→ v̄(t) and thus

G(uε(n)(t), v(t))→ Ḡ(t) a.e. as n→∞.

3. Because G(u, v) is strictly increasing in u from our assumption, the above convergence implies that

uε(n)(t) converges a.e. to some ū(t) ∈ L∞(0, T ). The end follows immediately.

Exercise. (Morris-Lecar system) Consider the system{εdgε(t)dt = G(vε(t))− gε(t),dvε(t)dt = g

(vε(t)

)+ gε(t)(V − vε(t)).

Identify the limiting system and show convergence.

Exercise. (Bistable case) Consider the case when, for a smooth function a : R→ (0, 1) we have

f(0, v) = f(1, v) = f(a(v), v) = 0,

f(u, v) =

< 0 for 0 < u < a(v),> 0 for a(v) < u < 1.1. Prove that

∫ T0|f(uε(t), vε(t))|dt ≤ C(T )ε.

2. Show that uε is uniformly with bounded total variation on [0, T ].

3. Show that, for a subsequence, as k →∞, vεk(t)→ v̄ (uniformly), and uεk(t)→ ū(t) (in Lp(0, T ), with1 ≤ p


2. Show that, for a subsequence, as k →∞, vεk(t)→ v̄ (uniformly), and uεk(t)→ ū(t) (in Lp(0, T ), with1 ≤ p

3.3. ENZYMATIC REACTIONS 19

of the complex is a reversible reaction, but the conversion to P +E is an irreversible reaction, leading to

the representation

S + Ek+−⇀↽−k−

C = SEkp−→ P + E.

Still using the law of mass action, this leads to the equations (we do not consider the space variable and

molecular diffusion, this is not the point here)ddtnS = k−nC − k+nSnE ,

ddtnE = (k− + kp)nC − k+nSnE ,

ddtnC = k+nSnE − (k− + kp)nC ,

ddtnP = kpnC .

This reaction comes with the initial data n0S > 0, n0E > 0 and n

0C = n

0P = 0. One can easily verify that

the substrate is entirely converted into the product

limt→∞

nP (t) = n0S , lim

t→∞nS(t) = 0, lim

t→∞nE(t) = n

0E , lim

t→∞nC(t) = 0.

A conclusion that is incorrect if n0E = 0. This is a fundamental observation in enzymatic reactions that

a very small amount of enzyme is sufficient to convert the substrate into the product.

Two conservation laws hold true in these equations and this helps us to understand the above limit,nE(t) + nC(t) = n

0E + n

0C = n

0E ,

nS(t) + nC(t) + nP (t) = n0S + n

0C + n

0P = n

0S .

(3.9)

Because the properties of the dynamic do not depend on nP , the system is equivalent to a 2×2 systemddtnS = k−nC − k+nS(n

0E − nC),

ddtnC = k+nS(n

0E − nC)− (k− + kp)nC .

(3.10)

Briggs and Haldane(1925) arrived to the conclusion that this system can be further simplified with the

quasi-static approximation on nC . That means that nC(t) can be computed algebraically from nS(t),

thus leading to a single ODE for nS(t)ddtnS = k−nC − k+nS(n

0E − nC),

0 = k+nS(n0E − nC)− (k− + kp)nC ,

(3.11)

still with the initial data nS(t = 0) = n0S (but no initial condition on nC). We can of course write this in

a more condensed way. Since nC =k+nSn

0E

k−+k++k+nS, and after adding the two equations, we find the typical

enzymatic reaction kinetic equation

d

dtnS = −n0E

kpk+nSk− + kp + k+nS

(3.12)

In other words, the law of mass action does not apply in this approximation and one finds the so-called

Michaelis-Menten law for the irreversible reactions, a Hill function rather than a polynomial.

Although not obvious at first glance, this result makes rigorous sense and we have the


Proposition 3.4 (Validity of Michaelis-Menten law) Assume n0E is sufficiently small. The solu-

tions of (3.10) and of (3.12) satisfy

supt≥0|nS(t)− nS(t)| ≤ C(n0S) n0E

for some constant C(n0S) independant of n0E.

We recall that nC(t = 0) = uεC(t = 0) = 0 and nS(t = 0) = u

εS(t = 0) > 0.

Proof. We reduce the system to slow-fast dynamic (see [13] for an extended presentation of the subject).

To simplify the notations, we set ε := n0E and we define the slow time and new unknowns as

ε := n0E , s = εt, uεS(s) = nS(t) ≥ 0, uεC(s) =

nC(t)

n0E≥ 0.

Then the two systems are respectively written asddsu

εS = k−u

εC − k+uεS(1− uεC), ε ddsu

εC = k+u

εS(1− uεC)− (k− + kp)uεC ,

ddsuS = k−uC − k+uS(1− uC), 0 = k+uS(1− uC)− (k− + kp)uC .

(3.13)

With these notations the result in Proposition 3.4 is equivalent to

sups≥0|uεS(s)− uS(s)| ≤ Cε.

The situation is the favorable case of Section 3.1 with the addition that the convergence is uniform

globally and not only locally.

The details of the proof are left to the reader because it is a general conclusion from the theory of

slow-fast dynamic. We shall make use of the two consequences of (3.13)

d

ds[uεS + εu

εC ] = −kpuεC ,

d

dsuS = −kpuC .

We have from (3.9) and uεC(t = 0) = 0,

(i) 0 ≤ uεC ≤ 1, 0 ≤ uC ≤ 1,(ii) 0 ≤ uεS ≤ n0S , 0 ≤ uS ≤ n0S ,(iii) 0 ≤ uεC ≤ uM < 1, 0 ≤ uC ≤ uM < 1 with uM :=

k+n0S

k+n0S+k−+kpindependent of ε,

(iv) | ddsuεC(s)| ≤ K for some K(n0S),

(v) From step (iv), introduce the bounded quantity rε(s) := − ddsuεC(s) + k+u

εC(s)

2, and

Rε(s) = uεS(s) + εuεC(s)− uS(s).

Compute that

uεC(s) =εrε(s)− k+uεCRε + k+uεS

k+uS + k− + kp.

uεC − uC =εrε + k+(1− uεC)Rε − εk+uεC

k+uS + k− + kp.

3.4. BELOUSOV-ZHABOTINSKII REACTION 21

Conclude thatd

dsRε(s) + kp

k+(1− uεC)k+uS + k− + kp

Rε(s) = εkp−rε + k+uεC

k+uS + k− + kp.

(vi) Using steps (iii) and (v), conclude that |Rε(s)| ≤ Cε for a constant C.Hints. (ii) uεS + u

εC decreases.

(iii) Write ddsuεC ≤ k+n0S(1− uεC)− (k− + kp)uεC .

(iv) Write the equation for z(s) = ddsuεC(s) as ε

ddsz+ λ(s)z(s) = µ(s) with λ > k−+ kp and µ bounded.

(vi) Write 12ddsR

ε(s)2 +ARε(s)2 ≤ εB|Rε(s)|, with A > 0, B > 0 two constants.

3.4 Belousov-Zhabotinskii reaction

Figure 3.2: Periodic regime of the system of ODEs (3.14) that mimics the Belousov-Zhabotinskii reaction.The parameters are kA = kB = 1 and kC = 2 (Left), kC = 20 (Right). The solution nB is the top (red) curve,

nA is in blue and nC in green. In abscissae is time.

For all complex chemical reactions, the detailed description of all elementary reactions is not realistic.

Then, as for the enzymatic reaction, one simplifies the system by assuming that some reactions are much

faster than others, or that some components are in higher concentrations than others. These manipu-

lations may violate mass conservation and entropy inequality may be lost; this is a condition to obtain

pattern formation as in the example of the CIMA reaction in Section 7.5.2.

The famous Belousov-Zhabotinskii reaction is known as the first historical example to produce peri-

odic patterns. This was discovered in 1951 by Belousov, it remained unpublished because no respectable

chemist at that time could accept this idea. Belousov and Zhabotinskii received the Lenin Prize in 1980,

a decade after Belousov’s death and the discovery in the USA of other periodic reactions, simpler to

reproduce.


We borrow a simple example from A. Turner2, this avoids entering the details of a real chemical reaction

(refer to [32] for a complete treatment). The simple example consists of three reactants denoted as A,

B, C, and three irreversible reactions (because of this feature, entropy dissipation and relaxation do not

hold).

A+BkA−→ 2A, B + C kB−→ 2B, C +A kC−→ 2C.

Therefore, following the general formalism in (2.6), the system is written asddtnA = nA(kAnB − kCnC),ddtnB = nB(kBnC − kAnA),ddtnC = nC(kCnA − kBnB).

(3.14)

Notice that

nA(t) > 0, nB(t) > 0, nC(t) > 0, nA(t) + nB(t) + nC(t) = n0A + n

0B + n

0C .

Numerical simulations are presented in Figure 3.2. For kc = 20, which is very large compared to kA

and kB one can estimate that the equations imply that kCnAnC = O(1) than thus nAnC ≈ 0. This isoberved on the simulation on the right.

2A. Turner, A Simple Model of the Belousov-Zhabotinsky Reaction from First Principles, 2009,

http://www.aac.bartlett.ucl.ac.uk/processing/samples/bzr.pdf

http://www.aac.bartlett.ucl.ac.uk/processing/samples/bzr.pdf

Chapter 4

Some material for parabolic

equations

4.1 Boundary conditions

When working in a domain Ω (a connected open set with sufficiently smooth boundary), we encounter

two elementary types of boundary conditions. The reaction-diffusion systems (1.1) or (2.1) are completed

either by the Dirichlet or the Neumann boundary conditions.

• Dirichlet boundary condition on ∂Ω.

ni = 0 on ∂Ω.

This boundary condition means that individuals or molecules, when they reach the boundary are defini-

tively removed from the population Ω, which therefore diminishes. This interpretation stems from the

Brownian motion underlying the diffusion equation (see below). But we can see that indeed, if we consider

the conservative chemical reactions (2.1) with (2.3), then

d

dt

∫Ω

I∑i=1

ni(t, x)dx =

I∑i=1

Di

∫∂Ω

∂

∂νni,

with ν the outward normal to the boundary. But with ni(t, x) ≥ 0 in Ω and ni(t, x) = 0 in ∂Ω, we have∂∂νni ≤ 0, therefore the total mass diminishes, for all t ≥ 0,

d

dt

∫Ω

I∑i=1

ni(t, x)dx ≤ 0,∫

Ω

I∑i=1

ni(t, x)dx ≤∫

Ω

I∑i=1

n0i (x)dx.

• Neumann boundary condition on ∂Ω.

∂

∂νni = 0 on ∂Ω,

still with ν the outward normal to the boundary. This means that individuals or molecules are reflected

when they hit the boundary. In the computation above for (2.1) with (2.3), the normal derivative vanishes

24 Some material for parabolic equations

and we find directly mass conservation

d

dt

∫Ω

I∑i=1

ni(t, x)dx = 0,

∫Ω

I∑i=1

ni(t, x)dx =

∫Ω

I∑i=1

n0i (x)dx.

• Mixed or Robin boundary condition on ∂Ω.

αni +∂

∂νni = 0 on ∂Ω, α ≥ 0.

When reaching the boundary, there is either reflexion or removal with a propability related to α (α = 0

no removal, α =∞ no reflexion). The mass balance reads

d

dt

∫Ω

I∑i=1

ni(t, x)dx =

I∑i=1

Di

∫∂Ω

∂

∂νni = −α

I∑i=1

Di

∫∂Ω

ni < 0.

• Full space. There is a large difference between the case of the full space Rd and the case of a boundeddomain. This can be seen by the results of spectral analysis in Section 4.2, which do not hold in the same

form in Rd. The countable spectral basis wk(x) is replaced by e−ix.ξ, and leads to the Fourier transform.

4.2 The spectral decomposition of Laplace operators (Dirichlet)

The spectral decomposition of the Laplace operator is going to play an important role in subsequent

chapters, therefore we introduce some material now. We begin with the Dirichlet boundary condition −∆u = f in Ω,u = 0 on ∂Ω. (4.1)Theorem 4.1 (Dirichlet) Consider a bounded connected open set Ω, then there is a spectral basis

(λk, wk)k≥1 for (4.1), that is,

(i) λk is a non-decreasing sequence with 0 < λ1 < λ2 ≤ λ3 ≤ ... ≤ λk ≤ ... and λk −→∞k→∞

,

(ii) (λk, wk) are eigenelements, i.e., for all k ≥ 1 we have −∆wk = λkwk in Ω,wk = 0 on ∂Ω,(iii) (wk)k≥1 is an orthonormal basis of L

2(Ω),

(iv) we have w1(x) > 0 in Ω and the first eigenvalue λ1 is simple, and for k ≥ 2, the eigenfunction wkchanges sign and can be multiple.

Remark 1. The hypothesis that Ω is connected is simply used to guarantee that the first eigenvalue

is simple and the corresponding eigenfunction is positive in Ω. Otherwise, we have several additional

nonnegative eigenfunctions with a first eigenfunction in one component and 0 in the others.

Remark 2. The sequence wk is also orthogonal in H10 (Ω) (for Dirichlet boundary conditions) and H

1(Ω)

4.2. THE SPECTRAL DECOMPOSITION OF LAPLACE OPERATORS (DIRICHLET) 25

for Neumann boundary conditions (see below). Indeed, if wk is orthogonal to wj in L2(Ω), then from the

Laplace equation for wk and the Stokes formula∫Ω

∇wj .∇wk = λk∫

Ω

wjwk = 0.

Therefore, orthogonality in L2 implies orthogonality in H10 or H1.

Remark 3. Note that for k ≥ 2, the eigenfunction wk changes sign because∫

Ωw1wk = 0 and w1 has a

sign.

Remark 4. Notice the formula∞∑i=1

wi(x)wi(x′) = δ(x− x′).

Indeed, this means that for all test function in ϕ ∈ C(Ω̄) ∩ L2(Ω),

∞∑i=1

wi(x)

∫Ω

wi(x′)ϕ(x′)dx′ =

∫Ω

ϕ(x′)δ(x− x′)dx′ = ϕ(x),

which is the decomposition of ϕ on the orthonormal basis (wi)i≥1

Proof of Theorem 4.1. We only sketch the proof. For additional matters see [15] Ch. 5, [6] p. 96. The

result is based on two ingredients. (i) The spectral decomposition of self-adjoint compact linear mappings

on Hilbert spaces is a general theory that extends the case of symmetric matrices. (ii) The simplicity

of the first eigenvalue with a positive eigenfunction is also a consequence of the Krein-Rutman theorem

(infinite dimension version of the Perron-Froebenius theorem).

First step. 1st eigenelements. In the Hilbert space H = L2(Ω), we consider the linear subspace V =

H10 (Ω). Then we define the minimum on V

λ1 = min∫Ω|u|2=1

∫Ω

|∇u|2dx. (4.2)

This minimum is achieved because a minimizing sequence (un) will converge strongly inH and weakly in V

to w1 ∈ V with∫

Ω|w1|2 = 1, by the Rellich compactness theorem (see [10, 6]). Therefore,

∫Ω|∇w1|2dx ≤

lim infn→∞∫

Ω|∇un|2dx = λ1. This implies equality and that λ1 > 0. The variational principle associated

to this minimization problem says that

−∆w1 = λ1w1,

which implies that w1 is smooth in Ω (by elliptic regularity)

Second step. Positivity. Because in V ,∣∣∇|u|∣∣2 = |∇u|2 a.e., the construction above tells us that |w1| is

also a first eigenfunction and we may assume that w1 is nonnegative. By the strong maximum principle

applied to the Laplace equation, we obtain that w1 is positive inside Ω (because it is connected). This

also proves that all the eigenfunctions associated with λ1 have a sign in Ω because, in a connected open

set, w1 cannot satisfy the three properties (i) be smooth, (ii) change sign and (iii) |w1| be positive also.


Third step. Simplicity. Finally, we can deduce the simplicity of this eigenfunction because if there were

two independent eigenfunction, a linear combination would allow us to build one which changes sign (by

orthogonality to w1 for example) and this is impossible by the above positivity argument.

Fourth step. Other eigenelements. We may iterate the construction. Denote Ek the finite dimensional

subspace generated by the k-th first eigenspaces. We work on the closed subspace E⊥k of H, and we may

define

λk+1 = minu∈E⊥k ∩V,

∫Ω|u|2=1

∫Ω

|∇u|2dx.

This minimum is achieved by the same reason as before. The variational form gives that the minimizers

are solutions of the k + 1-th eigenproblem. They can form a multidimensional space but always finite

dimensional; otherwise we would have an infinite dimensional subspace of L2(Ω), whose unit ball is

compact thanks to the Rellich compactness theorem, since∫

Ω|∇u|2dx ≤ λk+1 in this ball

Also λk → ∞ as k → ∞ because one can easily build (with oscillations or sharp gradients) functionssatisfying

∫Ω|un|2 = 1 and

∫Ω|∇un|2dx ≥ n.

4.3 The spectral decomposition of Laplace operators (Neumann)

The same method applies to the Neumann boundary condition −∆u = f in Ω,∂u∂ν = 0 on ∂Ω.

(4.3)

We state without proof the

Theorem 4.2 (Neumann) Consider a C1 bounded connected open set Ω, then there is a spectral basis

(λk, wk)k≥1 for (4.3), i.e.,

(i) λk is a non-decreasing sequence with 0 = λ1 < λ2 ≤ λ3 ≤ ... ≤ λk ≤ ... and λk −→∞k→∞

,

(ii) (λk, wk) are eigenelements, i.e., for all k ≥ 1 we have −∆wk = λkwk in Ω,∂wk∂ν = 0 on ∂Ω,

(iii) (wk)k≥1 is an orthonormal basis of L2(Ω)

(iv) w1(x) =1

|Ω|1/2 > 0, and for k ≥ 2, the eigenfunction wk changes sign.

4.4 The Poincaré and Poincaré-Wirtinger inequalities

As an outcome of the proof in Section 4.2, we recover the Poincaré inequality for ω a bounded domain

λD1

∫Ω

|u|2 ≤∫

Ω

|∇u|2dx, ∀u ∈ H10 (Ω), (4.4)

4.5. RECTANGLES: EXPLICIT SOLUTIONS 27

where λD1 denotes the first eigenvalue of the Laplace-Dirichlet equation (4.1). This inequality is just

another statement equivalent to (4.2).

The Poincaré-Wirtinger inequality states that

λN2

∫Ω

|u− 〈u〉|2 ≤∫

Ω

|∇u|2dx, ∀u ∈ H1(Ω), (4.5)

where λN2 is the first positive eigenvalue for the Laplace-Neumann equation (4.3) and

〈u〉 = 1Ω

∫Ω

u(x)dx,

is the L2 projection on the first eigenfunction with Neumann boundary conditions (a constant). Therefore

this statement is nothing but the characterization of the second eigenvalue given in the proof in Section 4.2.

4.5 Rectangles: explicit solutions

In one dimension, one can compute explicitly the spectral basis because the solutions of −u′′ = λu areall known. On Ω = (0, 1) we have

wk = ak sin(kπx), λk = (kπ)2, (Dirichlet)

wk = bk cos((k − 1)πx

), λk =

((k − 1)π

)2, (Neuman)

and ak and bk are normalization constants that ensure∫ 1

0|wk|2 = 1.

In two dimensions, on a rectangle (0, L1) × (0, L2) we see that the family is better described by twoindices k ≥ 1 and l ≥ 1 and we have

wkl = akl sin(kπx1L1

) sin(lπx2L2

), λkl =((k

L1)2 + (

l

L2)2)π2, (Dirichlet)

wkl = bkl cos((k − 1)π x

L1

)cos((l − 1)π x

L2

), λkl =

((k − 1L1

)2 + (l − 1L2

)2)π2, (Neuman)

These examples indicate that

• The first positive eigenvalue is of the order of 1/max(L1, L2)2 = 1diam(Ω)2 . For a large domain (evenwith a large aspect ratio) we can expect that the first eigenvalues are close to zero and that the eigenvalues

are close to each other.

• Except the first one, eigenvalues can be multiple (take L1/L2 ∈ N).• Large eigenvalues are associated with highly oscillating eigenfunctions.


Figure 4.1: Two sample paths of one dimensional Brownian motion according to the approximation (4.6).Abscissae tk, ordinates Xk.

Figure 4.2: Two sample paths of two dimensional Brownian motion simulated according to (4.6).

4.6 Brownian motion and the heat equation

The relationship between Brownian motion and the heat equation explains, in a simple framework, why

a random walk of individuals leads to the terms −D∆n for the population density n(t, x). It is ratherdifficult to construct rigorously Brownian motion. However, it is easy to give an intuitive approximation,

which is sufficient to build the probability law of Brownian trajectories.

To do so, we follow the Euler numerical method for an ODE with a time step ∆t that uses discrete

times tk = k∆t.

In a probability space with a measure denoted dP (ω), the initial position X0 ∈ Rd being given with aprobability law n0(x), we define iteratively a discrete trajectory Xk(ω) ∈ Rd as follows. We set

Xk+1(ω) = Xk(ω) +√

∆t Y k(ω) (4.6)

where Y k(ω) denotes a d-dimensional random variable independent of Xk (one speaks of independent

4.6. BROWNIAN MOTION AND THE HEAT EQUATION 29

increments) and with a normal law N(y). We recall that ‘independent’ means

Ef(Xk, Y k) :=∫f(Xk(ω), Y k(ω)

)dP (ω) =

∫Rd×Rd

f(x, y)dnk(x) N(y)dy (4.7)

with

• N(y) = 1(2π)d/2

e−|y|2/2, the normal law,

• dnk(x) the law of the process Xk, defined as P(Φ(Xk(ω))

)=∫

Φ(x)dnk(x).

Two simulations are presented. Figure 4.1 depicts, in the one dimensional case (tk, Xk), two iterates

Xk (for two different ω). Figure 4.2 shows two iterates Xk in two dimensions.

Our purpose is to compute the law dnk(x) at the limit ∆t → 0. To do so, we first use a C3 functionu, with u, Du, D2u and D3u bounded. We use the Taylor expansion of u to compute

u(Xk+1) = u(Xk) +√

∆tDu(Xk).Y k +∆t

2D2u(Xk).(Y k, Y k) +O

(∆t|Y k|

)3/2,

and then

Eu(Xk+1) = Eu(Xk) +∆t

2ED2u(Xk).(Y k, Y k) +O

(∆t)3/2

. (4.8)

Indeed, because N(·) is radially symmetric, the formula (4.7) used with f(x, y) = Du(Xk).Y k yields

EDu(Xk).Y k = EDu(Xk).∫RdyN(y)dy = 0.

Because∫Rd yiyjN(y)dy = δij , a further use of the formulas (4.8), (4.7) and the definition of the

probability law mentioned earlier, give∫Rdu(x)dnk+1(x) =

∫Rdu(x)dnk(x) +

∆t

2

∫Rd

∆u(x)dnk(x) +O(∆t)3/2

.

After dividing by ∆t and integration by parts of the term ∆u, we may rewrite this as∫Rdu[dnk+1 − dnk

∆t− 1

2∆dnk

]= O

(∆t)1/2

.

This holds true for any smooth function u and this means that, in the weak sense,

dnk+1 − dnk

∆t− 1

2∆dnk = O

(∆t)1/2

.

At the limit in the sense of distributions, as ∆t → 0, we obtain a probability law with a density n(t, x)that satisfies

∂∂tn(t, x)−

12∆n(t, x) = 0,

n(0, x) = n0(x).

In particular, even though n0 is a probability measure, it follows from the regularizing effects of the heat

equation that n(t, x) is a (smooth) function and not only a measure as soon as t > 0.


The equation for n(t, x) (the heat equation here) is generically called the Kolmogorov equation for the

limiting process (Brownian motion).

Exercise. Prove that the density of the probability law nk satisfies the integral equation

nk+1(x)− nk(x) =∫ [

nk(x+√

∆t y)− nk(x)]N(y)dy. (4.9)

Derive the heat equation for the limit n(t, x), as ∆t→ 0, if it exists. Show that this derivation uses onlytwo y-moments of N and not the full normal law.

See also the scattering equation in Chapter ??.

Chapter 5

Relaxation, perturbation and

entropy methods

Solutions of nonlinear parabolic equations and systems presented in Chapter 1 can exhibit various and

sometimes complex behavior, a phenomena usually called pattern formation. In which circumstances can

such complex behavior happen? A first answer is given in this chapter by indicating some conditions for

relaxation to trivial steady states; then nothing interesting can happen! We present relaxation results by

perturbation methods (small nonlinearity) and entropy methods. Indeed, before we can understand how

patterns occur in parabolic systems, a necessary step is to understand why patterns should not appear

in principle! Solutions of parabolic equations undergo regularization effects that lead them to constant

or simple states. Several asymptotic results can be stated in this direction and we present some of them

in this chapter. Sections 5.1 and 5.2 have been very much influenced by the book [21].

5.1 Asymptotic stability by perturbation methods (Dirichlet)

The simplest possible long term behavior for a semilinear parabolic equation is simply the relaxation

towards a stable steady state (that we choose to be 0 here). This is possible when the following two

features are combined

• the nonlinear part is a small perturbation of a main (linear differential) operator,• this main linear operator has a positive dominant eigenvalue.Of course this simple relaxation behavior is rather boring and appears as the opposite of pattern forma-

tion when Turing instability occurs, and this will be described later, see Chapter 7.

To illustrate this, we consider, in a bounded domain Ω, the semi-linear heat equation with a Dirichlet


boundary condition∂∂tui(t, x)−Di∆ui(t, x) = Fi(t, x;u1, ..., uI), 1 ≤ i ≤ I, x ∈ Ω,

ui(t, x) = 0, x ∈ ∂Ω,

ui(t = 0, x) = u0i (x) ∈ L2(Ω).

(5.1)

We assume that F (t, x; 0) = 0, so that u ≡ 0 is a steady state solution. Is it a stable and attractive state?Is the number of individuals leaving the domain through ∂Ω compensated by the birth term?

We shall use here a technical result. The Laplace operator (with Dirichlet boundary condition) has a

first eigenvalue λ1 > 0, associated with a positive eigenfunction, w1(x), which is unique except multipli-

cation by a constant,

−∆w1 = λ1 w1, w1 ∈ H10 (Ω). (5.2)

This eigenvalue is characterized as being the best constant in Poincaré inequality (see Section 4.2 or the

book [10])

λ1

∫Ω

|v(x)|2 ≤∫

Ω

|∇v|2, ∀v ∈ H10 (Ω),

with equality only for v = µw1, µ ∈ R.

Theorem 5.1 (Asymptotic stability) Assume miniDi = D > 0 and that there is a (small) constant

L > 0 such that ∀u ∈ RI , t ≥ 0, x ∈ Ω,

|F (t, x;u)| ≤ L|u|, or more generally, F (t, x;u) · u ≤ L |u|2 , (5.3)

δ := Dλ1 − L > 0, (5.4)

then, ui(t, x) vanishes exponentially as t→∞, namely,∫Ω

|u(t, x)|2 ≤ e−2δt∫

Ω

|u0(x)|2. (5.5)

Proof. We multiply the parabolic equation (5.1) by ui and integrate by parts

1

2

d

dt

∫Ω

ui(t)2 +Di

∫Ω

∣∣∇ui(t)∣∣2 = ∫Ω

ui(t)Fi(t, x;u(t)

),

and using the characterization (5.2) of λ1, we conclude

1

2

d

dt

∫Ω

I∑i=1

ui(t)2 +Dλ1

∫Ω

I∑i=1

ui(t)2 ≤ L

∫Ω

I∑i=1

ui(t)2.

The result follows from the Gronwall lemma.

Exercise. Consider a smooth bounded domain Ω ⊂ Rd, a real number λ > 0 and two smooth andLipschitz continuous functions R(u, v), Q(u, v), such that R(0, 0) = Q(0, 0) = 0. Let

(u(t, x), v(t, x)

)be

solutions of the system ∂∂tu−∆u = R(u(t, x), v(t, x)), t ≥ 0, x ∈ Ω,

u(t, x) = 0 sur ∂Ω,

∂∂tv + λv = Q(u(t, x), v(t, x)).

5.2. ASYMPTOTIC STABILITY BY PERTURBATION METHODS (NEUMAN) 33

1. Recall the Poincaré inequality for u.

2. Assume |R(u, v)| ≤ L(|u| + |v|) and |Q(u, v)| ≤ L(|u| + |v|), give a size condition for L such that forall initial data, the solution (u, v) converges exponentially to (0, 0) for t→∞.

Solution. A simple answer is min(λ1, µ) − 2L =: δ > 0. A more elaborate condition is based on thepositive real number such that λ1 − µ = a− a−1 and is λ1 − L− a−1 = µ− L− a =: δ > 0.

5.2 Asymptotic stability by perturbation methods (Neuman)

The next simplest possible long term behavior for a parabolic equation is relaxation towards an homoge-

neous (i.e., independent of x) solution, which is not constant in time. This is possible when two features

are combined

• the nonlinear part is a small perturbation of a main (differential) operator,• this main operator has a non-empty kernel (0 is the first eigenvalue).

Consider again, in a bounded domain Ω with outward unit normal ν, the semi-linear parabolic equation

with a Neumann boundary condition∂∂tui(t, x)−Di∆ui(t, x) = Fi(t;u1, ..., uI), 1 ≤ i ≤ I, x ∈ Ω,∂

∂ν(x)ui(t, x) = 0, x ∈ ∂Ω,

ui(t = 0, x) = u0i (x) ∈ L2(Ω).

(5.6)

The Laplace operator (with a Neumann boundary condition) has λ1 = 0 as a first eigenvalue, associated

with the constants w1(x) = 1/√|Ω| as eigenfunction. We shall use its second eigenvalue λ2, characterized

by the Poincaré-Wirtinger inequality (see [10] and Section 4.2)

λ2

∫Ω

|v(x)− 〈v〉|2 ≤∫

Ω

|∇v|2, ∀v ∈ H1(Ω), (5.7)

with the average of v(x) defined as

〈v〉 = 1|Ω|

∫Ω

v.

Note that this is also the L2 projection on the eigenspace spanned by w1.

Theorem 5.2 (Relaxation to a homogeneous solution) Assume miniDi = D > 0 and(F (u)− F (v)

)· (u− v) ≤ L |u− v|2 , ∀u, v ∈ RI , (5.8)

δ := Dλ2 − L > 0, (5.9)

then, ui(t, x) tends to become homogeneous with an exponential rate, namely,∫Ω

|u(t, x)− 〈u(t)〉|2 ≤ e−2δt∫

Ω

|u0(x)− 〈u0〉|2. (5.10)


Proof. Integrating in x equation (5.6), we find

d

dt〈ui〉 = 〈Fi(t;u)〉,

therefore,d

dt[ui − 〈ui〉]−Di∆[ui − 〈ui〉] = Fi(t;u)− 〈Fi(t;u)〉.

Thus, using assumption (5.8), we find

12ddt

∫Ω|u− 〈u〉|2 +D

∫Ω|∇(u− 〈u〉

)|2 =

∫Ω

(F (t;u)− 〈F (t;u)〉

)· (u− 〈u〉)

=∫

ΩF (t;u) · (u− 〈u〉)

=∫

Ω

(F (t;u)− F (t; 〈u〉)

)· (u− 〈u〉)

≤ L∫

Ω|u− 〈u〉|2.

Therefore, with notation (5.9)

d

dt

∫Ω

|u− 〈u〉|2 ≤ −2δ∫

Ω

|u− 〈u〉|2.

The result (5.10) follows directly.

Exercise. Explain why we cannot allow a dependency on x in the nonlinearity Fi(t;u).

Exercise. Let v ∈ H2(Ω) satisfy ∂v∂ν = 0 on ∂Ω (Neumanncondition).1. Prove, using the Poincaré-Wirtinger ineqality (5.7), that

λ2

∫Ω

|∇v|2 ≤∫

Ω

|∆v|2. (5.11)

2. In the context of Theorem5.2, assume that

I∑i,j=1

DjFi(t;u)ξiξj ≤ L|ξ|2, ∀ξ ∈ RI . Using the above

inequality, prove that ∫Ω

∑i

|∇ui(t, x)|2 ≤ e−2δt∫

Ω

∑i

|∇u0i (x)|2. (5.12)

3. Deduce a variant of Theorem 5.2.

Hints. 1. Integrate by parts the expression∫

Ω|∇v|2. 2. Use the equation for ddt

∂ui∂xl

.

5.3 Entropy and relaxation

We have seen in Lemma 2.3 that reaction kinetic equations as in (2.10) are endowed with an entropy

(2.12). This originates from the microscopic N -particle stochastic systems from which reaction kinetics

are derived (see Section ??).

5.3. ENTROPY AND RELAXATION 35

This entropy inequality is also very useful because it can be used to show relaxation to the steady

state, independently of the size of the constants k1, k2. To do that we consider a Neumann boundary

condition in a bounded domain Ω∂∂tn1 −D1∆n1 + k1n1 = k2(n2)

2, t ≥, x ∈ Ω,∂∂tn2 −D2∆n2 + 2k2(n2)

2 = 2k1n1,

∂∂νn1 =

∂∂νn2 = 0 on ∂Ω.

(5.13)

Theorem 5.3 The solutions of (5.13), with n0i ≥ 0, n0i ∈ L1(Ω), n0i ln(n0i ) ∈ L1(Ω), satisfy that

ni(t, x)→ Ni, as t→∞

with Ni the constants defined uniquely by

2N1 +N2 =

∫Ω

[2n01(x) + n02(x)]dx, k2(N2)

2 = k1N1.

Proof. Then S(t, x) = n1[

ln(k1n1)− 1]

+ n2[

ln(k1/22 n2)− 1

]satisfies, following Lemma 2.3,

ddt

∫ΩS(t, x)dx = −

∫Ω

[D1|∇n1|2n1

+D2|∇n2|2n2

]dx

−∫

Ω

[ln(k2 n

22)− ln(k1 n1)

][k2(n2)

2 − k1n1]dx.

And, because S is bounded from below, it also gives a bound on the entropy dissipation∫∞0

∫Ω

[D1|∇n1|2n1

+D2|∇n2|2n2

]dxdt

+∫∞

0

∫Ω

[ln(k2 n

22)− ln(k1 n1)

][k2(n2)

2 − k1n1]dxdt ≤ C(n01, n02).

(5.14)

This is again a better integrability estimate in x than the L1log estimate (derived from mass conservation)

because of the quadratic term (n2)2.

From a qualitative point of view, this says that the chemical reaction should lead the system to a

spatially homogeneous equilibrium state. Indeed, formally at least, the integral (5.14) can be bounded

only if

∇n1 = ∇n2 ≈ 0 as t→∞,

k2(n2)2 ≈ k1n1, ∇n1 = ∇n2 ≈ 0 as t→∞.

The first conclusion says that the dynamic becomes homogeneous in x (but may depend on t). The second

conclusion, combined with the mass conservation relation (2.11) shows that there is a unique possible

asymptotic homogeneous state because the constant state satisfies k2(N2)2 = k1N1 = k1(

M|Ω|−N2), which

has a unique positive root.

Exercise. Consider (2.10) with a Neumann boundary condition and set

S(t, x) =1

k1Σ1(k1n1(t, x)

)+

1

2k1/22

Σ2(k

1/22 n2(t, x)

).


1. We assume that Σ1, Σ2 satisfy Σ′1(u) = Σ

′2(u

1/2). Show that Σ1 is convex if, and only if, Σ2 is convex.

2. Under the conditions in question 1, show that the equation (2.10) dissipates entropy.

3. Adapt Stampacchia’s method to prove L∞ bounds on n1, n2 (and which is the appropriate quantity

for the maximum principle). What are the intuitive Lp bounds.

Exercise. Consider (2.10) with Dirichlet boundary conditions and n0i ≥ 0.1. Using the inequality

∂nj∂ν ≤ 0, which holds because nj ≥ 0, show that M(t) decreases where

M(t) =

∫Ω

[2n1(t, x) + n2(t, x)

]dx.

2. Consider the entropies of the previous exercise with the additional condition Σ′i(0) = 0. Show that

the equation (2.10) dissipates entropy.

(iii) Show that solutions of (2.10) with Dirichlet boundary conditions tend to 0 as t→∞.

5.4 Entropy: chemostat and SI system of epidemiology

Examples of entropy also appear in biological models. We treat here an example that arises as a first

modeling stage in two applications: 1. ecology and the model of the chemostat (the u represents a

nutrient, v a population that consumes the nutrient), 2. epidemiology with the celebrated Suceptible-

Infected model (SI system). In both cases the model isddtu = B − µuu− ruv,ddtv = ruv − µvv,

(5.15)

with B > 0, r > 0, µu > 0 and µv > 0 parameters.

In the chemostat B represents the renewal of nutrients u, µ the removal or degradation of nutrients

and r the consumption rate by the population and µv is the mortality and removal rate of the population.

In epidemiology, B represents the newborn, µu the mortality rate, r the encounter rate between sus-

ceptible and infected individuals (these encounters are responsible of new infected), µv is the mortality

of the infected (and recovery rate in the SIR model).

There is always a trivial (healthy) steady state

ū0 = B/µu, v̄0 = 0,

and a positive steady state

ū = µv/r, v̄ =rB − µuµv

rµvif rB > µuµv. (5.16)

Depending on the sign of v̄ we may have two different Lyapunov functionals (entropies)

Lemma 5.4 A ssume rB > µuµv and define

S̄(u, v) = −ū ln(u)− v̄ ln(v) + u+ v,

5.5. THE LOTKA-VOLTERRA PREY-PREDATOR SYSTEM WITH DIFFUSION (PROBLEM) 37

we haved

dtS̄ = − 1

u(√ūB − u

√v̄r + µu)

2. (5.17)

Lemma 5.5 We assume rB ≤ µuµv and define

S(u, v) = −ū0 ln(u) + u+ v,

we haved

dtS = − v

µu(µvµu − rB)−

1

µuu(B − µuu)2. (5.18)

We leave the proofs of these lemmas to the reader and go directly to the conclusion

Proposition 5.6 Solutions of the system (5.15) behave as follows

If Br > µuµv, then the entropy S̄ is convex and all solutions with v0 > 0 converge as t → ∞ to the

positive steady state.

If Br ≤ µuµv, then solutions become extinct (they converge to the trivial steady state as t → ∞).

The proof is standard and left to the reader. The steps are (i) u(t) is bounded, (ii) S(t) decreases,

in the case v̄ < 0, the limit can be −∞ and this means that v(t) vanishes and the result (ii) follows,otherwise S stays bounded and converges to a finite value, (iii) we conclude thanks to the right hand side

of (5.17).

5.5 The Lotka-Volterra prey-predator system with diffusion (Prob-

lem)

In the case of the Lotka-Volterra prey-predator system we can show relaxation towards a homogeneous

solution. The coefficients of the model need not be small as is required in Theorem 5.2. This is because

the model comes with a physically relevant quantity (such as entropy), which gives a global control.

Exercise. Consider the prey-predator Lotka-Volterra system without diffusion∂∂tn1 = n1[ r1 − an2],∂∂tn2 = n2[−r2 + bn1],

where r1, r2, a and b are positive constants and the initial data n0i are positive.

1. Show that there are local solutions and that they remain positive.

2. Show that the entropy (Lyapunov functional)

E(t) = −r1 lnn2 + an2 − r2 lnn1 + bn1,


is constant. Show that E is bounded from below and that E → ∞ as n1 + n2 → ∞. Conclude thatsolutions are global.

3. What is the unique steady state solution?

4. Show, using question 2, that the solutions are periodic (trajectories are closed).

Solution 1. By local Lipschitz regularity of the right hand side, the Cauchy-Lipschitz theorem asserts

that there is a local solution, i.e., defined on a maximal interval [0, T ∗]. We can write

n1(t) = n01e

∫ t0

[r1−an2(s)]ds > 0,

and, same thing for n2.

2. Set ϕi = lnni and write ∂∂tϕ1 = r1 − an2 = r1 − ae

ϕ2 ,

∂∂tϕ2 = −r2 − bn1 = −r2 + be

ϕ1 .

This is a Hamiltonian system and the Hamiltonian is constant along trajectories

H(t) = −r1ϕ2(t) + aeϕ2(t) − r2ϕ1(t) + beϕ1(t)] = H(0).

This proves that the ϕi(t) remain bounded from above and below, and thus that solutions are global.

3. n̄1 = r2/b, n̄2 = r1/a.

4. In each quadrant defined by the origin (n̄1, n̄2), we can write n2 as a function of n1 (or vice versa)

and conclude from that.

Exercise. Let Ω be a smooth bounded domain. Consider smooth positive solutions of the Lotka-Volterra

equation with diffusion and a Neumann boundary condition∂∂tn1 − d1∆n1 = n1[ r1 − an2],∂∂tn2 − d2∆n2 = n2[−r2 + bn1],∂ni∂ν = 0 on ∂Ω, i = 1, 2,

where d1, d2, r1, r2, a and b are positive constants and the initial data n0i are positive.

1. Consider the quantity m(t) =∫

Ω[bn1(t, x) + an2(t, x)]dx. Show that m(t) ≤ m(0)ert and find the

value r.

2. Show that the convex entropy

E(t) =

∫Ω

[−r1 lnn2 + an2 − r2 lnn1 + bn1]dx,

a) is bounded from below, b) is decreasing.

Conclude that m(t) is bounded.

3. What finite integral do we obtain from the entropy dissipation?

3. Assume that the quantities ∇ lnni(t, x) converge, as t→∞,a. What are their limits?

b. What can you conclude about the behavior of ni(t, x) as t→∞?

5.6. PROBLEM 39

5.6 Problem

The goal of this problem is to show that for rank-1 nonlinearities, the long term behaviour is determinded

by simple states in x without a size condition.

Let Ω a smooth bounded domain. Consider a smooth positive solution of the Lotka-Volterra system

with diffusion and Neumann boundary condition∂∂tni −Di∆ni + ai(x)ni = ni ri(t), t ≥ 0, x ∈ Ω, i = 1, 2, ..., I,∂ni∂ν = 0 on ∂Ω, i = 1, 2, ..., I,

ni(t = 0, x) = n0i (x),

with the nonlinearity defined, for some given positive and smooth functions(ψi(x)

)i=1,...,I

, by

ri(t) = Ri( ∫

Ω

ψ1(x)n1(t, x)dx, ...,

∫Ω

ψI(x)nI(t, x)dx).

1. We also define the first eigenfunctions Ni(x) > 0 for the eigenvalue λi, defined by −Di∆Ni + ai(x)Ni = λiNi, x ∈ Ω,∂Ni∂ν = 0 on ∂Ω,

∫ΩNi(x)

2dx = 1, i = 1, 2, ..., I.

Explain why the pair (Ni, λi) exists and give the corresponding Poincaré-Wirtinger inequality.

2. We consider ñi the solution of∂∂t ñi −Di∆ñi + ai(x)ñi = λiñi, t ≥ 0, x ∈ Ω, i = 1, 2, ..., I,∂ñi∂ν = 0 on ∂Ω, i = 1, 2, ..., I,

ñi(t = 0, x) = n0i (x).

Prove that∫

Ωñi(t, x)Ni(x)dx =

∫Ωñ0i (x)Ni(x)dx. Find the constant αi such that, as t→∞,

‖ñi(t, x)− αiNi(x)‖L2(Ω) ≤ ‖ñ0i (x)− αiNi(x)‖L2(Ω)e−µit

and identify µi.

3. We write the solution of (5.6) as ni(t, x) = ρi(t)ñi(t, x). Identify the evolution equation givingddtρi(t)

in terms of the ρj and ñj .

4. Assume that for some M > 0 we have Ri(Y1, ...YI

)< λi whenever Yi > M , i = 1, ..., I. Show that the

ρi(t) are bounded and that for some µ > 0 we have

‖ni(t, x)− ρi(t)αiNi(x)‖L2(Ω) ≤ C0e−µt.

5. In dimension 1, assuming R(+∞) < λ, R(−∞) > λ and R′ < 0. Show that the long term dynamic isgiven by that of the equation

%̇(t) = %(t) R(%(t)α

∫ψ(x)N(x)dx

),

and that n(t, x) converges generically to a steady state ρ̄N(x) as t→∞.

[Hint.] ρ̇i(t) = ρi(t)(ri − λi)

Chapter 6

Blow-up and extinction of solutions

We know from Chapter 5 that for ‘small’ nonlinearities, the solutions of parabolic systems relax to an

elementary state. When the nonlinearity is too large, it is possible that solutions of nonlinear parabolic

equations do not exist for all times or simply vanish, two scenarios that are the first signs of visible

nonlinear effects.

The mechanisms can be seen in simple ordinary differential equations. Blow-up means that the solution

becomes pointwise larger and larger and eventually becomes infinite in finite time. To see this, consider

the equation

ż(t) = z(t)q, z(0) > 0, q > 1.

Its solution z(t) > 0 is given by

z(t)q−1 = z(0)q−1/(1− (q − 1)tz(0)q−1

)and thus it tends to infinity at t = T ∗ := 1(q−1)z(0)q−1 . For q = 2, it is a typical illustration of the

alternative arising in the Cauchy-Lipschitz Theorem; solutions can only tend to infinity in finite time

(case z(0) > 0), or they are globally defined (case z(0) < 0).

The mechanism of extinction is illustrated by the equation with the opposite sign

ż(t) = −z(t)q, z(0) > 0, q > 1.

Its solution

z(t)q−1 = z(0)q−1/(1 + (q − 1)tz(0)q−1

)(6.1)

vanishes as t→∞.The purpose of this Chapter is to study in which respect these phenomena can occur or not when

diffusion is added.

We begin with the semilinear parabolic equation with Dirichlet boundary conditions

∂

∂tu−∆u = u2.

We present several methods for proving blow-up in finite time. The first two methods are for semi-linear

parabolic equations, the third method is illustrated by the Keller-Segel system for chemotaxis; this is

41

42 Blow-up and extinction of solutions

more interesting because it blows-up in all Lp norms, for all p > 1, but the L1 norm is conserved (this

represents the total number of cells in a system where cell multiplication is ignored).

The last section is devoted to a counter-intuitive result with respect to extinction.

6.1 Semilinear equations; the method of the eigenfunction

To study the case of nonlinear parabolic equations, we consider the model∂∂tu−∆u = u

2, t ≥ 0, x ∈ Ω,

u(x) = 0, x ∈ ∂Ω,

u(t = 0, x) = u0(x) ≥ 0.

(6.2)

Here we treat the case when Ω is a bounded domain. To define a distributional solution is not completely

obvious because the right hand side u2 should be well defined, that is why we require that u belongs to

L2; then it remains to solve the heat equation with a right hand side in L1. That is why we call solution

of (6.2), a function satisfying for some T > 0,

u ∈ L2((0, T )× Ω

), u ∈ C

([0, T );L1(Ω)

). (6.3)

The question then is to know if effects of diffusion are able to overcome the effects of quadratic non-

linearity and (6.2) could have global solutions. The answer is given by the next theorems

Theorem 6.1 Assume that Ω is a bounded domain, u0 ≥ 0, u0 is sufficiently large (in a weighted L1

space introduced below), then there is a time T ∗ for which the solution of (6.2) satisfies

‖u‖L2((0,T )×Rd) −→T→T∗

∞.

Of course, this result means that u(t) also blows up in all Lp norms, 2 ≤ p ≤ ∞ because we work in abounded domain. It is also possible to prove that the blow-up time is the same for all these norms [35].

Proof of Theorem 6.1. We are going to arrive at a contradiction in the hypothesis that a function

u ∈ L2((0, T )× Ω

)can be a solution of (6.2) when T exceeds an explicit value computed later.

First we, notice that u(t, x) ≥ 0 because u0 ≥ 0.

Because we are working in a bounded domain, the smallest eigenvalue of the operator −∆ exists andis associated with a positive eigenfunction (see Section 4.2) −∆w1 = λ1w1, w1 > 0 in Ω,w1(x) = 0, x ∈ ∂Ω, ∫Ω(w1)2 = 1. (6.4)

6.1. SEMILINEAR EQUATIONS; THE METHOD OF THE EIGENFUNCTION 43

For a solution, we can multiply equation (6.2) by w1 and integrate by parts. We arrive at

ddt

∫Ωu(t, x) w1(x) dx =

Documents

Mathematical methods in biology Master of Mathematics ...perthame/MathMethodBiol.pdfMathematical methods in biology Master of Mathematics, second year Beno^ t Perthame Sorbonne Universit