Persistence theory in population dynamicshalsmith/PersistenceTutorial.pdfOutline 1 Introduction to Persistence Theory 2 Some Dynamical Systems Theory 3 Persistence deﬁnitions 4 Fundamental

Persistence theory in population dynamics

Hal Smith

A R I Z O N A S T A T E U N I V E R S I T Y

H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 1 / 28

Outline

1 Introduction to Persistence Theory

2 Some Dynamical Systems Theory

3 Persistence definitions

4 Fundamental Results of Persistence Theory

5 Example: Three Species Food Chain


Introduction to Persistence Theory

The problem

What can a mathematician tell a biologist about a very large system?

Hi = bacteria strain i, Vj = virus strain j, 1 ≤ i , j ≤ n ≈ 100

dHi

dt= riHi

(

1 −

∑nk=1 Hk

K

)

︸︷︷︸

growth and competition

−Hi

n∑

j=1

MijφjVj

︸︷︷︸

infection by virus

dVj

dt= βjφjVj

n∑

i=1

MijHi

︸︷︷︸

virus reproduction

− djVj︸︷︷︸

virus decay

, 1 ≤ i , j ≤ n

Mij =

1 Vj infects Hi

0 Vj does not infect Hi

.



Does species X survive?

Persistence: Freedman and Waltman (1977)x(0) > 0† ⇒ lim sup x(t) > 0.

Permanence: Schuster, Sigmund, Wolff (1979)∃m,M > 0, independent of x(0) > 0, som ≤ lim inf x(t) ≤ lim sup x(t) ≤ M.

Uniform Persistence: Hofbauer (1981)∃ǫ > 0, independent of x(0) > 0, so ǫ ≤ lim inft→∞ x(t)††.

† typically, x(0) = 0 implies x(t) = 0, t ≥ 0

†† ideally, ǫ would be specified in terms of parameters.



Volterra Predator-Prey Model: a, b, c, d > 0.

x ′ = x(a − by), y ′ = y(−c + dx)Do x and y persist? (yes) ; are they permanent? (no); do they persist uniformly? (no)



LPA Flour Beetle Model*

Life-cycle stages of (feeding) larva (x1), pupa (x2), and adult (x3):

x1(n + 1) = d x3(n)exp(−ax1(n)− bx3(n)),

x2(n + 1) = p x1(n),

x3(n + 1) = q x2(n)exp(−cx3(n)) + rx3(n).

r = adult survival probability,p = transition/survival probability from the larval to the pupal stage,q = transition/survival probability from the pupal to the adult stage,Coefficients a,b, and c are related to cannibalism and d to fecundity.

We want persistence theory to apply to discrete-time dynamicalsystems!

*Constantino, Cushing, Dennis, Desharnais, Nature 375(1995)


Some Dynamical Systems Theory

Abstract dynamical system

J = [0,∞) (continuous-time) or J = Z+ (discrete-time).X a metric space. Φ : J × X → X describes the dynamics:

Φ(t , x) = state of system at time t that was at state x at t = 0.

It must satisfy the conditions for a semidynamical system:1 Φ(0, x) = x ,2 Φ(t ,Φ(s, x)) = Φ(t + s, x), t , s ∈ J, x ∈ X .3 Φ is continuous.

Example: X = Rn+ = x ∈ R

n : xi ≥ 0,1 ≤ i ≤ n with usual metric.

x ′(t) = F (x(t)), t ∈ [0,∞)

Φ(t , x(0)) = x(t)

x(t + 1) = F (x(t)), t ∈ Z+

Φ(t , x(0)) = x(t)H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 7 / 28


Example: Bacteria B consuming Nutrient N in a Tube

Nt = dNNxx − cNB

Bt = dBBxx + rNB, 0 < x < 1, t > 0

with boundary conditions

− dNNx(t ,0) = F0, Bx(t ,0) = 0

dNNx(t ,1) + rNN(t ,1) = 0

dBBx(t ,1) + rBB(t ,1) = 0, t > 0

and initial conditions

(N(0, •),B(0, •)) = (N0(•),B0(•)) ∈ X ≡ C([0,1],R+)2

Φ(t , (N0,B0)) = (N(t , •),B(t , •)) ∈ X .



Dynamical Systems definitions

x0 ∈ X is an equilibrium if Φ(t , x0) = x0, ∀t ≥ 0.

A ⊂ X is invariant if Φ(t ,A) = A, t ∈ JA ⊂ X is positively invariant if Φ(t ,A) ⊂ A, t ∈ J.

invariant set A ⊂ X is an isolated invariant set if ∃ open set Vcontaining A such that K ⊂ A for every invariant set K ⊂ V .

The omega limit set of x ∈ X is

ω(x) = y : limn→∞

Φ(tn, x) = y , some sequence tn with tn → ∞

ω(x) is a closed (possibly empty) set that is positively invariant.

If Φ(t , x) : t ≥ 0 has compact closure, then ω(x) 6= ∅ is compact,connected, invariant, and dist(Φ(t , x), ω(x)) → 0, t → ∞.



Dynamical Systems definitions continued

The stable set of A ⊂ X is

W s(A) = x ∈ X : Φ(t , x) → A, t → ∞

where “Φ(t, x) → A, t → ∞” means dist(Φ(t, x),A) → 0 as t → ∞.

ψ : J ∪ (−J) → X is a total trajectory for Φ if

ψ(t + s) = Φ(t , ψ(s)), t ∈ J, s ∈ J ∪ (−J)

Key fact: If A is invariant and a ∈ A, then there is a total trajectoryψ : J ∪ (−J) → A with ψ(0) = a.


Persistence definitions

Persistence Definitions

Let ρ : X → [0,∞) be a nontrivial, continuous “persistence function”.ρ−1(0) is the “extinction set".

Φ is uniformly weakly ρ-persistent, if ∃ǫ > 0 such that

lim supt→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.

Φ is uniformly (strongly) ρ-persistent, if ∃ǫ > 0 such that

lim inft→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.

Equivalently, ∃ǫ > 0 such that ∀x ∈ X with ρ(x) > 0:

∃Tx > 0, ρ(Φ(t , x)) > ǫ, t > Tx .

“uniform” conveys that ǫ is independent of initial data x






lim supt→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


lim inft→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


∃Tx > 0, ρ(Φ(t , x)) > ǫ, t > Tx .







lim supt→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


lim inft→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


∃Tx > 0, ρ(Φ(t , x)) > ǫ, t > Tx .







lim supt→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


lim inft→∞

ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.


∃Tx > 0, ρ(Φ(t , x)) > ǫ, t > Tx .




Persistence Function geometry

“persistence function”: ρ : R2+ → [0,∞)

“extinction set” = ρ−10 = x ∈ R2+ : ρ(x) = 0

x1

x2

•

ρ = x1 + x2

ρ = ǫ

x1

x2

ρ = ǫ

ρ = minx1, x2



Uniform Persistence: Mental Image

A1

A0

ρ = ε

ρ = 0



Example: Bacteria B consuming Nutrient N in a Tube

Nt = dNNxx − cNB

Bt = dBBxx + rNB, 0 < x < 1, t > 0

with boundary conditions

− dNNx(t ,0) = F0, Bx(t ,0) = 0

dNNx(t ,1) + rNN(t ,1) = 0

dBBx(t ,1) + rBB(t ,1) = 0, t > 0

and initial conditions

(N(0, •),B(0, •)) = (N0(•),B0(•)) ∈ X ≡ C([0,1],R+)2

Φ(t , (N0,B0)) = (N(t , •),B(t , •)) ∈ X .

a natural persistence function is ρ(N,B) =∫ 1

0 B(x)dx , the populationsize.


Fundamental Results of Persistence Theory

Three Fundamental Results of Persistence Theory

1 Weak Uniform Persistence implies Strong Uniform Persistence.2 A necessary condition for uniform persistence is the existence of a

“coexistence equilibrium”, x0, with ρ(x0) > 0.3 The Acyclicity Persistence Theorem.



Weak ⇒ Strong Persistence

Theorem†: Suppose that x : ρ(x) > 0 is positively invariant and:

(H): ∃ compact B ⊂ X such that ρ(x) > 0 ⇒ Φ(t , x) → B.

Then, uniform weak persistence implies uniform strong persistence:

If ∃η > 0 such that

lim supt→∞ρ(Φ(t , x)) > η ∀x ∈ X , ρ(x) > 0,

then ∃ǫ > 0 such that

lim inft→∞ρ(Φ(t , x)) > ǫ ∀x ∈ X , ρ(x) > 0.

The conservative Volterra Predator-Prey model shows that the assumption(H) cannot be dropped!† Freedman& Moson (1990), Thieme(1993), Magal & Zhao (2005), H.S. & Thieme (2011)



It’s easier to prove weak uniform persistence

Negation of:

∃ǫ > 0 such that lim supt→∞ ρ(Φ(t , x)) > ǫ, ∀x ∈ X , ρ(x) > 0.

implies that:

∀ǫ > 0,∃x , ρ(x) > 0, and T > 0 such that ρ(Φ(t , x)) ≤ ǫ, t ≥ T .

Letting y = Φ(T , x), then:

ρ(Φ(t , y)) = ρ(Φ(t ,Φ(T , x))) = ρ(Φ(T + t , x)) ≤ ǫ, t ≥ 0.

so we obtain an exploitable relation:

0 < ρ(Φ(t , y)) ≤ ǫ, t ≥ 0.



A “Coexistence Equilibrium” is Necessary Conditionfor Persistence

Theorem: [Existence of Coexistence Equiibrium]Assume

X is a closed, convex subset of a Banach space.

Φ has a compact attractor A of bounded sets in X .

ρ is continuous and concave.

Φ is uniformly weakly ρ-persistent.

Φ(t , •) is compact for some t > 0.

Then there exists an equilibrium x∗ with ρ(x∗) > 0.

Magal & Zhao 2005, Smith & Thieme 2011



Acyclicity Definition

X0 = x ∈ X : ρ(Φ(t , x)) = 0, t ≥ 0

is the largest positively invariant subset of the ”extinction set”, ρ−1(0).

Let C,B ⊆ X0 be invariant sets. C is chained to B in X0, writtenC 7→ B, if ∃ a total trajectory φ : (−J) ∪ J → X0 with φ(0) 6∈ C ∪ B andφ(−t) → C and φ(t) → B as t → ∞.

A finite collection M1, · · · ,Mk of subsets of X0 is cyclic if, onrenumbering, M1 7→ M1 in X0 or M1 7→ M2 7→ · · · 7→ Mj 7→ M1 in X0 forsome j ∈ 2, · · · , k. Otherwise it is acyclic.




X0 = x ∈ X : ρ(Φ(t , x)) = 0, t ≥ 0







X0 = x ∈ X : ρ(Φ(t , x)) = 0, t ≥ 0






Acyclicity Theorem

Theorem: Assume ∃ compact B ⊂ X such that Φ(t , x) → B, x ∈ X .Let X0 = x ∈ X : ρ(Φ(t , x)) = 0, t ≥ 0 and define

Ω = ∪x∈X0ω(x) ⊂ X0.

Assume there is a finite collection of pairwise disjoint compact invariantsets M1,M2, · · · ,Mk in X0 such that:

1 Ω ⊂ ∪ki=1Mi

2 Mi is an isolated invariant set.3 M1,M2, · · · ,Mk is acyclic.4 ∀i , W s(Mi) ⊂ X0.

Then Φ is uniformly ρ-persistent.

Butler & Waltman; Butler, Waltman, Freedman 1986; Hale & Waltman 1989


Example: Three Species Food Chain


Z eats Y , Y eats X :

x ′ = rx(1 − x/K )− yg(x)

y ′ = yg(x)− kyy − zh(y)

z′ = z(h(y) − kz)

g(x) =mx

k + x, h(y) =

MyL + y

Chaos in open octant-Teacup attractor (Hastings, Kuznetzov)

Can all three species persist?



Teacup attractor for Food Chain

00.5

1

00.10.20.30.40.57

7.5

8

8.5

9

9.5

10

10.5

X

teacup attractor

Y

Z



Boundary Dynamics of Food Chain

Persistence function ρ(x , y , z) ≡ minx , y , z means that X0 is theboundary of the positive orthant. Equilibria in X0 consist ofE0 = (0,0,0),Ex = (a,0,0),Exy = (c,d ,0).

∃! cycle P; it attracts open positive quadrant of xy plane except Exy .

Boundary Attractor: Ω = E0 ∪ Ex ∪ Exy ∪ P

Connections: E0 Ex P Exy so E0,Ex ,Exy ,P is acyclic.

Figure: Dynamics on boundary of R3+.



Verify Hypotheses of Acyclicity Theorem

x ′ = rx(1 − x/K )− yg(x) = xf1y ′ = yg(x)− ky y − zh(y) = yf2z′ = z(h(y) − kz) = zf3

E0 is a hyperbolic saddle point, hence isolated†, andW s(E0) = x = 0⊂ X0 if f1(E0) > 0.

Ex is a saddle point and W s(Ex ) = y = 0, x > 0 if f2(Ex ) > 0.

Exy is a repeller and W s(Exy ) = Exy if f3(Exy ) = h(d)− kz > 0.

P = (x(t), y(t),0) : 0 ≤ t ≤ T is hyperbolic saddle andW s(P) = z = 0, x , y > 0 \ Exy if Floquet exponent∫

P f3dµ = 1T

∫ T0 h(y(t))dt − kz > 0.

† Hartman Grobman Theorem



The May-Leonard Lotka-Volterra Competition Model

N ′1 = N1[1 − N1 − αN2 − βN3]

N ′2 = N2[1 − βN1 − N2 − αN3]

N ′3 = N3[1 − αN1 − βN2 − N3]

with0 < α < 1 < β, α+ β > 2,

implies no 2-species equilibria and the coexistence equilibrium isunstable. Equilibria E1 = (1,0,0), E2 = (0,1,0), E3 = (0,0,1) areunstable.

Permanence fails. Every solution with positive initial conditionsconverges to a heteroclinic cycle:

E1 E3 E2 E1

† May & Leonard, SIAM J. Appl. Math. (1975)H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 25 / 28


May-Leonard Dynamics

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N1N2

N3



Acyclic Coverings for May-Leonard System

Persistence function ρ(N) ≡ mini Ni means that X0 is the boundary ofthe positive orthant since it is invariant. Therefore:

Ω = ∪N∈X0ω(N) = E0,E1,E2,E3

E0,E1,E2,E3 is not an acyclic covering of Ω since there is a cycleE1 E2 E3 E1

E0,HC is an acyclic covering of Ω but W s(HC) is not contained inX0!

Acyclicity Theorem does not apply.H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 27 / 28




Ω = ∪N∈X0ω(N) = E0,E1,E2,E3







Ω = ∪N∈X0ω(N) = E0,E1,E2,E3





THANK YOU

These results, and more, contained in:

Dynamical Systems and Population PersistenceAmerican Mathematical SocietyGraduate Studies in Mathematics, vol 118, 2011Hal L. Smith and Horst R. Thieme

Dynamical Systems in Population BiologyCMS Books in MathematicsSpringer, 2003Xiao-Qiang Zhao


Documents

Persistence theory in population dynamicshalsmith/PersistenceTutorial.pdfOutline 1 Introduction to Persistence Theory 2 Some Dynamical Systems Theory 3 Persistence deﬁnitions 4 Fundamental