Persistence theory in population dynamicshalsmith/PersistenceTutorial.pdfOutline 1 Introduction to...
Transcript of Persistence theory in population dynamicshalsmith/PersistenceTutorial.pdfOutline 1 Introduction to...
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 2 / 28
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
.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 3 / 28
Introduction to Persistence Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 4 / 28
Introduction to Persistence Theory
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)
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 5 / 28
Introduction to Persistence Theory
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)
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 6 / 28
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
Some Dynamical Systems Theory
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 .
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 8 / 28
Some Dynamical Systems Theory
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 → ∞.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 9 / 28
Some Dynamical Systems Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 10 / 28
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 11 / 28
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 11 / 28
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 11 / 28
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 11 / 28
Persistence definitions
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 12 / 28
Persistence definitions
Uniform Persistence: Mental Image
A1
A0
ρ = ε
ρ = 0
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 13 / 28
Persistence definitions
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 14 / 28
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 15 / 28
Fundamental Results of Persistence Theory
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)
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 16 / 28
Fundamental Results of Persistence Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 17 / 28
Fundamental Results of Persistence Theory
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 18 / 28
Fundamental Results of Persistence Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 19 / 28
Fundamental Results of Persistence Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 19 / 28
Fundamental Results of Persistence Theory
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.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 19 / 28
Fundamental Results of Persistence Theory
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 20 / 28
Example: Three Species Food Chain
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?
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 21 / 28
Example: Three Species Food Chain
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 22 / 28
Example: Three Species Food Chain
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+.
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 23 / 28
Example: Three Species Food Chain
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 24 / 28
Example: Three Species Food Chain
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
Example: Three Species Food Chain
May-Leonard Dynamics
0
0.2
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0.8
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0
0.2
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0.6
0.8
10
0.1
0.2
0.3
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1
N1N2
N3
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 26 / 28
Example: Three Species Food Chain
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
Example: Three Species Food Chain
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
Example: Three Species Food Chain
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
Example: Three Species Food Chain
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
H.L. Smith (ASU) Persistence Theory July , 2015, Guangzhou, China 28 / 28