Assume that (A1) and (A2) hold

(a) Ifω

0 F₀(t, 0)dt≤ 0, then limt→∞φ(t, 0, u) = 0,∀u ∈ R+; (b) Ifω

0 F₀(t, 0)dt > 0, then limt→∞(φ(t, 0, u)− u^∗(t)) = 0, ∀u ∈ R₊\ {0}, where u^∗(t) is the unique positive ω-periodic solution of the periodic Kol-mogorov equation ^du_dt = uF₀(t, u).

Proof. Let φ₀(t, s, u), t ≥ s ≥ 0, be the unique solution of the ω-periodic Kolmogorov equation

du

dt = uF₀(t, u), u∈ R+, (5.4)

with φ₀(s, s, u) = u∈ R+. We first claim that the following threshold result on the global asymptotics of (5.4) holds:

(i) Ifω

0 F₀(t, 0)dt≤ 0, then limt→∞φ₀(t, 0, u) = 0,∀u ∈ R₊; (ii) Ifω

0 F₀(t, 0)dt > 0, then (5.4) admits a unique positive periodic solution u^∗(t), and limt→∞(φ₀(t, 0, u)− u^∗(t)) = 0,∀u ∈ R+\ {0}.

Note that if F₀(t, u) is continuously differentiable with respect to u, the above claim is a simple corollary of Theorem3.1.2with n = 1. But we need to prove it under the assumption that F₀(t, u) is continuous and locally Lipschitz in u uniformly for t ∈ [0, ω]. Let Q : R₊ → R₊ be the Poincar´e map associated with the periodic system (5.4). For any u > 0, we have u(t) := φ₀(t, 0, u) >

0,∀t > 0, and hence the strict monotonicity of F₀(t, u) for u > 0 implies that du

dt = u(t)F₀(t, u(t)) < u(t)F₀(t, 0), ∀t > 0.

136 5 N -Species Competition in a Periodic Chemostat

By the comparison theorem, it then follows that u(t) < u(0)e^0tF0(s,0)ds, ∀t > 0.

In the case whereω

0 F₀(t, 0)dt≤ 0, the Poincar´e map Q : R+→ R+satisfies Q(u) = u(ω) < u(0)e^0ωF(s,0)ds≤ u(0) = u, ∀u > 0,

which implies that Q :R₊→ R₊ admits no positive fixed point, and that for any u > 0,

0 < Qⁿ⁺¹(u) < Qⁿ(u), ∀n ≥ 0.

Thus, there exists ¯u≥ 0 such that limn→∞Qⁿ(u) = ¯u. Since ¯u = Q(¯u), the nonexistence of positive fixed points of Q implies ¯u = 0. Then limn→∞Qⁿ(u) = 0,∀u > 0, and hence limt→∞u(t) = 0. In the case where ω

0 F₀(t, 0)dt > 0, Lemma 5.1.1with n = 1 implies that {0} is an isolated invariant set for Q, and W^s(0)∩ int(R+) =∅. By Theorem1.3.1and Remark1.3.1, as applied to Q : X :=R+→ X with X0= int(R+) and ∂X₀={0}, {0} is a strong repeller in the sense that there exists δ > 0 such that ω(u)≥ δ, ∀u > 0. It is easy to see that Q : R+ → R+ is strongly monotone and strictly subhomogeneous.

By Theorem2.3.2, it then follows that Q admits a unique positive fixed point u^∗, and lim_n→∞Qⁿ(u) = u^∗,∀u > 0. Thus, the conclusion in (ii) holds with u^∗(t) = φ₀(t, 0, u^∗).

By conditions (A1) and (A2), it easily follows that for any u ∈ R₊ and s≥ 0, φ(t, s, u) and φ₀(t, s, u) exist globally on [s,∞), and solutions of (5.3) and (5.4) are uniformly bounded. By Proposition3.2.2, φ(t, s, u) is asymptotic to the ω-periodic semiflow T (t) := φ₀(t, 0,·) : R+ → R+, and hence Tn(u) :=

φ(nω, 0, u), n ≥ 0, is an asymptotically autonomous discrete process with limit discrete semiflow Qⁿ :R+ → R+, n≥ 0. By Theorem 3.2.1, it suffices to prove in case (a) that limn→∞Tn(u) = 0,∀u ∈ R+, and in case (b) that limn→∞Tn(u) = u^∗(0), ∀u ∈ R+\ {0}.

In case (a), by conclusion (i) above, u = 0 is a global attractor for Q : R+→ R+. Thus, Theorem 1.2.1implies that for any u∈ R+, ω(u) = 0, and hence limn→∞Tn(u) = 0.

In case (b), by conclusion (ii) above, u = u^∗(0) is a globally attractive fixed point of Q inR+\ {0}. Thus, the only fixed points of Q in R+are 0 and u^∗(0); both are isolated invariant sets, and there is no Q-cyclic chain among them. Then Theorem 1.2.2implies that for any u∈ R₊, either ω(u) = 0 or ω(u) = u^∗(0). By Lemma 5.1.2with n = 1, we have &W^s(0)∩ (R⁺\ {0}) = ∅;

that is, ω(u)= 0, ∀u > 0. Consequently, for any u > 0, ω(u) = u^∗(0), and hence lim_n→∞T_n(u) = u^∗(0).

Now we consider a single population growth model in a periodic chemostat dS(t)

dt = (S⁰(t)− S(t))D₀(t)− x(t)P (t, S(t)), dx(t)

dt = x(t)(P (t, S(t))− D1(t)).

(5.5)

5.2 Single Population Growth 137 Here S(t) denotes the concentration of the nutrient, x(t) denotes the biomass of the species at time t, P (t, s) represents the specific per capita nutrient uptake function, S⁰(t) and D₀(t) are the input nutrient concentration and the dilution rate, respectively, and D₁(t) represents the specific removal rate of the species. We assume that S⁰(t), D₀(t), and D₁(t) are all continuous, ω-periodic, positive functions, and that P (t, s) : R²₊ → R₊ is continuous, ω-periodic in t, and satisfies

(B1) P (t, s) is locally Lipschitz in s;

(B2) P (t, 0) = 0, ∀t ≥ 0, and for each t ≥ 0, P (t, s) is strictly increasing for s∈ R₊.

Let D(t) :R₊ → R₊ be a continuous, ω-periodic, and positive function.

For the linear periodic equation dV (t)

dt = S⁰(t)D₀(t)− D(t)V (t), (5.6) it easily follows that (5.6) admits a unique positive ω-periodic solution V^∗(t) such that every solution V (t) of (5.6) with V (0)≥ 0 satisfies limt→∞(V (t)− V^∗(t)) = 0. Moreover, V^∗(t) can be expressed explicitly as

V^∗(t) = e^−0tD(s)ds -ω

0 e^0^sD(u)duS⁰(s)D₀(s)ds e^0^ωD(s)ds− 1 +

 t 0

e^0sD(u)duS⁰(s)D₀(s)ds

! .

Let D(t) = max(D₀(t), D₁(t)) and D(t) = min(D₀(t), D₁(t)). Then, D(t) and D(t) : R₊ → R₊ are continuous, ω-periodic, and positive functions. Let V₁^∗(t) and V₂^∗(t) be the unique positive ω-periodic solutions of (5.6) with D(t) replaced by D(t) and D(t), respectively. By the comparison theorem and the global attractivity of each V_i^∗(t), it easily follows that V₂^∗(t)≤ V₁^∗(t),∀t ≥ 0.

Theorem 5.2.2. Let (B1) and (B2) hold. Then the following threshold dy-namics hold:

(a) If ω

0 (P (t, V₂^∗(t))− D1(t))dt > 0, then system (5.5) admits a positive (componentwise) ω-periodic solution, and there exist α > 0 and β > 0 such that every solution (S(t), x(t)) of (5.5) with S(0)≥ 0 and x(0) > 0 satisfies

α≤ lim inf

t→∞ x(t)≤ lim sup

t→∞ x(t)≤ β.

(b) Ifω

0(P (t, V₁^∗(t))− D1(t))dt≤ 0, then every solution (S(t), x(t)) of (5.5) with S(0)≥ 0 and x(0) ≥ 0 satisfies limt→∞x(t) = 0.

Interpreting the predictions of the model biologically, Theorem5.2.2 im-plies that in case (a) the model system admits a periodic coexistence state and the species is uniformly persistent, but in case (b) the species ultimately goes to extinction.

138 5 N -Species Competition in a Periodic Chemostat

Proof. Let ˆP (t, s) : R₊× R → R be a continuous extension of P (t, s) on R₊× R₊ toR₊× R such that ˆP (t, s) is ω-periodic in t and locally Lipschitz in s, and for any t≥ 0, ˆP (t, s) is strictly increasing for s∈ R.

In case (a), since V₁^∗(t)≥ V₂^∗(t),∀t ∈ [0, ω], and ˆP (t, V_i^∗(t)) = P (t, V_i^∗(t)),

∀t ∈ [0, ω], 1 ≤ i ≤ 2, Theorem5.2.1(in the periodic case) implies that the periodic equation

dx(t)

dt = x(t)( ˆP (t, V_i^∗(t)− x(t)) − D₁(t))

admits a unique positive ω-periodic solution x^∗_i(t), and x^∗_i(t) is globally at-tractive inR+\ {0}, 1 ≤ i ≤ 2. By the comparison theorem, it easily follows that x^∗₁(t)≥ x^∗₂(t),∀t ∈ [0, ω]. We further claim that V₁^∗(t) > x^∗₁(t),∀t ∈ [0, ω].

Indeed, let x^∗₁(t₁) = max_0≤t≤ωx^∗₁(t), t₁∈ [0, ω]. Then ^dx∗1_dt^(t1)= 0, and hence P (tˆ ₁, V₁^∗(t₁)− x^∗₁(t₁)) = D₁(t₁) > 0.

Since ˆP (t₁, s) is strictly increasing for s ∈ R, V₁^∗(t₁) > x^∗₁(t₁). Let y(t) = V₁^∗(t)− x^∗₁(t). Then y(t) satisfies the periodic differential equation

dy

dt = S⁰(t)D₀(t)− D(t)V₁^∗(t)− (V₁^∗(t)− y)( ˆP (t, y)− D₁(t)). (5.7) Since y(t₁) > 0 and

dy dt



y=0

= S⁰(t)D₀(t) + (D₁(t)− D(t))V₁^∗(t)≥ S⁰(t)D₀(t) > 0,

it follows that y(t) > 0,∀t ≥ t₁. Thus, the ω-periodicity of y(t) implies that y(t) > 0,∀t ≥ 0, that is, V₁^∗(t) > x^∗₁(t),∀t ≥ 0.

For any (S₀, x₀) ∈ R²₊ with S₀ ≥ 0 and x₀ > 0, let (S(t), x(t)) be the unique solution of (5.5) satisfying S(0) = S₀and x(0) = x₀ with [0, β) as its maximal existence interval. It then easily follows that S(t) > 0 and x(t) >

0,∀t ∈ (0, β). Let V (t) = S(t) + x(t). Then S⁰(t)D₀(t)− D(t)V (t) ≤ dV (t)

dt ≤ S⁰(t)D₀(t)− D(t)V (t), ∀t ∈ [0, β).

Let V (t) be the unique solution of the linear ω-periodic equation dV

dt = S⁰(t)D₀(t)− D(t)V

satisfying V (0) = V (0), and let V (t) be the unique solution of the linear ω-periodic equation

dV

dt = S⁰(t)D₀(t)− D(t)V

5.2 Single Population Growth 139

satisfying V (0) = V (0). Then the standard comparison theorem implies that V (t)≤ V (t) ≤ V (t), ∀t ∈ [0, β). (5.8) Since V (t) and V (t) exist globally on [0,∞), β = ∞. Therefore, x(t) satisfies

x(t)

P (t, V (t)ˆ − x(t)) − D1(t)

≤dx(t)

dt ≤ x(t)

P (t, V (t)ˆ − x(t)) − D1(t) 

for all t≥ 0. Then, by the comparison theorem,

x(t)≤ x(t) ≤ x(t), ∀t ≥ 0, (5.9)

where ¯x(t) is the unique solution of the nonautonomous equation dx(t)

dt = x(t)

P (t, V (t)ˆ − x(t)) − D1(t) 

, (5.10)

with ¯x(0) = x₀, and x(t) is the unique solution of the nonautonomous equation dx(t)

dt = x(t)

P (t, V (t)ˆ − x(t)) − D1(t) 

, (5.11)

with x(0) = x₀. Since limt→∞(V (t)− V₁^∗(t)) = 0 and limt→∞(V (t)− V₂^∗(t))

= 0, we have

tlim→∞( ˆP (t, V (t)− x) − ˆP (t, V₁^∗(t)− x)) = 0 and

tlim→∞( ˆP (t, V (t)− x) − ˆP (t, V₂^∗(t)− x)) = 0 uniformly for x in any bounded subset ofR+. In case (a), since

 ω 0

( ˆP (t, V₁^∗(t))− D1(t))dt≥

 ω 0

( ˆP (t, V₂^∗(t))− D1(t))dt,

=

 ω 0

(P (t, V₂^∗(t))− D1(t))dt > 0,

Theorem5.2.1(b) implies that

tlim→∞(¯x(t)− x^∗₁(t)) = 0 and lim

t→∞(x(t)− x^∗₂(t)) = 0.

By (5.9), it then follows that lim inf

t→∞ (x(t)− x^∗₂(t))≥ lim

t→∞(x(t)− x^∗₂(t)) = 0 (5.12) and

lim sup

t→∞ (x(t)− x^∗₁(t))≤ lim_t

→∞(x(t)− x^∗₁(t)) = 0, (5.13)

140 5 N -Species Competition in a Periodic Chemostat

and hence there exist α > 0 and β > 0 such that x(t) satisfies α≤ lim inf

t→∞ x(t)≤ lim sup

t→∞ x(t)≤ β.

In case (b), since

 ω 0

( ˆP (t, V₂^∗(t))− D₁(t))dt≤

 ω 0

( ˆP (t, V₁^∗(t))− D₁(t))dt,

=

 ω 0

P (t, V₁^∗(t))− D1(t))dt≤ 0,

Theorem5.2.1(a) implies that limt→∞x(t) = 0 and lim¯ t→∞x(t) = 0. By (5.9), we get limt→∞x(t) = 0.

In case (a), it remains to prove the existence of a positive periodic solution of (5.5). Under the abstract setting of periodic semiflows, this can be done by using Theorem1.3.8 as in the latter part of the proof of Theorem5.3.1.

Instead, we give an alternative, more elementary proof. Let V = S + x. Then the system (5.5) is transformed into the following ω-periodic system

dV

dt = S⁰(t)D₀(t)− D0(t)(V − x) − D1(t)x, dx

dt = x

P (t, Vˆ − x) − D1(t) 

.

(5.14)

Then the positive invariance of R²₊ with respect to (5.5) implies that the closed and convex set W :={(V, x) : V ≥ x ≥ 0} ⊂ R²₊ is positively invariant with respect to (5.14). Moreover, for any S₀ ≥ 0 and x₀ > 0, since the first equation of (5.5) implies that S^(t)|_S=0= S⁰(t)D₀(t) > 0, the unique solution (S(t), x(t)) of (5.5) with S(0) = S₀ and x(0) = x₀ satisfies S(t) > 0 and x(t) > 0,∀t > 0. That is, for any V₀≥ x₀> 0, the unique solution (V (t), x(t)) of (5.14) with V (0) = V₀ and x(0) = x₀ satisfies V (t) > x(t) > 0,∀t > 0. Let G : W → W be the Poincar´e map associated with (5.14); that is, for every (V₀, x₀) ∈ W, G(V0, x₀) = (V (ω), x(ω)). Clearly, the continuous dependence of solutions on initial data implies that G : W → W is continuous. Let

W₀:={(V, x) ∈ W : V₂^∗(0)≤ V ≤ V₁^∗(0), x^∗₂(0)≤ x ≤ x^∗₁(0)} . Since 0 < x^∗₁(t) < V₁^∗(t),∀t ∈ [0, ω], (V₁^∗(0), x^∗₁(0)) is in the interior of W , and hence W₀ is a nonempty, closed, bounded, and convex subset of R²₊. For each (V₀, x₀)∈ W0, the corresponding solution (V (t), x(t)) of (5.14) with V (0) = V₀ and x(0) = x₀ satisfies

(V (t), x(t))∈ W, ∀t ≥ 0; (5.15)

that is, V (t)≥ x(t) ≥ 0, ∀t ≥ 0. Then V (t) satisfies S⁰(t)D₀(t)− D(t)V (t) ≤dV (t)

dt ≤ S⁰(t)D₀(t)− D(t)V (t), ∀t ≥ 0.

5.2 Single Population Growth 141

Since V₂^∗(0)≤ V₀≤ V₁^∗(0), the comparison theorem implies that

V₂^∗(t)≤ V (t) ≤ V₁^∗(t), ∀t ≥ 0. (5.16) Therefore, x(t) satisfies

x(t)

P (t, Vˆ ₂^∗(t)− x(t)) − D1(t)

≤ dx(t)

dt , ∀t ≥ 0,

and dx(t)

dt ≤ x(t)

P (t, Vˆ ₁^∗(t)− x(t)) − D₁(t) 

, ∀t ≥ 0.

Since x^∗₂(0)≤ x0≤ x^∗₁(0), again by the comparison theorem we get

x^∗₂(t)≤ x(t) ≤ x^∗₁(t), ∀t ≥ 0. (5.17) Then (5.16) and (5.17) imply that

V₂^∗(0) = V₂^∗(ω)≤ V (ω) ≤ V₁^∗(ω) = V₁^∗(0),

x^∗₂(0) = x^∗₂(ω)≤ x(ω) ≤ x^∗₁(ω) = x^∗₁(0). (5.18) By (5.15) and (5.18), it follows that G(V₀, x₀) = (V (ω), x(ω))∈ W₀, and hence G(W₀)⊂ W₀. By the Brouwer fixed point theorem, there exists (V^∗, x^∗)∈ W₀ such that G(V^∗, x^∗) = (V^∗, x^∗). Clearly, the unique solution (V^∗(t), x^∗(t)) of (5.14) with (V^∗(0), x^∗(0)) = (V^∗, x^∗) is an ω-periodic solution of (5.14). Since V^∗ ≥ x^∗ > 0, by the previous claim, we have V^∗(t) > x^∗(t) > 0, ∀t > 0.

By the ω-periodicity of V^∗(t) and x^∗(t), it then follows that V^∗(t) > x^∗(t) >

0,∀t ≥ 0. Consequently, (S^∗(t), x^∗(t)) = (V^∗(t)− x^∗(t), x^∗(t)) is a positive (componentwise) ω-periodic solution of system (5.5).

In the case that D₀(t) = D₁(t), ∀t ∈ [0, ω], it is easy to see that V₁^∗(t) = V₂^∗(t), x^∗₁(t) = x^∗₂(t),∀t ∈ [0, ω]. Thus, (5.12) and (5.13) imply the following threshold dynamics for the model system.

Corollary 5.2.1. Let (B1) and (B2) hold and assume that D₀(t) = D₁(t),

∀t ∈ [0, ω]. Then the following statements are valid:

(a) If ω

0 (P (t, V₁^∗(t))− D1(t)) > 0, then system (5.5) admits a unique pos-itive, periodic solution (S^∗(t), x^∗₁(t)) = (V₁^∗(t)− x^∗₁(t), x^∗₁(t)), and ev-ery solution (S(t), x(t)) of (5.5) with S(0) ≥ 0 and x(0) > 0 satisfies limt→∞(S(t)− S^∗(t)) = 0 and limt→∞(x(t)− x^∗(t)) = 0.

(b) Ifω

0 (P (t, V₁^∗(t))− D1(t))≤ 0, then every solution (S(t), x(t)) of (5.5) with S(0) ≥ 0 and x(0) ≥ 0 satisfies limt→∞(S(t)− V₁^∗(t)) = 0 and lim_t→∞x(t) = 0.

142 5 N -Species Competition in a Periodic Chemostat

5.3 N-Species Competition

In this section we consider the n-species competition model in the periodic chemostat

dS(t)

dt = (S⁰(t)− S(t))D₀(t)−

n i=1

Pi(t, S(t))xi(t), dxi(t)

dt = xi(t)(Pi(t, S(t))− Di(t)), 1≤ i ≤ n.

(5.19)

Here S(t) denotes the concentration of the nutrient, xi(t) denotes the biomass of the ith species at time t, P_i(t, s) represents the specific per capita nutrient uptake function of the ith species, S⁰(t) and D₀(t) are the input nutrient concentration and the dilution rate, respectively, and Di(t) represents the specific removal rate, or washout rate, of species xi. We assume that S⁰(t) and Di(t), 1≤ i ≤ n, are all continuous, ω-periodic, and positive functions, and that each Pi(t, s) satisfies conditions (B1) and (B2).

Let

Pi(t, s) =



P_i(t, s) if t≥ 0, s ≥ 0, 0 if t≥ 0, s ≤ 0.

Then each Pi:R₊× R → R is a continuous extension of Pi(t, s) onR₊× R₊ toR+× R. Let

D(t) = max(D₀(t), D₁(t), . . . , Dn(t)) and

D(t) = min(D₀(t), D₁(t), . . . , Dn(t)).

Then D(t) and D(t) : R+ → R+ are continuous, ω-periodic, and positive functions. Let V₁^∗(t) and V₂^∗(t) be the unique positive ω-periodic solutions of (5.6) with D(t) replaced by D(t) and D(t), respectively. As shown in the previous section, V₂^∗(t)≤ V₁^∗(t),∀t ≥ 0.

We are now in a position to prove the main result of this section.

Theorem 5.3.1. Assume that

In document Dynamical Systems in Population Biology (Page 144-151)