Abstract
In recent work of Ballyk et al. [27], it is argued that the plug-flow reactor, aside from its importance in chemical and bioengineering, is a good candidate as a surrogate model of the mammalian large intestine. In that work, a model of competition between different strains of microorganisms for a scarce nutrient in a plug-flow reactor, formulated by Kung and Baltzis [207], was studied with special attention given to the effects of random motility of the organisms on their ability to persist in the reactor and be good competitors in a mixed culture. The growth-limiting nutrient is assumed to enter the reactor tube at constant concentration at the upstream end of the reactor, so that the model equations take the form of a time-independent system of reaction–advection–diffusion equations. However, if the plug-flow reactor is to stand as a surrogate model of the intestine, then it is much more realistic to consider input nutrient concentration as being time-dependent. In the present chapter we consider this competition model with periodically varying input nutrient concentration, including pulsed input where the concentration may fall to zero over part of the cycle.
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In recent work of Ballyk et al. [27], it is argued that the plug-flow reactor, aside from its importance in chemical and bioengineering, is a good candidate as a surrogate model of the mammalian large intestine. In that work, a model of competition between different strains of microorganisms for a scarce nutrient in a plug-flow reactor, formulated by Kung and Baltzis [207], was studied with special attention given to the effects of random motility of the organisms on their ability to persist in the reactor and be good competitors in a mixed culture. The growth-limiting nutrient is assumed to enter the reactor tube at constant concentration at the upstream end of the reactor, so that the model equations take the form of a time-independent system of reaction–advection–diffusion equations. However, if the plug-flow reactor is to stand as a surrogate model of the intestine, then it is much more realistic to consider input nutrient concentration as being time-dependent. In the present chapter we consider this competition model with periodically varying input nutrient concentration, including pulsed input where the concentration may fall to zero over part of the cycle.
In Section 8.1 we briefly introduce the model and then discuss the well-posedness of the initial–boundary value problem and the positivity of its solutions. Section 8.2 is devoted to the special case of the model system with identical diffusivities and vanishing cell death rates. After consideration of the washout solution, we establish a conservation principle. We then consider single-population growth in the reactor, showing that when the washout solution is linearly stable, then it is globally stable, and when it is unstable, there is a unique, single-population periodic solution that attracts all solutions with nonzero initial data and is asymptotically stable in the linear approximation. Finally, we show that for two competing populations, where each single population periodic solution is unstable to invasion by the other population, we have persistence of both populations and the existence of a positive periodic solution representing coexistence. Section 8.3 is devoted to the perturbed system with different diffusivities and inclusion of cell death rates. We carry over the bulk of the results of Section 8.2 to the case where the random motility coefficients do not differ much from the diffusion constant of the nutrient and the cell death rates are small.
8.1 The Model
The plug-flow reactor may be thought of as a tube, of length L, through which a liquid medium flows with constant (small) velocity v. At the upstream end of the tube, x = 0, the nutrient concentration in the medium is maintained at the periodically varying concentration S 0(t) = S 0(t +ω). Downstream, bacteria consume nutrient, grow, divide, and die or leave the reactor at x = L. Bacteria are assumed to be motile, but their motility is random in the sense that it is modeled by an effective diffusion coefficient and is independent of nutrient concentration (chemotaxis is not considered here). The concentrations of nutrient S and microbial strains u i , i = 1, 2, are governed by the equations (we have scaled variables so that L = 1)
where the d i are the random motility coefficients of strain u i , k i is its death rate, and f i (S) is its uptake and growth rate. The quantity d 0 is the diffusion constant for nutrient S. Since the rate of change of the total nutrient concentration equals the difference between the inflow and outflow rates of the nutrient minus the consumption of the nutrient, we have
On the other hand,
It then follows that
Since the rate of change of the total concentration of species u i is the difference between the natural growth and death rates of the species minus the washout rate of the species, we have
On the other hand,
Thus we get
Consequently, we impose on the model system the boundary conditions
and nonnegative initial conditions
Next we discuss the well-posedness of the initial–boundary value problem (8.1)–(8.3) and the positivity of its solutions. Assume that the initial data in (8.3) satisfy \((S_{0},u_{01},u_{02}) \in X^{+} = C([0,1], \mathbb{R}_{+}^{3})\), the positive cone in the Banach space \(X = C([0,1], \mathbb{R}^{3})\) with uniform norm. For local existence and positivity of solutions in the space X +, we follow [243], where existence and uniqueness and positivity are treated simultaneously, ignoring issues related to time delays treated there. The idea is to consider mild solutions of the system of abstract integral equations (we set u 0 = S and u 00 = S 0 to simplify notation)
where u(t) = (u 0(t), u 1(t), u 2(t)) ≡ (S(⋅ , t), u 1(⋅ , t), u 2(⋅ , t)) ∈ X +. T i (t) is the positive, nonexpansive, analytic semigroup on \(C([0,1], \mathbb{R})\) (see [326, Chapter 8]) such that u = T i (t)u 0i satisfies the linear initial value problem
V (t, s), t > s, is the family of affine operators on \(C([0,1], \mathbb{R})\) such that u = V (t, s)u 00 satisfies the linear system with inhomogeneous, periodic boundary conditions, with start time s, given by
Due to the periodicity of the inhomogeneity in the boundary conditions, S 0(t +ω) = S 0(t), we have that V (t, s) = V (t +ω, s +ω) for t > s. The nonlinear operator \(B_{i}: C([0,1], \mathbb{R}_{+}) \rightarrow C([0,1], \mathbb{R})\) is defined by
The result [243, Theorem 1] can be used to give local existence and positivity of noncontinuable solutions of (8.1)–(8.3), although the elliptic operator in that setting is slightly different. The reason is that the semigroups T i and evolution operator V defined above have the same properties as those in [243] (so [243, Corollary 4] may be applied). Indeed, V (t, s) satisfies \(V (t,s)C([0,1], \mathbb{R}_{+}) \subset C([0,1], \mathbb{R}_{+})\) for t > s, by standard maximum principle arguments, and similarly (see [326, Chapter 8]), \(T_{i}(t)C([0,1], \mathbb{R}_{+}) \subset C([0,1], \mathbb{R}_{+})\) for t > 0. The operator V and semigroup T 0 are related as below (1.9) in [243] on setting γ(x, t) = S 0(t). Since we assume that f i (0) = 0, it follows that B i (u)(x) = 0 whenever u i (x) = 0 for some x; hence, B = (B 0, B 1, B 2) is quasi-positive. Thus, [243, Theorem 1 and Remark 1.1] imply that (8.1)–(8.3) has a unique nonnegative noncontinuable solution that satisfies (8.1)–(8.2) in the classical sense for t > 0.
8.2 Unperturbed Model
Consider the system of equations with identical diffusivities and vanishing cell death rates
with boundary conditions
and initial value conditions
where d 0 > 0, v > 0, and \(S^{0}(\cdot ) \in C^{2}(\mathbb{R}_{+}, \mathbb{R})\), with S 0(t) ≥ 0, S 0(⋅ ) ≢ 0, S 0(t +ω) = S 0(t) for some real number ω > 0, and \(f_{i}(\cdot ) \in C^{2}(\mathbb{R}_{+}, \mathbb{R}_{+})\) satisfies
-
(H)
\(f_{i}(0) = 0,\quad f_{i}'(S) > 0,\,\forall S \in \mathbb{R}_{+},\,i = 1,2\).
Let n be the outward normal to the boundary of (0, 1). Clearly, for any \(\phi (\cdot ) \in C^{1}([0,1], \mathbb{R})\),
Therefore, the boundary condition (8.8) is equivalent to the following one:
Let \(X^{+} = C([0,1], \mathbb{R}_{+}^{3})\). As mentioned in Section 8.1, [243, Theorem 1 and Remark 1.1] imply that for any ϕ = (S 0(⋅ ), u 01(⋅ ), u 02(⋅ )) ∈ X +, there exists a unique (mild) solution (S(x, t, ϕ), u 1(x, t, ϕ), u 2(x, t, ϕ)) of (8.7)-(8.8), defined on its maximal interval of existence [0, σ ϕ ), satisfying
Moreover, (S(x, t, ϕ), u 1(x, t, ϕ), u 2(x, t, ϕ)) is a classical solution of (8.7)–(8.8) for t ∈ (0, σ ϕ ).
8.2.1 Conservation Principle
Since we have scaled the u i in nutrient-equivalent units, the total nutrient W(x, t) = S(x, t) + u 1(x, t) + u 2(x, t) should eventually come into balance with the input S 0(t). Then W(x, t) satisfies the following scalar linear equation
Note that equations (8.7)–(8.8) reduce to (8.11) for W = S when u i = 0, i = 1, 2. In what follows, we use B ϕ = 0 to denote the homogeneous boundary conditions \(d_{0}\frac{\partial \phi (0)} {\partial n} + v\phi (0) = 0\) and \(\frac{\partial \phi (1)} {\partial n} = 0\).
Proposition 8.2.1.
System ( 8.11 ) admits a unique positive ω-periodic solution W ∗ (x,t) > 0, and for any \(W_{0}(\cdot ) \in C([0,1], \mathbb{R})\) , the unique (mild) solution W(x,t) of ( 8.11 ) with W(⋅,0) = W 0 (⋅) satisfies limt→∞ (W(x,t) − W ∗ (x,t)) = 0 uniformly for x ∈ [0,1].
Proof.
Let u(x, t) = W(x, t) − S 0(t) and \(S_{1}(t) = -\frac{dS^{0}(t)} {dt},\,t \geq 0\). Then u(x, t) satisfies
Since the boundary conditions in (8.12) are homogeneous, (8.12) can then be written as an abstract ordinary differential equation in \(C([0,1], \mathbb{R})\) given by
where A is the closure in \(C([0,1], \mathbb{R})\) of A 0 = d 0 ∂∕∂ x 2 − v ∂∕∂ x with
For any \(\phi \in C([0,1], \mathbb{R})\), the mild solution of (8.12) can be expressed as
where T(t) is the analytic semigroup generated by A in \(C([0,1], \mathbb{R})\) (see, e.g., [272] and [326, Chapter 7.1]). It easily follows that u(t) is an ω-periodic solution of (8.13) if and only if u 0 = u(0) satisfies
By an argument similar to that in [326, Section 8.1], it follows that σ = sup{Reλ: λ ∈ σ(A)} < 0. Then the radius of the spectrum of the compact operator T(ω) satisfies r(T(ω)) < 1, and hence (8.13) admits a unique ω-periodic solution u ∗(t). Let v(t) = u(t) − u ∗(t). Then v(t) satisfies
By [272, Theorem 4.4.3], there exist M > 0 and μ > 0 such that \(\left \|T(t)\right \| \leq Me^{-\mu t},t \geq 0\), and hence lim t → ∞ v(t) = 0 in \(C([0,1], \mathbb{R})\). Then lim t → ∞ (u(x, t) − u ∗(x, t)) = 0 uniformly for x ∈ [0, 1].
Let W ∗(x, t) = u ∗(x, t) + S 0(t), x ∈ [0, 1], t ≥ 0. It then follows that W ∗(x, t) is an ω-periodic solution of (8.11), and for any \(W_{0}(\cdot ) \in C([0,1], \mathbb{R})\), the unique (mild) solution W(x, t) of (8.11) with W(⋅ , 0) = W 0(⋅ ) satisfies
For any \(W_{0}(\cdot ) \in C([0,1], \mathbb{R}_{+})\), by [243, Theorem 1 and Remark 1.1], the unique solution W(x, t) of (8.11) with W(⋅ , 0) = W 0(⋅ ) satisfies
It remains to prove that W ∗(x, t) > 0, for all x ∈ [0, 1] and t ≥ 0. For any t ≥ 0, by (8.17) we have lim n → ∞ (W(x, t + n ω) −W ∗(x, t + n ω)) = lim n → ∞ (W(x, t + n ω) −W ∗(x, t)) = 0, uniformly for x ∈ [0, 1]. Then \(W^{{\ast}}(x,t) =\lim _{n\rightarrow \infty }W(x,t + n\omega ) \geq 0,\,\forall x \in [0,1],t \geq 0\). Since S 0(t) ≥ 0, S 0(⋅ ) ≢ 0, there exists t 0 > 0 such that S 0(t 0) > 0. It is easy to see that \(d_{0}\frac{\partial u^{{\ast}}(0,t_{ 0})} {\partial n} + vu^{{\ast}}(0,t_{ 0}) = 0\) implies u ∗(⋅ , t 0) ≢ − S 0(t 0). Then W ∗(⋅ , t 0) = u ∗(⋅ , t 0) + S 0(t 0) ≢ 0. By the standard parabolic maximum principle, it follows that
Then, by the ω-periodicity of W ∗(x, ⋅ ), we have W ∗(x, t) > 0, \(\forall x \in [0,1]\), t ≥ 0. ■
8.2.2 Single Species Growth
If only one microbial species is present in the reactor, we have the single species model
with boundary conditions
and initial conditions
where d 0 > 0, v > 0, \(f(\cdot ) \in C^{2}(\mathbb{R}_{+}, \mathbb{R}_{+})\) satisfies (H) and S 0(⋅ ) is as in (8.8). Let \(Y ^{+} = C([0,1], \mathbb{R}_{+}^{2})\). It then follows that for any ϕ = (S 0(⋅ ), u 0(⋅ )) ∈ Y +, (8.20)–(8.22) admits a unique (mild) solution (S(x, t, ϕ), u(x, t, ϕ)), defined on its maximal interval of existence [0, σ ϕ ), satisfying \(S(x,t,\phi ) \geq 0,u(x,t,\phi ) \geq 0,\,\forall x \in [0,1],t \in [0,\sigma _{\phi })\). By the conservation principle in Subsection 8.2.1, for each ϕ ∈ Y +, σ ϕ = ∞.
We determine stability of periodic solutions in the following way. For any \(m \in C^{1}([0,1] \times \mathbb{R}, \mathbb{R})\) with \(m(x,t+\omega ) = m(x,t),\forall x \in [0,1],\,t \in \mathbb{R}\), let μ(m(⋅ , ⋅ )) be the unique principal eigenvalue of the periodic–parabolic eigenvalue problem (see [152, Section II.14])
The main result of this subsection says that if the washout periodic solution (S, u) = (W ∗, 0) is stable or neutrally stable in the linear approximation then it is globally stable, while if it is unstable then there exists a unique positive periodic solution representing survival of the population to which all other solutions with u 0 ≠ 0 approach asymptotically.
Theorem 8.2.1.
Let W ∗ (x,t) be as in Proposition 8.2.1 .
-
(a)
If μ(f(W ∗ (x,t))) ≥ 0, then for any ϕ = (S 0 (⋅),u 0 (⋅)) ∈ Y +, limt→∞ (S(x,t,ϕ) − W ∗ (x,t)) = 0 and limt→∞ u(x,t,ϕ) = 0 uniformly for x ∈ [0,1];
-
(b)
If μ(f(W ∗ (x,t))) < 0, then (8.20)–(8.21) admits a unique positive ω-periodic solution (S ∗ (x,t),u ∗ (x,t)) and for any ϕ = (S 0 (⋅),u 0 (⋅)) ∈ Y + with u 0 (⋅)≢0, limt→∞ (S(x,t,ϕ) − S ∗ (x,t)) = 0 and limt→∞ (u(x,t,ϕ) − u ∗ (x,t)) = 0 uniformly for x ∈ [0,1]. Moreover, (S ∗ (x,t),u ∗ (x,t)) is linearly asymptotically stable for (8.20)–(8.21).
Proof.
Let \(\hat{f}(\cdot ): \mathbb{R} \rightarrow \mathbb{R}\) be a smooth extension of \(f(\cdot ): \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}\) such that \(\hat{f}(0) = 0,\hat{f}'(s) > 0\), \(\forall s \in \mathbb{R}\), and \(\hat{f}(s) = f(s),\,\forall s \in \mathbb{R}_{+}\). Let W = S + u. Then system (8.20) with (8.21) is equivalent to the system
with boundary conditions
Given ϕ = (S 0(⋅ ), u 0(⋅ )) ∈ Y +, let (W(x, t), u(x, t)) be the unique solution of (8.24)–(8.25) satisfying (W(x, 0), u(x, 0)) = (S 0(x) + u 0(x), u 0(x)), x ∈ [0, 1]. Then U(x, t) = u(x, t +ω), x ∈ [0, 1], t ≥ 0, satisfies the nonautonomous scalar parabolic equation
By the conservation principle, lim t → ∞ (W(x, t) − W ∗(x, t)) = 0 uniformly for x ∈ [0, 1], and hence (8.26) is asymptotic to the following periodic scalar parabolic equation
Let 1 < p < ∞, and let X 0 = L p(0, 1) and X 1 = W p, B 2(0, 1). For β ∈ (1∕2 + 1∕(2p), 1), let X β be the fractional power space of X 0 with respect to (A 0, B) (see, e.g., [150]). Then X 1 ⊂ X β ⊂ X 0 and X β ↪ C 1+λ[0, 1] for some λ > 0. Clearly, U(⋅ , 0) = u(⋅ , ω) ∈ X 1 ⊂ X β. By Theorem 3.2.2, it follows that
-
(a)
If μ(f(W ∗(x, t))) ≥ 0, lim t → ∞ U(x, t) = 0, and hence lim t → ∞ u(x, t) = 0, uniformly for x ∈ [0, 1];
-
(b)
If μ(f(W ∗(x, t))) < 0, (8.27) admits a unique positive ω-periodic solution u ∗(x, t) and lim t → ∞ (U(x, t) − u ∗(x, t)) = 0, and hence lim t → ∞ (u(x, t) − u ∗(x, t)) = lim t → ∞ (U(x, t −ω) − u ∗(x, t −ω)) = 0, uniformly for x ∈ [0, 1].
In case (a), lim t → ∞ (S(x, t) − W ∗(x, t)) = lim t → ∞ [(W(x, t) − W ∗(x, t)) − u (x, t)] = 0 uniformly for x ∈ [0, 1]; In case (b), let S ∗(x, t) = W ∗(x, t) − u ∗ (x, t). Then lim t → ∞ (S(x, t) −S ∗(x, t)) = lim t → ∞ [(W(x, t) −W ∗(x, t)) − (u(x, t) −u ∗(x, t))] = 0uniformly for x ∈ [0, 1]. We further claim that (S ∗(x, t), u ∗(x, t)) is a positive ω-periodic solution of (8.20)–(8.21). It then suffices to prove that \(W^{{\ast}}(x,t) > u^{{\ast}}(x,t),\,\forall x \in [0,1],t \geq 0\). Since \(d_{0}\frac{\partial W^{{\ast}}(0,t)} {\partial n} + vW^{{\ast}}(x,t) = vS^{0}(t) \geq 0\) and \(\frac{\partial W^{{\ast}}(1,t)} {\partial n} = 0,t > 0\), W ∗(x, t) is an upper solution of (8.27). Let u 0(x, t) be the unique solution of (8.27) with u 0(⋅ , 0) = W ∗(⋅ , 0). Then \(u_{0}(x,t) \leq W^{{\ast}}(x,t),\,\forall x \in [0,1],\,t \geq 0\). It then follows that
Since lim t → ∞ (u 0(x, t) − u ∗(x, t)) = 0 uniformly for x ∈ [0, 1], letting n → ∞ in (8.28), we have
Let t 0 > 0 be such that S 0(t 0) > 0. Clearly, the boundary conditions for W ∗(x, t) and u ∗(x, t) in (8.11) and (8.27) imply that u ∗(⋅ , t 0) ≢ W ∗(⋅ , t 0). Then, by the parabolic maximum principle, we get
and hence by the ω-periodicity of u ∗(x, t) and W ∗(x, t),
Let P: Y + → Y + be the Poincaré map associated with (8.20)–(8.21); that is, \(P(\phi ) = (S(\cdot,\omega,\phi ),u(\cdot,\omega,\phi )),\,\forall \phi = (S_{0}(\cdot ),u_{0}(\cdot )) \in Y ^{+}\). Let ϕ 0 = (S ∗(⋅ , 0), u ∗(⋅ , 0)). Clearly, P(ϕ 0) = ϕ 0. It remains to prove the linear asymptotic stability of the positive periodic solution (S ∗(x, t), u ∗(x, t)) in the sense that r(D ϕ P(ϕ 0)) < 1. Let \(\bar{S} = S - S^{{\ast}},\bar{u} = u - u^{{\ast}}\). We then get the linearization of (8.20)–(8.21) at (S ∗(x, t), u ∗(x, t)) given by
with homogeneous boundary conditions
Let \(\overline{U}(t,s),\,t \geq s \geq 0\), be the evolution operator of linear system (8.29)–(8.30). It easily follows that \(D_{\phi }P(\phi _{0}) = \overline{U}(\omega,0)\). Under the change of variables \(w =\bar{ S} +\bar{ u},z =\bar{ u}\), that is,
(8.29)–(8.30) is then transformed into the system
with boundary conditions
Let U 1(t, s), t ≥ s ≥ 0, be the evolution operator of the linear equation
and let U 2(t, s), t ≥ s ≥ 0, be the evolution operator of the periodic linear equation
Then
is the evolution operator of periodic linear system (8.31)–(8.32). In particular,
As claimed in Subsection 8.2.1, r(U 1(ω, 0)) < 1. By the definition of principal eigenvalue (see [152, Proposition 14.4]), we have
Since (S ∗(x, t), u ∗(x, t)) is an ω-periodic solution of (8.20)–(8.21), u ∗(x, t) satisfies the periodic linear equation
Then, by the uniqueness of the principal eigenvalue, we have μ(f(S ∗(x, t))) = 0. Since f(S ∗(x, t)) − u ∗(x, t)f′(S ∗(x, t)) < f(S ∗(x, t)), by the monotonicity of the principal eigenvalue ([152, Lemma 15.5]),
Therefore, r(U 2(ω, 0)) < 1. Clearly, \(U(\omega,0): Y = C([0,1],R) \times C([0,1], \mathbb{R}^{2}) \rightarrow Y\) is a compact and positive operator. We further claim that r(U(ω, 0)) < 1. Indeed, let α = r(U(ω, 0)). If α = 0, obviously we have r(U(ω, 0)) < 1. In the case where α > 0, by the Krein–Rutman theorem (see, e.g., [152, Theorem 7.1]), there exists \(\phi = \left (\begin{array}{*{10}c} \phi _{1}\\ \phi _{ 2} \end{array} \right ) > 0\) in Y such that
Then U 1(ω, 0)ϕ 1 = α ϕ 1. If ϕ 1 > 0, then α = r(U 1(ω, 0)) < 1. If ϕ 1 = 0, then ϕ 2 > 0 and U 2(ω, 0)ϕ 2 = α ϕ 2, and hence, α = r(U 2(ω, 0)) < 1. Clearly,
It then follows that \(r(D_{\phi }P(\phi _{0})) = r(\overline{U}(\omega,0)) = r(U(\omega,0)) < 1\). ■
8.2.3 Two-Species Competition
For any ϕ = (S 0(⋅ ), u 01(⋅ ), u 02(⋅ )) ∈ X +, let
be the unique (mild) solution of (8.7)–(8.8) with Φ(⋅ , 0, ϕ) = ϕ. Then \(S(x,t) \geq 0,u_{i}(x,t) \geq 0,\,\forall x \in [0,1],\,t \in [0,\sigma _{\phi }),\,i = 1,2\). By the conservation principle, for each ϕ ∈ Y +, σ ϕ = ∞.
In the case where μ(f i (W ∗(x, t))) < 0, i = 1, 2, according to Theorem 8.2.1, let (S i ∗(x, t), u i ∗(x, t)) be the unique positive ω-periodic solution of (8.20)–(8.21) with f(⋅ ) = f i (⋅ ), i = 1, 2, respectively. As shown in the proof of Theorem 8.2.1, for each 1 ≤ i ≤ 2,
and u i ∗(x, t) is the unique positive ω-periodic solution of the periodic-parabolic equation
We now show that if each population can survive in the bioreactor in the absence of competition and if each population can invade the other’s single-population periodic solution, then there exist two, not necessarily distinct, positive periodic solutions, each representing coexistence of the two populations, and system (8.7)–(8.8) is uniformly persistent.
Theorem 8.2.2.
Assume that
-
(1)
\(\mu (f_{i}(W^{{\ast}}(x,t))) < 0,\,\forall i = 1,2\);
-
(2)
μ(f 1 (S 2 ∗ (x,t))) < 0 and μ(f 2 (S 1 ∗ (x,t))) < 0.
Then system (8.7)–(8.8) admits two positive ω-periodic solutions \((\bar{S}_{1}^{{\ast}}(x,t),\bar{u}_{1{\ast}}(x,t)\) , \(\bar{u}_{2}^{{\ast}}(x,t))\) and \((\bar{S}_{2}^{{\ast}}(x,t),\bar{u}_{1}^{{\ast}}(x,t),\bar{u}_{2{\ast}}(x,t))\) with
and for any ϕ = (S 0 (⋅),u 01 (⋅),u 02 (⋅)) ∈ X + with \(u_{0i}(\cdot )\not\equiv 0,\,\forall i = 1,2\) , Φ(x,t,ϕ) = (S(x,t),u 1 (x,t),u 2 (x,t)) satisfies
Proof.
For each 1 ≤ i ≤ 2, let \(\hat{f}_{i}(\cdot ): \mathbb{R} \rightarrow \mathbb{R}\) be a smooth extension of \(f_{i}(\cdot ): \mathbb{R}_{+} \rightarrow \mathbb{R}\) such that \(\hat{f}_{i}(0) = 0,\,\hat{f}_{i}'(s) > 0,\,\forall s \in \mathbb{R},\,\mbox{ and}\,\hat{f}_{i}(s) = f_{i}(s),\,\forall s \in \mathbb{R}_{+}\). Let W = S + u 1 + u 2. Then system (8.7) with (8.8) is equivalent to the system
with boundary conditions
Given ϕ = (S 0(⋅ ), u 01(⋅ ), u 02(⋅ )) ∈ X +, let
and let
Then (U 1(x, t), U 2(x, t)) satisfies the following nonautonomous parabolic system
By the conservation principle, lim t → ∞ (W(x, t) − W ∗(x, t)) = 0 uniformly for x ∈ [0, 1], and hence (8.40) is asymptotic to the following periodic-parabolic system
Let X β be as in the proof of Theorem 8.2.1, let Z = X β × X β, and let Z + be the usual positive cone of Z. Since (U 1(⋅ , 0), U 2(⋅ , 0)) = (u 1(⋅ , ω), u 2(⋅ , ω)) ∈ Z, we consider systems (8.40) and (8.41) with initial values in Z +. Let Δ = { (t, s): 0 ≤ s ≤ t < ∞}. Define \(\tilde{\varPhi }:\varDelta \times Z^{+} \rightarrow Z^{+}\) by \(\tilde{\varPhi }(t,s,\psi ) =\tilde{ u}(\cdot,t,s,\psi ),t \geq s \geq 0,\psi \in Z^{+}\), where \(\tilde{u}(x,t,s,\psi ) = (\tilde{u}_{1}(x,t,s,\psi ),\tilde{u}_{2}(x,t,s,\psi ))\) is the unique solution of (8.40) with \(\tilde{u}(\cdot,s,s,\psi ) =\psi\). Define T n : Z + → Z +, n ≥ 0, by \(T_{n}(\psi ) =\tilde{\varPhi } (n\omega,0,\psi ),\psi \in Z^{+}\). Let T(t): Z + → Z +, t ≥ 0, be the periodic semiflow generated by periodic system (8.41), i.e., T(t)ψ = u(⋅ , t, ψ), where u(x, t, ψ) is the unique solution of (8.41) with u(⋅ , 0, ψ) = ψ. Clearly, Q = T(ω): Z + → Z + is the Poincaré map associated with the periodic system (8.41). Then, by Proposition 3.2.1, \(\tilde{\varPhi }:\varDelta \times Z^{+} \rightarrow Z^{+}\) is an asymptotically periodic semiflow with limit ω-periodic semiflow T(t): Z + → Z +, t ≥ 0, and hence T n : Z + → Z +, n ≥ 0, is an asymptotically autonomous discrete process with limit Q: Z + → Z +. Moreover, for any ψ ∈ Z +, γ +(ψ) = { T n (ψ): n ≥ 0} is precompact in Z +. Let (Z, P) be the ordered Banach space with the positive cone P = X + β × (−X + β), where X + β is the usual positive cone of X β, and denote its order by ≤ P . It then follows that T(t): Z + → Z +, t ≥ 0, is monotone with respect to ≤ P in the sense that if ϕ, ψ ∈ Z + with ϕ ≤ P ψ, then \(T(t)\phi \leq _{P}T(t)\psi,\,\forall t \geq 0\).
Clearly, condition (2) implies μ(f 1(W ∗(x, t) − u 2 ∗(x, t))) = μ(f 1(S 2 ∗(x, t))) < 0 and μ(f 2(W ∗(x, t) − u 1 ∗(x, t))) = μ(f 2(S 1 ∗(x, t))) < 0. By Theorem 2.4.2, as applied to the Poincaré map associated with (8.41), or an argument similar to that in [152, Theorem 33.3], it then follows that (8.41) admits two positive ω-periodic solutions \((\bar{u}_{1{\ast}}(x,t),\bar{u}_{2}^{{\ast}}(x,t))\) and \((\bar{u}_{1}^{{\ast}}(x,t),\bar{u}_{2{\ast}}(x,t))\) with
such that the compressive dynamics stated in Theorem 8.2.2 holds for (8.41) on Z +. Let \(E_{1}^{{\ast}} = (\bar{u}_{1{\ast}}(\cdot,0),\bar{u}_{2}^{{\ast}}(\cdot,0))\) and \(E_{2}^{{\ast}} = (\bar{u}_{1}^{{\ast}}(\cdot,0),\bar{u}_{2{\ast}}(\cdot,0))\). Clearly, Q(E i ∗) = E i ∗, i = 1, 2. Let Z 0: = { (ϕ 1, ϕ 2) ∈ Z +: ϕ i (⋅ ) ≢ 0, i = 1, 2} and ∂ Z 0: = Z + ∖ Z 0. Clearly, Q: Z 0 → Z 0, and Q: ∂ Z 0 → ∂ Z 0. It then follows that Q: Z 0 → Z 0 admits a global attractor A 0 ⊂ [E 1 ∗, E 2 ∗] P . Let M 0 = (0, 0), M 1 = (u 1 ∗(⋅ , 0), 0), and M 2 = (0, u 2 ∗(⋅ , 0)). It is easy to see that ∪ i = 0 2 M i is an isolated and acyclic covering of \(\cup _{\phi \in \partial Z_{0}}\omega (\phi )\) for Q: ∂ Z 0 → ∂ Z 0. By our assumptions and Proposition 3.2.3, we have \(\tilde{W}^{s}(M_{i}) \cap Z_{0} =\emptyset,\ \forall i = 0,1,2\), where \(\tilde{W}^{s}(M_{i})\) is the stable set of M i with respect to T n : Z + → Z +, n ≥ 0. By Lemma 1.2.2, every ω-limit set ω(ϕ) of γ +(ϕ) = { T n (ϕ): n ≥ 0} is internally chain transitive for Q: Z + → Z +. By Theorem 1.2.1, it then follows that \(\omega (\psi ) \subset A_{0},\,\forall \psi \in Z_{0}\). By Theorem 3.2.1, \(\lim _{t\rightarrow \infty }d(\tilde{u}(\cdot,t,0,\psi ),T(t)A_{0}) = 0\). Since E 1 ∗ ≤ P A 0 ≤ P E 2 ∗, by the monotonicity of T(t): Z + → Z +, t ≥ 0, we have
Note that
For any ϕ = (S 0(⋅ ), u 01(⋅ ), u 02(⋅ )) ∈ X + with u 0i (⋅ ) ≢ 0, \(\forall i = 1,2\), since (U 1(⋅ , 0), U 2(⋅ , 0)) ∈ Z 0, we have
uniformly for x ∈ [0, 1].
Let \(\bar{S}_{1}^{{\ast}}(x,t) = W^{{\ast}}(x,t) -\bar{ u}_{1{\ast}}(x,t) -\bar{ u}_{2}^{{\ast}}(x,t)\) and \(\bar{S}_{2}^{{\ast}}(x,t) =\) \(W^{{\ast}}(x,t) -\bar{ u}_{1}^{{\ast}}(x,t) -\bar{ u}_{2{\ast}}(x,t)\). We need to confirm that
are two positive ω-periodic solutions of (8.7)–(8.8). It suffices to prove that
for all x ∈ [0, 1] and t ≥ 0. Since u 1 ∗(⋅ , 0) ≪ P W ∗(⋅ , 0), we can choose ψ 0 = (ψ 1 0, ψ 2 0) ∈ Z 0 such that
Let (u 1 0(x, t), u 2 0(x, t)) be the unique solution of (8.41) with (u 1 0(⋅ , 0), u 2 0(⋅ , 0)) = ψ 0, and let \(\bar{f}(s) =\max \{\hat{ f}_{1}(s),\hat{f}_{2}(s)\},\,\forall s \in \mathbb{R}\). Then
satisfies \(V (x,0) \leq W^{{\ast}}(x,0),\,\forall x \in [0,1]\), and
Note that W ∗(x, t) satisfies
for 0 < x < 1 and t > 0. By the standard comparison theorem, it follows that
By Theorem 2.4.2, Q n(ψ 0) = (u 1 0(⋅ , n ω), u 2 0(⋅ , n ω)) → E 2 ∗ as n → ∞, and hence
uniformly for x ∈ [0, 1]. By (8.45), we have
for all x ∈ [0, 1] and t ≥ 0. Letting n → ∞ in (8.46), we get
Since \(V ^{{\ast}} =\bar{ u}_{1}^{{\ast}}(x,t) +\bar{ u}_{2{\ast}}(x,t)\) satisfies (8.43) and W ∗(x, t) satisfies (8.11), as argued in the proof that W ∗(x, t) > u ∗(x, t) in Subsection 8.2.2, we further have \(\bar{u}_{1}^{{\ast}}(x,t) +\bar{ u}_{2{\ast}}(x,t) < W^{{\ast}}(x,t),\,\forall x \in [0,1],\,t \geq 0\). Similarly, we can prove that \(\bar{u}_{1{\ast}}(x,t) +\bar{ u}_{2}^{{\ast}}(x,t) < W^{{\ast}}(x,t),\,\forall x \in [0,1],\,t \geq 0\). ■
8.3 Perturbed Model
In order to apply abstract perturbation-type results to periodic systems with parameters, we first consider the weak repellers uniform in parameters and the continuity of solutions on parameters uniformly for initial values.
8.3.1 Periodic Systems with Parameters
Let Λ be a subset of \(\mathbb{R}^{l}\). We consider the periodic-parabolic system with parameter (E λ ):
where \(u = (u_{1},\mathop{\ldots },u_{m}) \in \mathbb{R}^{m},\lambda \in \varLambda\), \(B_{i}u_{i} = \frac{\partial u_{i}} {\partial n} +\alpha _{i}u_{i},\,\alpha _{i} \geq 0\), A i (λ) are uniform elliptic operators with coefficients continuous in (x, λ), F i are smooth functions, and for some real number \(\omega > 0,\,F_{i}(x,t+\omega,u,\lambda ) = F_{i}(x,t,u,\lambda ),\,\forall 1 \leq i \leq m\). We assume that for any \(\phi = (\phi _{1},\mathop{\ldots },\phi _{m}) \in C^{+} = C(\overline{\varOmega },R_{+}^{m})\), the unique (mild) solution u(x, t, ϕ, λ) of (E λ ) with u(⋅ , 0, ϕ, λ) = ϕ exists globally on [0, ∞) and \(u_{i}(x,t,\phi,\lambda ) \geq 0,\,\forall x \in \overline{\varOmega },\,t \geq 0,\,1 \leq i \leq m\).
For each 1 ≤ i ≤ m and any \(m \in C^{1}([0,1] \times \mathbb{R}, \mathbb{R})\) with \(m(x,t+\omega ) = m(x,t),\,\forall x \in [0,1],\,t \in \mathbb{R}\), let μ(A i (λ), m(⋅ , ⋅ )) be the unique principal eigenvalue of the periodic–parabolic eigenvalue problem (see [152, Chapter II])
Then we have the following result on the uniform weak repeller .
Proposition 8.3.1.
Let λ 0 ∈Λ be fixed. Assume that there exists some 1 ≤ i ≤ m such that F i (x,t,u,λ) = u i G i (x,t,u,λ), and (E λ ) admits a nonnegative periodic solution
with μ(A i (λ 0 ),G i (x,t,u 0 ∗ (x,t),λ 0 )) < 0. Then there exist η > 0 and δ > 0 such that for any |λ −λ 0 | < δ and any ϕ ∈ C + with ϕ i (⋅)≢0, we have
Proof.
Let M = u 0 ∗(⋅ , 0) and let B(M, r) denote the open ball in \(C = C(\overline{\varOmega },R^{m})\) centered at the point M and with radius r. By the definition of the principal eigenvalue in [152, Proposition 14.4] and the continuous dependence of evolution operators on parameters (see, e.g., [13] and [89, Section III.11]), we have
Then there exists δ 0 > 0 such that for any | λ −λ 0 | < δ 0,
Let \(\epsilon _{0} = -\frac{1} {2}\mu \left (A_{i}(\lambda _{0}),G_{i}(x,t,u_{0}^{{\ast}}(x,t),\lambda _{ 0})\right )\). Then for any | λ −λ 0 | < δ 0,
Let \(r =\max _{x\in \overline{\varOmega },\,t\in [0,\omega ]}\vert u_{0}^{{\ast}}(x,t)\vert + 1\). Therefore, the uniform continuity of G i (x, t, u, λ) on the compact set \(\overline{\varOmega } \times [0,\omega ] \times \overline{B(0,r)} \times \overline{B(\lambda _{0},\delta _{0})}\) implies that there exist δ 1 ∈ (0, δ 0) and η 1 ∈ (0, 1) such that for any \(u,v \in \overline{B(0,r)}\) with | u − v | < η 1 and | λ −λ 0 | < δ 1,
Since \(\lim _{(\phi,\lambda )\rightarrow (M,\lambda _{0})}u(\cdot,t,\phi,\lambda ) = u(\cdot,t,M,\lambda _{0}) = u_{0}^{{\ast}}(\cdot,t)\) in C uniformly for t ∈ [0, ω], there exist δ 2 ∈ (0, δ 1) and η 2 > 0 such that for any ϕ ∈ B(M, η 2) ⊂ C, | λ −λ 0 | < δ 2,
We claim that for any | λ −λ 0 | < δ 2 and ϕ ∈ B(M, η 2) ∩ C + with ϕ i (⋅ ) ≢ 0, there exists n 0 = n 0(λ, ϕ) ≥ 1 such that
Assume, by contradiction, that there exist ϕ 0 ∈ B(M, η 2) ∩ C + with ϕ 0i (⋅ ) ≢ 0 and | λ 1 −λ 0 | < δ 2 such that for all n ≥ 1,
For any t ≥ 0, let t = n ω + t′, where t′ ∈ [0, ω) and n = [t∕ω] is the greatest integer less than or equal to t∕ω. Then we have
for all \(x \in \overline{\varOmega }\), and hence
for all t ≥ 0 and \(x \in \overline{\varOmega }\). Therefore, by (8.49) and the ω-periodicity of G i (x, t, u, λ 1) with respect to t,
Let ψ i (x, t) be a positive eigenfunction corresponding to the principal eigenvalue μ = μ(A i (λ 1), G i (x, t, u 0 ∗(x, t), λ 0)); that is, ψ i (x, t) satisfies
Then ψ(⋅ , 0) ≫ 0 in \(C(\overline{\varOmega }, \mathbb{R})\). Let
Since ϕ 0i (⋅ ) > 0 in \(C(\overline{\varOmega }, \mathbb{R})\), by applying the parabolic maximum principle to the ith component of \((E_{\lambda _{1}})\), we have u i (⋅ , t, ϕ 0, λ 1) ≫ 0 in \(C(\overline{\varOmega }, \mathbb{R})\) for all t > 0. Let u i (x, t) = u i (x, t +ω, ϕ 0, λ 1). Then u i (⋅ , 0) = u i (⋅ , ω, ϕ 0, λ 1) ≫ 0 in \(C(\overline{\varOmega }, \mathbb{R})\), and hence there exists k > 0 such that u i (⋅ , 0) ≥ k ψ i (⋅ , 0). Therefore, by (8.53), u i (x, t) satisfies
By (8.54), it easily follows that \(v(x,t) = ke^{(-\mu -\epsilon _{0})t}\psi _{i}(x,t)\) satisfies
By (8.55), (8.56), and the standard comparison theorem, we get
Then lim t → ∞ u i (x, t) = ∞ for any x ∈ Ω, which contradicts (8.51). It follows that for any | λ −λ 0 | < δ 2 and any ϕ ∈ C + with ϕ i (⋅ ) ≢ 0, limsup n → ∞ d(u(⋅ , n ω, ϕ, λ), M) ≥ η 2. ■
By the continuous dependence of the evolution operator on parameters (see, e.g., [13] and [89, Section III.11]), the variation of constants formula, and a generalized Gronwall’s inequality argument (see, e.g., [152, Lemma 19.4] and the proof of Proposition 3.2.1), we can prove the following result on the continuity of solutions on parameters uniformly for initial values.
Proposition 8.3.2.
Assume that solutions of (E λ ) are uniformly bounded uniformly for λ ∈Λ; that is, for any r > 0, there exists B = B(r) > 0 such that for any ϕ ∈ C + with \(\left \|\phi \right \| \leq r\), \(\left \|u(\cdot,t,\phi,\lambda )\right \| \leq B(r)\) , \(\forall t \geq 0,\,\lambda \in \varLambda\) . Then for any λ 0 ∈Λ and any integer k > 0,
uniformly for t ∈ [ω,kω] and ϕ in any bounded subset of C +.
8.3.2 Single Species Growth
Consider the single species growth model with not necessarily identical diffusivities and nonvanishing cell death rate
with boundary conditions
where d 0 > 0, d > 0, v > 0, and k ≥ 0, and S 0(⋅ ) and f(⋅ ) are as in (8.20)–(8.21). Let \(Y ^{+} = C([0,1], \mathbb{R}_{+}^{2})\). Let d 0 > 0 and v > 0 be fixed and let λ = (d, k), d > 0, k ≥ 0. As argued in Section 8.1, [243, Theorem 1 and Remark 1.1] imply that for any ϕ = (S 0(⋅ ), u 0(⋅ )) ∈ Y +, (8.57)–(8.58) has a unique solution \(\left (S(x,t,\phi,\lambda ),u(x,t,\phi,\lambda )\right )\), defined on its maximal interval of existence [0, σ ϕ ), satisfying \(\left (S(\cdot,0,\phi,\lambda ),u(\cdot,0,\phi,\lambda )\right ) =\phi\). Moreover,
We further have the following result.
Lemma 8.3.1.
Let \(\varLambda =\{ (d,k):\, \frac{d_{0}} {2} \leq d \leq 2d_{0},\,k \geq 0\}\) . Then for each λ ∈Λ, ϕ ∈ Y + , (S(x,t,ϕ,λ),u(x,t,ϕ,λ)) exists globally on [0,∞), and solutions of (8.57)–(8.58) are uniformly bounded and ultimately bounded uniformly for λ ∈Λ.
Proof.
Given ϕ = (S 0(⋅ ), u 0(⋅ )) ∈ Y +, for convenience, let
Then S(x, t) satisfies
By the parabolic comparison theorem, we have
where \(\bar{S}(x,t)\) is the unique solution of (8.11) with \(\bar{S}(x,0) = S(x,0)\). By Proposition 8.2.1, \(\bar{S}(x,t)\) exists globally on [0, ∞) and \(\lim _{t\rightarrow \infty }(\bar{S}(x,t) - W^{{\ast}}(x,t)) = 0\) uniformly for x ∈ [0, 1].
Let μ be the unique positive solution to equation \(\tan \mu = \frac{v} {2\mu d_{0}}\) on the interval \([0, \frac{\pi } {2})\). Clearly, sin(μ x) ≥ 0, cos(μ x) > 0, x ∈ [0, 1]. Then for any λ ∈ Λ, by using (8.57) and (8.58) and integration by parts, we have
and
Let \(y(t) =\int _{ 0}^{1}(S(x,t) + u(x,t))\cos (\mu x)dx,\,\forall t \in [0,\sigma _{\phi })\). Then we get
By the standard comparison theorem for ordinary differential equations, it then follows that for all t ∈ [0, σ ϕ ),
where y ∗(t) is the unique positive ω-periodic solution of linear ordinary differential equations
Since S(x, t) ≥ 0, u(x, t) ≥ 0, and \(\cos (\mu x) \geq \min _{x\in [0,1]}\cos (\mu x) = m > 0,\,\forall x \in [0,1]\), (8.63) implies that for all t ∈ [0, σ ϕ ),
Then u(⋅ , t) is L 1-bounded on [0, σ ϕ ). By (8.60), (8.64), and an argument similar to that in [6, Theorem 3.1], [186, Lemma 3.13], and [215, Proposition 2.4 and Theorem 2.5], it follows that for each ϕ ∈ Y +, (S(x, t, ϕ, λ), u(x, t, ϕ, λ)) is L ∞-bounded, and hence σ ϕ = ∞, and solutions of (8.57)–(8.58) are uniformly L ∞-bounded and ultimately L ∞-bounded uniformly for λ ∈ Λ. ■
Now we show that the hypothesis of Theorem 8.2.1(b) for the unperturbed system implies the existence of a globally attracting single-population periodic solution for the perturbed system at least when the perturbation is small.
Theorem 8.3.1.
Let λ = (d,k), λ 0 = (d 0 ,0), and W ∗ (x,t) and μ(m(⋅,⋅)) be as in Section 8.2. Assume that μ(f(W ∗ (x,t))) < 0 and let (S ∗ (x,t),u ∗ (x,t)) be as in Theorem 8.2.1 . Then there exists δ 0 > 0 such that for any |λ −λ 0 | < δ 0, (8.57)–(8.58) admits a unique positive ω-periodic solution (S ∗ (x,t,λ),u ∗ (x,t,λ)) with
and such that the map λ → (S ∗ (⋅,⋅,λ),u ∗ (⋅,⋅,λ)) is continuous. Moreover, for any (S 0 (⋅),u 0 (⋅)) ∈ Y + with u 0 (⋅)≢0,
uniformly for x ∈ [0,1].
Proof.
Let k 0 > 0 be given and let \(\varLambda _{0} =\{ (d,k):\, \frac{d_{0}} {2} \leq d \leq 2d_{0},0 \leq k \leq k_{0}\}\). For any λ ∈ Λ 0, let S λ = S(λ, ⋅ ): Y + → Y + be the Poincaré map associated with (8.57)–(8.58); that is, S(λ, ϕ) = (S(⋅ , ω, ϕ, λ), u(⋅ , ω, ϕ, λ)), ϕ ∈ Y +. Then S(⋅ , ⋅ ): Λ 0 × Y + → Y + is continuous. By Lemma 8.3.1, it follows that for each λ ∈ Λ 0, S λ : Y + → Y + is compact and point dissipative uniformly for λ ∈ Λ 0; that is, there exists a bounded and closed subset B 0 of Y +, independent of λ ∈ Λ 0, such that for any ϕ ∈ Y +, λ ∈ Λ 0, there exists N = N(ϕ, λ) such that S λ n(ϕ) ∈ B 0 for all n ≥ N. Then, by Theorem 1.1.3, for each λ ∈ Λ 0, there exists a global attractor A λ for S λ : Y + → Y +. Clearly, A λ ⊂ B 0. By a change of variables
the boundary conditions (8.58) then become the homogeneous ones
which is independent of parameter λ. By Lemma 8.3.1 and Proposition 8.3.2, when applied to the resulting system with parameter λ under the above change of variables and the above boundary conditions, it then follows that S(⋅ , ϕ): Λ 0 → Y + is continuous uniformly for ϕ in any bounded subset of Y +. We further have the following claim:
Claim.
For any bounded subset B ⊂ Y +, \(\overline{ \cup _{\lambda \in \varLambda _{0}}S_{\lambda }(B)}\) is compact in Y +.
Indeed, for any sequence {ψ n } in \(\cup _{\lambda \in \varLambda _{0}}S_{\lambda }(B)\), we have \(\psi _{n} = S_{\lambda _{n}}(\phi _{n}),\,\lambda _{n} \in \varLambda _{0},\phi _{n} \in B,n \geq 0\). By the compactness of Λ 0, without loss of generality we can assume that for some λ 1 ∈ Λ 0, λ n → λ 1 as n → ∞. Since \(S_{\lambda _{1}}(B)\) is precompact, there exist ψ 0 ∈ Y + and a subsequence n k → ∞ such that \(S_{\lambda _{1}}(\phi _{n_{k}}) \rightarrow \psi _{0}\) as k → ∞. Combining the continuity of S(⋅ , ϕ): Λ 0 → Y + uniformly for ϕ ∈ B and the inequality
we get \(\psi _{n_{k}} \rightarrow \psi _{0},\,k \rightarrow \infty \). Therefore, \(\cup _{\lambda \in \varLambda _{0}}S_{\lambda }(B)\) is precompact.
Let
Then S λ : Y 0 → Y 0 and S λ : ∂ Y 0 → ∂ Y 0. Let ϕ 0 = (S ∗(⋅ , 0), u ∗(⋅ , 0)). Then \(S_{\lambda _{0}}(\phi _{0}) =\phi _{0}\). By Theorem 8.2.1, r(D ϕ S(λ 0, ϕ 0)) < 1, and \(\lim _{n\rightarrow \infty }S_{\lambda _{0}}^{n}\phi =\phi _{0}\) for every ϕ ∈ Y 0. For each λ ∈ Λ 0, by Proposition 8.2.1,
Clearly, M = (W ∗(⋅ , 0), 0) is a global attractor for S λ : ∂ Y 0 → ∂ Y 0. Note that (W ∗(x, t), 0) is a nonnegative ω-periodic solution of (8.57)–(8.58) and μ(f(W ∗(x, t))) < 0. By a change of variables
and Proposition 8.3.1, as applied to the resulting system, it then follows that there exist δ 1 > 0 and η 1 > 0 such that for any | λ −λ 0 | < δ 1 and any ϕ ∈ Y 0, we have
which implies that M is isolated for S λ : Y + → Y +, and W λ s(M) ∩ Y 0 = ∅, | λ −λ 0 | < δ 1, where W λ s(M) is the stable set of M with respect to S λ . By Theorem 1.3.1, S λ is uniformly persistent with respect to (Y 0, ∂ Y 0) for each | λ −λ 0 | < δ 1. Therefore, there exists a global attractor A λ 0 ⊂ Y 0 for S λ : Y 0 → Y 0 (see, e.g., Theorem 1.3.6). Clearly, A λ 0 ⊂ B 0, and \(\overline{ \cup _{\lambda \in \varLambda _{0},\phi \in Y ^{+}}\omega _{\lambda }(\phi )} \subset \overline{\cup _{\lambda \in \varLambda _{0}}S_{\lambda }(B_{0})}\). Then by the previous claim, \(\overline{ \cup _{\lambda \in \varLambda _{0},\phi \in Y ^{+}}\omega _{\lambda }(\phi )}\) is compact. By Theorem 1.4.2, it follows that there exist δ 2 > 0 and η 2 > 0 such that for any | λ −λ 0 | ≤ δ 2, ϕ ∈ Y 0, \(\liminf _{n\rightarrow \infty }d(S_{\lambda }^{n}\phi,\partial Y _{0}) \geq \eta _{2}\). Then there exists a bounded and closed subset B 0 ∗ of Y 0 such that A λ 0 ⊂ B 0 ∗ for all | λ −λ 0 | ≤ δ 2. Let \(\varLambda _{1} =\varLambda _{0} \cap \overline{B(\lambda _{0},\delta _{2})}\), where B(λ 0, δ 2) = {λ: | λ −λ 0 | < δ 2}. Since \(\overline{ \cup _{\lambda \in \varLambda _{1}}S_{\lambda }(A_{\lambda }^{0})} \subset \overline{\cup _{\lambda \in \varLambda _{1}}S_{\lambda }(B_{0}^{{\ast}})}\), by the previous claim \(\overline{ \cup _{\lambda \in \varLambda _{1}}S_{\lambda }(A_{\lambda }^{0})}\) is compact. Moreover, \(\overline{ \cup _{\lambda \in \varLambda _{1}}S_{\lambda }(A_{\lambda }^{0})} = \overline{ \cup _{\lambda \in \varLambda _{1}}A_{\lambda }^{0}} \subset \overline{B_{0}^{{\ast}}} = B_{0}^{{\ast}}\subset Y _{0}\). By applying Theorem 1.4.1 on the perturbation of a globally stable fixed point to S λ (⋅ ): Y + → Y + with U = Y 0 and B λ = A λ 0, λ ∈ Λ 1, we complete the proof. ■
8.3.3 Two-Species Competition
Consider two-species competition with unequal diffusivities and nonvanishing cell death rates
with boundary conditions
where d 0 > 0, v > 0, d i > 0, and k i ≥ 0, and S 0(⋅ ) and f i (⋅ ), i = 1, 2, are as in (8.7)–(8.8). Let \(X^{+} = C([0,1], \mathbb{R}_{+}^{3})\). Let d 0 > 0 and v > 0 be fixed and let λ = (d 1, d 2, k 1, k 2), d i > 0, k i ≥ 0, i = 1, 2. As mentioned in Section 8.1, for any ϕ = (S 0(⋅ ), u 01(⋅ ), u 02(⋅ )) ∈ X +, (8.65)–(8.66) has a unique solution (S(x, t, ϕ, λ), u 1(x, t, ϕ, λ), u 2(x, t, ϕ, λ)), defined on its maximal interval of existence [0, σ ϕ ), satisfying (S(⋅ , 0, ϕ, λ), u 1(⋅ , 0, ϕ, λ), u 2(⋅ , 0, ϕ, λ)) = ϕ. Moreover,
By a similar argument as in Lemma 8.3.1, we have the following result on the boundedness of solutions uniformly for λ.
Lemma 8.3.2.
Let \(\varLambda =\{ (d_{1},d_{2},k_{1},k_{2}):\, \frac{d_{0}} {2} \leq d_{i} \leq 2d_{0},\,k_{i} \geq 0,\,i = 1,2\}\) . Then for each λ ∈Λ, ϕ ∈ X + , (S(x,t,ϕ,λ),u 1 (x,t,ϕ,λ),u 2 (x,t,ϕ,λ)) exists globally on [0,∞), and solutions of (8.65)–(8.66) are uniformly bounded and ultimately bounded uniformly for λ ∈Λ.
Now we can state one of the main results of this chapter. It says that both species persist for the perturbed system and there exists a positive periodic solution when the hypotheses of Theorem 8.2.2 hold for the unperturbed system and the perturbation is sufficiently small.
Theorem 8.3.2.
Let λ = (d 1 ,d 2 ,k 1 ,k 2 ) and λ 0 = (d 0 ,d 0 ,0,0). Assume that all conditions in Theorem 8.2.2 hold. Then there exist δ > 0 and β > 0 such that for any |λ −λ 0 | < δ, (8.65)–(8.66) admits at least one positive ω-periodic solution, and for any ϕ = (S 0 (⋅),u 01 (⋅),u 02 (⋅)) ∈ X + with \(u_{0i}(\cdot )\not\equiv 0,\,\forall i = 1,2,\) there exists t 0 = t 0 (ϕ,λ) such that
Proof.
Let k 0 > 0 be given and let
For each λ ∈ Λ 0, let S λ (⋅ ) = S(λ, ⋅ ): X + → X + be the Poincaré map associated with (8.65)–(8.66); that is,
Then S(⋅ , ⋅ ): Λ 0 × X + → X + is continuous. By Lemma 8.3.2, for each λ ∈ Λ 0, S λ : X + → X + is compact and point dissipative uniformly for λ ∈ Λ 0, and hence, by Theorem 1.1.3, there exists a global attractor A λ for S λ : X + → X +. Let
and ∂ X 0: = X + ∖ X 0. Then S λ : X 0 → X 0 and S λ : ∂ X 0 → ∂ X 0. By Theorem 8.2.1, (S i ∗(x, t), u i ∗(x, t)) is the unique positive ω-periodic solution of (8.20)–(8.21) with f(⋅ ) = f i (⋅ ), i = 1, 2, respectively. Clearly, (W ∗(x, t), 0, 0), (S 1 ∗(x, t), u 1 ∗(x, t), 0) and (S 2 ∗(x, t), 0, u 2 ∗(x, t)) are nonnegative periodic solutions of (8.65)–(8.66) with λ = λ 0. Let
Then \(S_{\lambda _{0}}(M_{0}) = M_{0},\,S_{\lambda _{0}}(M_{i}^{0}) = M_{i}^{0},\,\forall i = 1,2.\) By a change of variables
and Proposition 8.3.1, as applied to the resulting system, it follows that there exist δ 0 > 0 and η 0 > 0 such that for any λ ∈ Λ 0 with | λ −λ 0 | < δ 0, and for any ϕ ∈ X 0,
By Proposition 8.2.1 and Theorem 8.2.1, it follows that M 0, M 1 0, and M 2 0 are acyclic for \(S_{\lambda _{0}}\) in ∂ X 0, and \(\cup _{\phi \in \partial X_{0}}\omega _{\lambda _{0}}(\phi ) = M_{0} \cup M_{1}^{0} \cup M_{2}^{0}\), where \(\omega _{\lambda _{ 0}}(\phi )\) is the omega limit set of ϕ for \(S_{\lambda _{0}}\). Moreover, (8.67) implies that M 0 ∪ M 1 0 ∪ M 2 0 is an isolated covering of \(\overline{ \cup _{\phi \in \partial X_{0}}\omega _{\lambda _{0}}(\phi )}\) for \(S_{\lambda _{0}}\) in ∂ X 0 and that \(W_{\lambda _{0}}^{s}(M_{0}) \cap X_{0} =\emptyset\) and \(W_{\lambda _{0}}^{s}(M_{i}^{0}) \cap X_{0} =\emptyset,\,\forall i = 1,2\), where \(W_{\lambda _{0}}^{s}(M)\) denotes the stable set of M with respect to \(S_{\lambda _{0}}\). By Theorem 1.3.1 and Remark 1.3.1, it follows that \(S_{\lambda _{0}}: X^{+} \rightarrow X^{+}\) is uniformly persistent with respect to (X 0, ∂ X 0), and hence there exists a global attractor \(A_{\lambda _{0}}^{0} \subset X_{0}\) for \(S_{\lambda _{0}}: X_{0} \rightarrow X_{0}\) (see, e.g., Theorem 1.3.6).
Let \(\varLambda _{1} =\varLambda _{0} \cap \overline{B(\lambda _{0},\delta _{0})}\). Again by a change of variables
Lemma 8.3.2, and Proposition 8.3.2, as applied to the resulting system, it follows that S λ : X + → X + is point dissipative uniformly for λ ∈ Λ 1 and S(⋅ , ϕ): Λ 1 → X + is continuous uniformly for ϕ in any bounded subset of X +. Therefore, by the same argument as in the claim in the proof of Theorem 8.3.1, for any bounded subset B of X +, \(\overline{ \cup _{\lambda \in \varLambda _{1}}S_{\lambda }(B)}\) is compact. It then follows that, as argued in Theorem 8.3.1, \(\overline{ \cup _{\lambda \in \varLambda _{1},\phi \in X^{+}}\omega _{\lambda }(\phi )}\) is compact. Therefore, by (8.67) and Theorem 1.4.2, there exist δ 1 ∈ (0, δ 0) and η > 0 such that for any λ ∈ Λ 0 with | λ −λ 0 | ≤ δ 1, and any ϕ ∈ X 0, \(\liminf _{n\rightarrow \infty }\,d(S_{\lambda }^{n}\phi,\partial X_{0}) \geq \eta\). Moreover, by Theorem 1.3.10, S λ admits a fixed point S λ (ϕ λ ) = ϕ λ ∈ X 0, and hence (8.65)–(8.66) with | λ −λ 0 | ≤ δ 1 admits a nonnegative ω-periodic solution (S(x, t, ϕ λ , λ), u 1(x, t, ϕ λ , λ), u 2(x, t, ϕ λ , λ)) with u i (⋅ , t, ϕ λ , λ) ≫ 0 in \(C([0,1], \mathbb{R}),\,\forall t \geq 0,i = 1,2\). By parabolic maximum principle and the fact that S 0(⋅ ) ≥ 0 with S 0(⋅ ) ≢ 0, it then easily follows that S(⋅ , t, ϕ λ , λ) ≫ 0 in \(C([0,1], \mathbb{R}),\,\forall t \geq 0\). Thus, (S(x, t, ϕ λ , λ), u 1(x, t, ϕ λ , λ), u 2(x, t, ϕ λ , λ)) is a positive ω-periodic solution of (8.65)–(8.66).
It remains to prove the practical persistence claimed in the theorem. Let \(\varLambda _{2} =\varLambda _{0} \cap \overline{B(\lambda _{0},\delta _{1})}\). By both the point dissipativity and the uniform persistence of S λ with respect to (X 0, ∂ X 0) uniformly for λ ∈ Λ 2, it follows that there exists a closed and bounded set B 0 ⊂ X 0, independent of λ, such that \(\underline{d}(B_{0},\partial X_{0}) =\inf _{\phi \in B_{0}}d(\phi,\partial X_{0}) > 0\) and B 0 attracts points in X 0. As argued in Theorem 8.3.1, for each λ ∈ Λ 2, S λ : X 0 → X 0 admits a global attractor A λ 0 ⊂ X 0, and hence A λ 0 attracts any compact subset of X 0. Clearly, for each λ ∈ Λ 2, A λ 0 ⊂ B 0, and hence B 0 attracts compact subsets of X 0 under S λ . Since for each λ ∈ Λ 2, S λ : X + → X + is compact, and for any bounded subset B of X +, as claimed in the previous paragraph, \(\cup _{\lambda \in \varLambda _{2}}S_{\lambda }(B)\) is precompact, it follows that {S λ : λ ∈ Λ 2} is collectively compact. By Theorem 1.1.5, it then follows that A λ 0 is upper semicontinuous in λ ∈ Λ 2. In particular,
Let Φ λ (t, ⋅ ): X + → X + be defined by
Clearly, S λ = Φ λ (ω, ⋅ ) and S λ n = Φ λ (n ω, ⋅ ). It then follows that \(\varPhi _{\lambda }: \mathbb{R}_{+} \times X^{+} \rightarrow X^{+}\) is a periodic semiflow. Moreover, by Theorem 3.1.1, lim t → ∞ d(Φ λ (t, ϕ), \(\tilde{A}_{\lambda }^{0}) = 0\), \(\forall \phi \in X_{0}\), where \(\tilde{A}_{\lambda }^{0} = \cup _{t\in [0,\omega ]}\varPhi _{\lambda }(t,A_{\lambda }^{0}) \subset X_{0}\). Since A λ 0 = S λ (A λ 0), \(\tilde{A}_{\lambda }^{0} = \cup _{t\in (0,\omega ]}\varPhi _{\lambda }(t,A_{\lambda }^{0})\). By the compactness of \(\tilde{A}_{\lambda _{0}}^{0}\) and the parabolic maximum principle, it then follows that there exists β 0 > 0 such that for any \(\phi = (\phi _{0},\phi _{1},\phi _{2}) \in \tilde{ A}_{\lambda _{0}}^{0}\), \(\phi _{i}(x) \geq \beta _{0},\,\forall x \in [0,1],\,i = 1,2\). By (8.68), we have \(\lim _{\lambda \rightarrow \lambda _{0}}\sup _{ \phi \in \tilde{A}_{\lambda }^{0}}d(\phi,\tilde{A}_{\lambda _{0}}^{0}) = 0\). Consequently, there exist δ 2 ∈ (0, δ 1) and β 1 > 0 such that for any | λ −λ 0 | < δ 2, and any \(\phi = (\phi _{0},\phi _{1},\phi _{2}) \in \tilde{ A}_{\lambda }^{0}\), we have \(\phi _{i}(x) \geq \beta _{1},\,\forall x \in [0,1],\,i = 1,2\). Now the global attractivity of \(\tilde{A}_{\lambda }^{0}\) in X 0 for Φ λ completes the proof. ■
Remark 8.3.1.
In the case where the velocity of the flow of medium in the bioreactor varies periodically as well, that is, v = v(t) = v(t +ω), a change of variables
results in the boundary conditions becoming homogeneous Neumann boundary conditions, and using similar ideas as in Sections 8.2 and 8.3, we can also discuss the global dynamics of the modified model systems.
Remark 8.3.2.
In the case of constant nutrient input, that is, S 0(⋅ ) ≡ S 0, it follows that the ω-periodic solutions in Sections 8.2 and 8.3 reduce to steady states of the corresponding autonomous reaction–diffusion systems, and hence we have the analogous results of Theorems 8.2.1, 8.2.2, 8.3.1, and 8.3.2.
8.4 Notes
This chapter is adapted from Smith and Zhao [336]. The model with constant nutrient input was formulated by Kung and Baltzis [207], and was studied in Ballyk, Le, Jones and Smith [27]. Smith and Zhao [341] established the existence of traveling waves for this model in the case of single species growth. The transformations in Section 8.3 converting Robin-type boundary conditions to Neumann boundary conditions were motivated by Pilyugin and Waltman [278]. Similar perturbation ideas as in Section 8.3 were used for two-species periodic competitive parabolic systems under perturbations in Zhao [437].
Hsu, Wang and Zhao [175] studied a periodically pulsed bioreactor model in a flowing water habitat with a hydraulic storage zone in which no flow occurs, and obtained sufficient conditions in terms of principal eigenvalues for the persistence of single population and the coexistence of two competing populations. Yu and Zhao [422] investigated the spatial dynamics of a periodic reaction–advection–diffusion model for a stream population, and established a threshold-type result on the global stability of either zero or the positive periodic solution in the case of a bounded domain.
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Zhao, XQ. (2017). A Periodically Pulsed Bioreactor Model. In: Dynamical Systems in Population Biology. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-56433-3_8
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