Volume 2012, Issue 1 423193

Research Article

Open Access

Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

Huilan Wang,

Corresponding Author

Huilan Wang

[email protected]

Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China

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Zhengqiu Zhang,

Zhengqiu Zhang

Department of Applied Mathematics, Hunan University, Changsha 410082, China hnu.edu.cn

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Weiping Zhou,

Weiping Zhou

Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China

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Huilan Wang,

Corresponding Author

Huilan Wang

[email protected]

Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China

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Zhengqiu Zhang,

Zhengqiu Zhang

Department of Applied Mathematics, Hunan University, Changsha 410082, China hnu.edu.cn

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Weiping Zhou,

Weiping Zhou

Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China

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First published: 05 April 2012

https://doi.org/10.1155/2012/423193

Academic Editor: Vimal Singh

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Abstract

By using continuation theorem of coincidence degree theory, sufficient conditions of the existence of positive periodic solutions are obtained for a generalized predator-prey system with diffusion and delays. In this paper, we construct a V-function to make the prior estimation for periodic solutions, which makes the discussion more concise. Moreover, to compute the mapping′s topological degree, a polynomial function matrix is constructed straightforwardly as a homotopic mapping for the generalized one, which improves the methods of computation on topological degree for a generalized mapping.

1. Introduction

In the study of dynamic population models, which are represented by differential equations, to elaborate the realistic factors into models, sometimes one needs to consider the effects caused by various of factors such as delays, diffusion, and others. The global characteristics (including persistent, stable or attractive, oscillatory, and chaotic behavior) of such differential system have been an intensive subject in biomathematics (see e.g., [1–6]). One important aspect is the existence of positive periodic solutions ([7–17]), which imply the system is periodic.

In the literature available, the methods of studying the existence of periodic solutions of a periodic system often fall into one of the following three categories: (1) combining the results of persistence of system with Horn fixed point theorem (usually for a system with delay) or Brower fixed point theorem (usually for a system without delay), the existence results of periodic solutions are obtained ([7, 8]); (2) employing other kinds of fixed point theorems (usually for one dimension system), the differential model is investigated by transforming it into equivalent integral one ([9–11]); (3) by virtue of the theory of topological degree, especially by using Mawhin’s Continuation theorem of coincidence degree (usually for high dimension system), much significant work has been done ([12–17]). However, for a generalized high-dimension system, no matter by fixed point theorems or by Mawhin’s continuation theorem of coincidence degree, it is very difficult to accomplish the work. The relevant literature is seldom to find. Zhang considered a generalized prey-predator system with delay ([16]), where the generalized mapping’s degree is computed step by step and a new technique of computation on topological degree is presented.

It is well known that the key and the difficult points are the prior estimation for solutions, and the topological degree’s computation of generalized mapping as Mawhin’s continuation theorem is applied. In this paper, we investigate a generalized model of one predator competing for two preys, which takes the nonautonomous form:

(1.1)

where x_i denotes the density of the prey in the ith patch; x₃ represents the total predator population for both patches; τ_i > 0 is a constant and D_i(t) is a positive continuous function and denotes the dispersal rate; p_i is the functional response of the predator population on the prey in the ith patch, and c_i is the conversion ratio of prey into predator; D_i(t), c_i(t), f_i(t, x_i, x₃) and g(t, x₃) are continuous functions in t ∈ [0, +∞) with a common period ω > 0; f_i(t, x_i, x₃) and g(t, x₃) are differentiable with the other variables; i = 1,2. The model is transformed from an autonomous one (see [6]). In [6], sufficient conditions for the permanence and the existence of positive attractive equilibrium are derived.

Our purpose is to obtain sufficient conditions for the existence of positive periodic solutions associated with system (1.1) and to consider whether delays and diffusion have effect on the results. What is more important is to present some new techniques in prior estimation of solutions and computation on topological degree. In this paper, we construct a V-function to make the prior estimation, which makes the proof more concise. Furthermore, we select a polynomial function matrix as a homotopic mapping to the generalized one, which plays a key role in the computation of topological degree.

In a biological sense, we take the initial conditions:

(1.2)

where τ = max {τ₁, τ₂}.

In system (1.1), we assume that

(H₁)
for t ≥ 0, ∂f_i(t, x_i, x₃)/∂x_i < 0, ∂f_i(t, x_i, x₃)/∂x₃ < 0, i = 1,2, and (∂g(t, x₃))/∂x₃ > 0;
(H₂)
p_i(x) is continuous and .

For a positive continuous ω-periodic function f(t), we define

(1.3)

In the following sections, we derived the sufficient conditions for the existence of positive periodic solutions and show that the delays and diffusion have no effect on the result.

2. Main Result and Proof

We first introduce some notion of the continuation theorem of coincidence degree theory and the lemma formulated in [18].

Let X, Z be Banach spaces, let L : Dom L ⊂ X → Z be a linear mapping, and let N : X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimkerL = codimimL < +∞ and Im L is closed in Z. Let P : X → X and Q : Z → Z be two projectors such that Im P = Ker L and Im L = Ker Q = Im (I − Q). It follows that L/(Dom L∩Ker P):(I − P)X → ImL is invertible. We denote the inverse of that map by K_P. For an open bounded subset Ω of X, the mapping N will be called L-compact on if is bounded and is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.

Lemma 2.1. Let L be a Fredholm mapping of index zero and N be L-compact on . Suppose

(a)
for each λ ∈ (0,1) and x ∈ ∂Ω, Lx ≠ λNx;
(b)
for each x ∈ Ker L ⋂ ∂Ω, QNx ≠ 0;
(c)
Brower degree deg _B(JQN, Ω∩Ker L, 0) ≠ 0.

Then Lx = Nx has at least one solution in

Theorem 2.2. Assume that there exist four positive constants M, N, m, n such that

(1)
for t ∈ R, f_i(t, M, 0) ≤ 0 and −g(t, N) + c₁(t)p₁(M) + c₂(t)p₂(M) < 0, i = 1,2;
(2)
for t ∈ R, f_i(t, m, N) ≥ 0 and −g(t, n) + c₁(t)p₁(m) + c₂(t)p₂(m) > 0, i = 1,2.

Then system (1.1) has at least one positive ω-periodic solution.

Proof. Denote x_i(t) = exp [u_i(t)] (i = 1,2, 3); then system (1.1) can be rewritten as

(2.1)

Clearly, if system (2.1) has an ω-periodic solution

, then system (1.1) has a positive ω-periodic solution

. We define

(2.2)

Then X and Z are Banach spaces with the norm ∥·∥.

Let

(2.3)

Then, L is a Fredholm mapping of index zero and the generalized inverse of L is K_P : Im L → Ker P∩Dom L, which is given by

(2.4)

Hence

(2.5)

where

(2.6)

Consequently,

(2.7)

By Arzela-Ascoli theorem, we can verify that the map

is compact. Thus, N is L− compact on

for any open bounded Ω ⊂ X. Since Im Q = Ker L, the isomorphism J is the identical mapping.Corresponding to the operator equation Lx = λN(x, λ), λ ∈ (0,1), we have

(2.8)

Assume that u = u(t) ∈ X is a solution of (2.8) for a certain λ ∈ (0,1). Then there exist ξ_i, η_i ∈ [0, ω] (i = 1,2, 3) such that

(2.9)

Therefore, we obtain the equations

(2.10)

and equations

(2.11)

To make estimation of u_i(ξ_i) (i = 1,2, 3), we define a V-function as

. Obviously,

and

. According to the hypothesis and equations (2.10), we have

(2.12)

which implies that

, that is,

(2.13)

The third equation of (2.10) gives

(2.14)

Thus

(2.15)

(2.16)

Similarly, we define

. Then

, and

. From (2.11), we have

(2.17)

which implies that

(2.18)

that is,

(2.19)

From (2.19), it follows that

(2.20)

Hence

(2.21)

that is,

(2.22)

Denote

(2.23)

In view of (2.13)–(2.22), we have

(2.24)

Clearly, R_i (i = 1,2) are independent of λ. Denote

, where R₀ is taken sufficiently large such that the solution

of the following system:

(2.25)

satisfies

, provided that system (2.25) has one or a number of solutions. Let

. It is easy to see that the condition (a) of Lemma 2.1 is satisfied.

When u ∈ ∂Ω∩Ker L = ∂Ω∩R³, u is a constant vector in R³ with If system (2.25) has one or a number of solutions, then

(2.26)

If system (2.25) has no solution, then naturally QN(u, 0) ≠ (0,0,0)^T. Hence, the condition (b) of Lemma 2.1 is satisfied.

Finally, we will prove that condition (c) of Lemma 2.1 is satisfied. To this end, we define a mapping ψ : Dom L × [0,1] → X by

(2.27)

where ν ∈ [0,1] is a parameter and a, b are two chosen numbers as follows:

(2.28)

In the following, we will show that when u ∈ ∂Ω∩Ker L, ψ(u, ν) ≠ (0,0,0)^T.

We consider three possible cases: (1) either of satisfies , (2) either of satisfies , and (3) both of satisfy .

Case 1. Either of satisfies since there exists a constant t₁ such that

(2.29)

then

(2.30)

Case 2. Either of satisfies we consider two subcases as follows.

Subcase 1 .. . There exists a constant t₂ and u_i (i = 1 or 2) such that

(2.31)

then

(2.32)

Subcase 2. . Because the other u_j (j = 2 or 1) must satisfy either or . The later one is discussed in Case 1. Then we only consider the subcase when and . Under these conditions, there exists a constant t₃ such that

(2.33)

Case 3. Both of satisfy : we also consider the following two subcases.

Subcase 3. . There exists a constant t₄ such that

(2.34)

Hence

(2.35)

Subcase 4. . If , then , which is a contradiction to . Hence, . When and , there exists a t₅ such that

(2.36)

To sum up, ψ(u, ν)≠(0,0, 0) ^T when u ∈ ∂Ω∩Ker L.

Since the algebraic equations

(2.37)

have a unique solution (x^*, y^*, z^*) such that

(2.38)

then

(2.39)

This completes the proof of Theorem 2.2.

Remark 2.3. Theorem 2.2 implies that the delays and the diffusion have no effect on the result provided (H₁)-(H₂) holds.

3. Application

Example 3.1. Consider the system:

(3.1)

where τ_i (i = 1,2) are positive constants and all the coefficients are positive continuous functions with period ω.

It is easy to see that

(3.2)

Obviously, the assumptions (H₁)-(H₂) are satisfied. Let

(3.3)

Then

(3.4)

Let

(3.5)

then −g(t, N) + c₁(t)M + c₂(t)M < 0. While

(3.6)

f_i(t, m, N) ≥ 0. Let

(3.7)

then −g(t, n) + c₁(t)m + c₂(t)m > 0. Therefore, the conditions of Theorem 2.2 are satisfied. We can draw the conclusion that system (3.1) has at least one positive periodic solution.

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Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

Abstract

1. Introduction

2. Main Result and Proof

3. Application

References

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Existence of Positive Periodic Solutions for a Nonautonomous Generalized Predator-Prey System with Time Delay in Two-Patch Environment

Abstract

1. Introduction

2. Main Result and Proof

3. Application

References

References

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