Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

Abstract

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.

1. Introduction

Three-species reaction-diffusion models with Holling-type II functional response have been a familiar subject for the analysis. Taking more practical factors into consideration, a model with a modified Leslie-Gower scheme is worthy to explore. Leslie-Gower’s scheme indicates that the carrying capacity of the predator is proportional to the population size of the prey. The existing works [1–3] are all about models with this scheme. As a matter of fact, predators prefer to prey on other prey in the event of a shortage of favorite prey, so the research of the modified Leslie-Gower model springs up. Aziz-Alaoui and Okiye [4] focused on a two-dimensional continuous time dynamical system modeling a predator-prey food chain and gave the main result of the boundedness of solutions, the existence of an attracting set, and the global stability of the coexisting interior equilibrium, which was based on a modified version of the Leslie-Gower scheme and Holling-type II scheme. Singh and Gakkhar [5] investigated the stabilization problem of the modified Leslie-Gower type prey-predator model with the Holling-type II functional response. The analysis of models with a modified Leslie-Gower scheme can be also found in [6–10].

Nonconstant positive steady states have received increasing attention in recent years, see [11–18] and references therein. Ko and Ryu [19] showed that the predator-prey model with Leslie-Gower functional response had no nonconstant positive solution in homogeneous environment, but the system with a general functional response might have at least one nonconstant positive steady state under some conditions. Zhang and Zhao [20] analyzed a diffusive predator-prey model with toxins under the homogeneous Neumann boundary condition, including the existence and nonexistence of nonconstant positive steady states of this model by considering the effect of large diffusivity. Shen and Wei [21] considered a reaction-diffusion mussel-algae model with state-dependent mussel mortality which involved a positive feedback scheme. Wang and his partners [22] considered a tumor-immune model with diffusion and nonlinear functional response and investigated the effect of diffusion on the existence of nonconstant positive steady states and the steady-state bifurcations. Hu and Li [23] were concerned about a strongly coupled diffusive predator-prey system with a modified Leslie-Gower scheme and established the existence of nonconstant positive steady states. Qiu and Guo [24] analyzed a stationary Leslie-Gower model with diffusion and advection.

Motivated by the mentioned above, we consider a three-species reaction-diffusion model with a modified Leslie-Gower and Holling-type II scheme under the homogeneous Neumann boundary condition as follows: where and represent the density of two competitors, respectively, while stands for the density of the predator who preys on . , , and are all positive as the intrinsic growth rates, and regard as influencing factors within diverse populations themselves while and are influencing factors between different populations. All of them are nonnegative. and are the modified Leslie-Gower scheme, and , , , and are positive. Applying the following scaling to (1), as well as assuming for simplicity of calculation: still using replace the following ODE system can be logically obtained: where .

It is clear that , and are nonnegative constant solutions of system (3). is a semitrivial solution when it satisfies . When , is a semitrivial solution where

System (3) yields that

If the following alternative conditions hold: there exists the unique positive equilibrium as where

Taking the diffusion into account, the corresponding PDE system can be written as

where is a smooth bounded domain, is the outward unit normal vector on , is the Laplace operator, and diffusion coefficients are

The rest of this paper is arranged as follows. In Section 2, the stability of constant solutions for the ODE system is discussed. In Section 3, the stability of constant solutions for the PDE system is studied. In Section 4, we focus on the priori estimates of positive steady states. In the last two sections, we have a discussion about the nonexistence and existence of nonconstant positive steady states under different conditions.

2. Stability of Constant Solutions for the ODE System

In this section, we discuss the stability of constant solutions with the condition of their existence for the ODE system.

Theorem 1. For the ODE system (3), let and . (i) and are all unconditionally unstable(ii)If satisfies , then is unstable; if holds, is local asymptotically stable(iii)If satisfies , then is unstable; if holds, is local asymptotically stable(iv)If and satisfy , then is unstable; if and holds, is local asymptotically stable

Proof. The Jacobian matrix of the ODE system (3) is Obviously, we can obtain at and its corresponding characteristic polynomial is so its eigenvalues are , , and Therefore, is unstable to system (3).

By the same manner, we know that , and are all unstable to ODE system (3).

The Jacobian matrix of the ODE system at is

The characteristic polynomial is

When the eigenvalue satisfies , it deduces that , so we can see that is unstable to ODE system (3). When , we consider that

Letand take value for as (4) and (5), we know that , . With the existence condition , and hold, such that equation (17) has two solutions with negative real parts.

Because of , holds, then if . So we can conclude that when , is local asymptotically stable to ODE system (3).

The Jacobian matrix of the ODE system at is

The characteristic polynomial is The corresponding eigenvalues are . If , is unstable. Otherwise, , is local asymptotically stable to ODE system (3).

The Jacobian matrix of the ODE system at is

The corresponding characteristic polynomial is , where

When satisfies , then , is unstable applying the Hurwitz criterion [25]. When , we can find . So is local asymptotically stable to ODE system (3).

The proof is complete.

3. Stability of Constant Solutions for the PDE System

In this section, the stability of the constant solutions with the condition of their existence for the PDE system is discussed.

Let as the eigenvalues of the operator over under the homogeneous Neumann boundary condition and be the corresponding eigenspace while is a set of the orthogonal basis of , , and . Then, .

Theorem 2. For the PDE system (11), let and . (i) and are all unconditionally unstable(ii)If satisfies , then is unstable; if and holds, is uniformly asymptotically stable(iii)If satisfies , then is unstable; if holds, is uniformly asymptotically stable(iv)If and satisfy , then is unstable; if and holds, is uniformly asymptotically stable

Proof. The linearization of (11) at the positive constant solution can be expressed by where and is the Jacobian matrix at . For each , is invariant under the operator . And is an eigenvalue of on if and only if is an eigenvalue of the matrix .
The Jacobian matrix of PDE system (11) is

According to the Theorem 1, are all unstable to ODE system (3). Hence, there exist the eigenvalue with positive real parts in the PDE system. It means that are all unstable to PDE system (11).

The Jacobian matrix of the PDE system at is The characteristic polynomial is

When the eigenvalue satisfies , it deduces that , there exists an eigenvalue with positive real part, and is unstable to PDE system (11).

It is clear that eigenvalue as . Then, we discuss the following equation emphatically:

Let

It shows that on account of . When , we know holds. So the eigenvalues all have negative real parts.

The Jacobian matrix of PDE system (11) at can be written as

The characteristic polynomial is The corresponding eigenvalues are and . If , there exists an eigenvalue with positive real part; is unstable to PDE system (11). On the contrary, if , the eigenvalues all have negative real parts.

The Jacobian matrix of the PDE system at is

Its characteristic polynomial is , where

When , there exists an eigenvalue with positive real part; is unstable to PDE system (11).

When and , holds. Similarly, since and . If , we have and . As a result of and , can be obtained. What is more, leads to . Thus, the eigenvalues all have negative real parts.

In the following, we shall prove that there exists a positive constant when the corresponding eigenvalues all have negative real parts, such that

Let , then

Since as , it follows that

Applying the Hurwitz criterion, the three roots of all have negative real parts. Thus, there exists a positive constant such that . By continuity, there exists such that the three roots of satisfy

Hence,

Let ,. Then, for ,

Therefore, the constant solutions are uniformly asymptotically stable when the corresponding eigenvalues all have negative real parts.

The proof is complete.

4. A Priori Estimates of Positive Steady States

The corresponding steady-state problem of system (11) is

Two lemmas are listed here for the preliminary.

Lemma 3. (Harnack inequality [26]).
Let be a positive solution to , where , satisfying the homogeneous Neumann boundary condition. Then, there exists a positive constant such that

Lemma 4. (maximum principle [27]).
Suppose that and (i)if satisfiesand then (ii)if satisfiesand then

The results of upper and lower bounds can be stated as follows.

Theorem 5. (upper bounds).
Assuming that is a positive solution of system (37), we get

Proof. Since and , such that according to Lemma 4. Because of it is evident that
The proof is complete.

Theorem 6. (lower bounds).
Fix and as positive constants. Assume that then there exists a positive constant who can make every positive solution of system (37) satisfy

Proof. Let In view of (41), (42), and (43), a positive constant can be easily found, such that where . Thus, , , and satisfy that According to the Harnack inequality in Lemma 3, there must be a positive constant such that Suppose that (45), (46), and (47) hold of no account.
There must be a sequence with such that the corresponding positive solutions of system (37) reach the qualification Then, we apply to the system of (37) and integrate by parts, so we obtain that There exists a subsequence of according to the -regularity theory and Sobolev embedding theorem, but we still use to represent for convenience. So there must be and as the limiting of and when . They can be written as follows: Let , we get that We now discuss the following three cases.

Case 1. . Since as , holds for every , so that which contradicts with (55).

Case 2. . Since as , holds for every , so that which contradicts with (55).

Case 3. . Since as , holds for every , so that which contradicts with (55).

The proof is complete.

5. Nonexistence of Nonconstant Positive Steady States

We prove the nonexistence of nonconstant positive steady states of system (37) in this section.

Theorem 7. Let is the smallest positive eigenvalue of operator over under the homogeneous Neumann boundary conditions and fixed positive constants satisfy and , then there exists a positive constant such that when , and , system (37) has no nonconstant positive steady states.

Proof. Assume that is the positive solution of (37). For any , let . The differential equation (37) multiplies and integrates by parts over to get Combine (59), (60), and (61), we have where are the arbitrary small positive constants arising from Young inequality. Meanwhile, applying the Poincaré inequality , we gain that for some positive constants