Coexistence states for a modified Leslie-Gower type predator-prey model with diffusion
© Shi and Li; licensee Springer 2012
Received: 30 June 2012
Accepted: 3 December 2012
Published: 19 December 2012
This paper is concerned with a modified Leslie-Gower predator-prey model with general functional response under homogeneous Robin boundary conditions. We establish the existence of coexistence states by the fixed index theory on positive cones. As an example, we apply the obtained results to this model with Holling-type II functional response. Our results show that the intrinsic growth rates and the principle eigenvalues of the corresponding elliptic problems with respect to the Robin boundary conditions play more important roles than other parameters for the existence of positive solutions.
MSC: 35J55, 37B25, 92D25.
where , , , , , c, e are positive constants in a biological viewpoint. More precisely, in , Aziz-Alaoui et al. investigated the boundedness of solutions, the existence of positive invariance attracting set and global stability of the coexisting interior equilibrium. Later, Nindjin et al.  gave the qualitative analysis of the corresponding delayed system.
In the evolutionary process of the species, the individuals do not remain fixed in space, and their spatial distribution changes continuously due to the impact of many reasons (the environment factors, food supplies, etc.). Therefore, spatial effects such as diffusion and dispersal should be introduced into population models. In particular, introducing the spatial effects is not trivial in many works. For example, the famous Turing instability was observed in many nature processes, and it was also proved in some mathematical models with diffusion. Clearly, such a Turing instability cannot be formulated by the ordinary differential equations.
In the above, u and v, respectively, stand for the population densities of prey and predator; is a bounded domain with smooth boundary ∂ Ω; ν denotes the outward unit normal vector of the boundary ∂ Ω; , are nonnegative constants. is the birth function of the prey u. It should be pointed out that system (1.4) has the function response with a general formation , which is different from the models in the related papers.
The rest of this paper is arranged as follows. In Section 2, we collect some known results including the eigenvalue problem and the fixed point index on positive cones. In Section 3, we establish the existence of positive solutions for system (1.5). In Section 4, as an example, we apply the obtained results to system (1.5) with Holling-type II functional response.
Let E be a real Banach space and be a closed convex set. Then W is called a wedge if for all , and a wedge W is said to be a cone if . For , define and . It is evident that is a wedge containing W, y, −y, while is a closed subspace of E containing y. In what follows, we always assume that . Let be a compact linear operator on E. We say that has property α on if there exist and such that . Suppose that is a compact operator with a fixed point . If ℱ is Fréchet differential at y, then the derivative has the property that . For an open subset , define , where I is the identity map. If y is an isolated fixed point of ℱ, then the fixed point index of ℱ at y related to W is defined by , herein is a small open neighborhood of y in W. The following results of fixed point index can be obtained from [16, 27–29].
Lemma 2.1 Assume that is invertible on .
(i) If has property α, then .
(ii) If does not have property α, then , where σ is the sum of multiplicities of all eigenvalues of which are greater than one.
In particular, we denote by for the sake of convenience. It is well known that is strictly increasing with respect to , namely, if and . Furthermore, the eigenfunction of (2.1) corresponding to the eigenvalue is unique and positive. In [30, 31], the authors discussed the eigenvalue problem (2.1) in detail and established the existence and comparison results for (2.1). Furthermore, we cite the following lemma on the eigenvalue of (2.1), which can be found in [16, 32].
Lemma 2.2 Let and , in Ω.
(a1) If , then .
(b1) If , then .
(c1) If , then .
In addition, if M is a positive constant such that on , then we have the following conclusions:
(c2) , where is the spectral radius of an operator.
Lemma 2.3 Assume that for all and on for some positive constant C.
(a) If , then (2.2) has no positive solutions. Moreover, the trivial solution is globally asymptotically stable.
(b) If , then (2.2) has a unique positive solution which is globally asymptotically stable and satisfies for all .
3 Existence of positive solutions for system (1.5)
In order to establish the existence of positive solutions of system (1.5), we give the following hypotheses.
(H1) , , , , for any , where the constants and .
(H2) , and , for any , where the constant .
has a unique positive solution . Hence, when and , system (1.5) has two semi-trivial solutions and . By virtue of the maximum principle and Hopf’s lemma, we obtain the following results on the boundedness of the nonnegative solutions of (1.5), of which the proof is omitted here.
From Proposition 3.1, we can see that the nonnegative solution of (1.5) must lie in .
are nonnegative for all .
Lemma 3.2 For any open set in W, .
Thus, it follows from Lemma 2.2 that , which indicates that is invertible on and does not have property α on . So, we may conclude that by Lemma 2.1. The proof is completed. □
Lemma 3.3 Assume that and . Then .
If , then by the lemma, which is a contradiction to the assumption. Thus, . Similarly, since , then . Therefore, is invertible on .
This implies that has property α. It follows from Lemma 2.1 that . The proof is completed. □
This yields a contradiction. Hence, . This indicates that is invertible on .
This implies that has property α. By Lemma 2.1, we know . The proof is completed. □
Similarly, we have the following lemma, the proof of which is a slight modification of the above.
Now, we establish the existence of positive solutions of the system based on the above results about the fixed index.
Theorem 3.6 Assume that , . Then system (1.5) has a positive solution.
and . Therefore, system (1.5) has a positive solution in . The proof is completed. □
By virtue of Lemma 2.3, we see that (4.2) has a unique positive solution with when , where ρ is a positive constant. Hence, system (4.1) admits two semi-trivial solutions and if and .
By virtue of the maximum principle and Hopf’s lemma, we can give a priori estimates of positive solutions of (4.1), the proof of which is omitted here.
From Proposition 4.1, we can see that the nonnegative solution of (4.1) must lie in .
Similar to the discussion in Section 3, we have the following lemmas.
Lemma 4.2 Assume that . Then
(i) , for an open set in W.
(ii) , if .
(iii) , if .
(iv) , if .
Lemma 4.3 Assume that holds. Then the following items hold:
(i) , if .
(ii) , if .
Theorem 4.4 For system (4.1), the following results hold:
(i) If , then (4.1) has no positive solution and, in addition, if , then (4.1) has no nonnegative nonzero solution.
(ii) If and , then (4.1) admits a positive solution.
(iii) If and (4.1) has a positive solution, then .
and so by Lemma 2.2. Using the comparison property of an eigenvalue, it follows that , which is a contradiction. Next, assume that is a nonnegative nonzero solution of (4.1). If and , then . Similarly, if and , then . A contradiction occurs. This completes the proof of (i).
and . So, (4.1) has a positive solution in , which shows that the second statement is true.
The proof is complete. □
The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the article. This study was supported by the Universities Natural Science Foundation of Jiangsu Province (No. 11KJB110003) and Young Talents Support Projects of Huaiyin Normal University (11HSQNZ02).
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