Global behavior for a diffusive predator-prey system with Holling type II functional response
© Zhao; licensee Springer 2012
Received: 6 April 2012
Accepted: 25 September 2012
Published: 9 October 2012
A strongly coupled self- and cross-diffusion predator-prey system with Holling type II functional response is considered. Using the energy estimate, Sobolev embedding theorem and bootstrap arguments, the global existence of non-negative classical solutions to this system in which the space dimension is not more than five is obtained.
MSC: 35K50, 35K55, 35K57, 92D40.
where Ω is a bounded region in () with a smooth boundary ∂ Ω; η is the outward normal on ∂ Ω, ; and are non-negative smooth functions and are not identically zero; u and v denote the population densities of predator and prey, respectively; α, β, r, a, K, m and c are positive constants, and ; and are the diffusion rates of the two species; () are given non-negative constants, and are self-diffusion rates; is the cross-diffusion rate. It means that the diffusion is from one species of high-density areas to the other species of low-density areas. See [1, 2] for more details on the ecological backgrounds of this system.
If , a semi-trivial solution is globally asymptotically stable;
If and , a unique positive constant solution is globally asymptotically stable;
If and , a positive constant solution is locally asymptotically stable.
In view of the study of dynamic behavior of a predator-prey reaction-diffusion system with Holling type II functional response, a natural problem is what the global behavior for a predator-prey cross-diffusion system (1.1) is. To the best of our knowledge, the existing results are very few. In this paper, we consider the space dimension to be less than six, and initial function and under some smooth conditions, using the energy estimate, Sobolev embedding theorem and bootstrap arguments, we consider the global existence of non-negative classical solutions for system (1.1).
We denote . means that u, , () and are in . . .
2 Auxiliary results
Choose . By use of the maximum principle, the non-negative solution of system (1.1) can be derived from the maximum principle, i.e., for all . This completes the proof of Lemma 2.1. □
From (2.5), we have . Hence, . Moreover, (2.4) can be obtained by use of the Sobolev embedding theorem. □
with the boundary condition on , where . Then ∇W is in .
The proof of the above lemma can be found in [, Proposition 2.1].
The following result can be derived from Lemma 2.3 and Lemma 2.4 of .
Finally, one proposes some standard embedding results which are important to obtain the and normal estimates of the solution for (1.1).
Lemma 2.5 There exists a constant such that .
By Lemma 2.3 with , , , we obtain the desired result. □
, , ;
, , ;
, , ,
where C is a positive constant dependent on q, n, Ω, T.
3 The existence of classical solutions
The main result about the global existence of non-negative classical solutions for the cross-diffusion system (1.1) is given as follows.
Theorem 3.1 Assume that and satisfy homogeneous Neumann boundary conditions and belong to for some . Then system (1.1) has a unique non-negative solution if the space dimension is .
-, -estimate and -estimate for v.
-estimate for v.
where , and is the Kronecker sign.
which implies that by Lemma 2.6.
Since , we have , i.e., . It means that . So, . From (2.1) and (3.14), .
The existence of classical solutions.
Under the conditions of Theorem 3.1, we consider above global solutions to be classical. By (3.20) and Lemma 2.6, we know , . It follows from Lemma 3.3 in  that . Since , we have .
Therefore, the result of Theorem 3.1 can be obtained for , namely . When , namely , we have . (3.24) and (3.26) are obtained by repeating the above bootstrap argument and the Schauder estimate. This completes the proof of Theorem 3.1. □
The author thanks the anonymous referee for a careful review and constructive comments. This work was supported by the Qinghai Provincial Natural Science Foundation (2011-Z-915).
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