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.
Keywords:predator-prey system; cross-diffusion; global solution
In this paper, we consider the global existence of non-negative classical solutions to the following diffusion predator-prey system with Holling type II functional response:
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.
Obviously, the non-negative equilibrium solutions of system (1.1) are and . For the reaction-diffusion problem of system (1.1), i.e., (), the global attraction, persistence and stability of non-negative equilibrium solutions are studied in . The main result can be summarized as follows:
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).
2 Auxiliary results
Using the Hölder inequality and Young inequality to estimate the right-hand side of (2.6), we have
where depends on and . So, we know . Since , it follows from the elliptic regularity estimate [, Lemma 2.3] that
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 .
whereCis a positive constant dependent onq, 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.
Firstly, integrating the first equation of (1.1) over Ω, we have
Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields
So, we get
Then multiplying both sides of the second equation of system (1.1) by v and integrating over Ω, we obtain
Estimating the first term on the right-hand side of (3.4),
Substituting (3.5) into (3.4) yields
From (3.11), we know
By Lemma 2.4 and (3.12), we know
By embedding theorem, we get
The second equation of system (1.1) can be written as the following divergence form:
In order to apply the maximum principle  to (3.16), we need to prove the following conditions:
Next, we will show that the above conditions (1) to (3) are satisfied for (3.16). When , it is easy to find that the condition (1) is satisfied by use of (3.15). Since for all , the condition (2) is verified. In view of the condition (3), we take appropriate q and r. Rewrite the first equation of system (1.1) as
From (3.14), we have . Therefore, all conditions of the Hölder continuity theorem [, Theorem 10.1] hold for (3.18). Hence,
We will discuss (2.5) which is the corresponding form of (3.18). It follows from (2.1) and (3.14) that , . From (3.19), we obtain . Thus, according to the parabolic regularity result of [, pp.341-342, Theorem 9.1], we can conclude that
Then the condition (3) and (3.17) are satisfied by choosing . According to the maximum principle [, p.181, Theorem 7.1], we can conclude that . From (2.1), there exists a positive constant such that
Therefore, the global solution to the problem (1.1) exists.
(iii) 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 .
Rewrite the second equation of system (1.1) as
Therefore, we can conclude that , u, v, ∇u and ∇v are all bounded. By the Schauder estimate , there exists such that
Furthermore, by the Schauder estimate, we obtain
Combining (3.24) and (3.26), we get
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 declares that he has no competing interests.
The work was realized by the author.
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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