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:
(1) If , a semi-trivial solution is globally asymptotically stable;
(2) If and , a unique positive constant solution is globally asymptotically stable;
(3) 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
Lemma 2.1Let be the solution of (1.1). There exists a positive constant such that
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. □
Lemma 2.2Let , for the solution to the following equation:
where , are positive constants and . Then there exists a positive constant , depending on and , such that
Proof From , it is easy to find that
where and . and are bounded in from (2.1). Multiplying (2.5) by and integrating by parts over yields
Using the Hölder inequality and Young inequality to estimate the right-hand side of (2.6), we have
with some . Substituting (2.7) into (2.6), we obtain
where depends on and . So, we know . Since , it follows from the elliptic regularity estimate [, Lemma 2.3] that
From (2.5), we have . Hence, . Moreover, (2.4) can be obtained by use of the Sobolev embedding theorem. □
Lemma 2.3Assume that is a bounded function satisfying
with the boundary condition on , where . Then ∇Wis 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 .
Lemma 2.4Let , . If
and there exist positive constants and such that ( ), there exists a positive constant , independent ofwbut possibly depending onn, Ω, p, βand , such that
Finally, one proposes some standard embedding results which are important to obtain the and normal estimates of the solution for (1.1).
Lemma 2.5There exists a constant such that .
Proof Let , . By Lemma 2.1, u is bounded. Therefore, X is also bounded. By Lemma 2.2, we have . Moreover, X satisfies
By Lemma 2.3 with , , , we obtain the desired result. □
Lemma 2.6Let be a fixed bounded domain and . Then for all with , one has
(1) , , ;
(2) , , ;
(3) , , ,
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.
Theorem 3.1Assume 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 .
(i) -, -estimate and -estimate for v.
Firstly, integrating the first equation of (1.1) over Ω, we have
Thus, for all , we can obtain
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
Integrating the above expression in yields
Estimating the first term on the right-hand side of (3.4),
Substituting (3.5) into (3.4) yields
Select and denote . Notice the positive equilibrium point of (1.1) exists under condition , then
By Lemma 2.5, . Integrating the above inequality and using the Gronwall inequality, we get
Hence, there exists a positive constant such that . Furthermore, we have
Secondly, multiplying both sides of the second equation of system (1.1) by ( ) and integrating over Ω, we have
Integrating the above equation over ( ), it is clear that
By Lemma 2.2, it can be found that . According to the Hölder inequality and Young inequality, we get
Choose an appropriate number ε satisfying . Substituting (3.10) into (3.9) and taking , we have
From (3.11), we know
When , it is easy to find that and . So,
Set . It follows from the -estimate for v that
By Lemma 2.4 and (3.12), we know
Obviously, and . It is easy to know that E is bounded by use of reduction to absurdity. Since , . So, is bounded, i.e., . Denote still as q. So,
Finally, when , with . For , taking in (3.9), it follows from (3.8) that there exists a positive constant such that
By embedding theorem, we get
(ii) -estimate for v.
The second equation of system (1.1) can be written as the following divergence form:
where , and is the Kronecker sign.
In order to apply the maximum principle  to (3.16), we need to prove the following conditions:
(1) is bounded;
where ν, are positive constants and
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
When , , it is clear that has an upper bound over by Lemma 2.1. Set
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
which implies that by Lemma 2.6.
Since , we have , i.e., . It means that . So, . From (2.1) and (3.14), .
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
Next, the regularity of v will be discussed. Set . So, satisfies
where . According to (3.22) to (3.24), we have , . Applying the Schauder estimate to (3.25), we know
From , we can see
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).
Dubey, B, Das, B, Hussain, J: A predator-prey interaction model with self and cross-diffusion. Ecol. Model.. 141, 67–76 (2001). Publisher Full Text
Ko, W, Ryu, K: Qualitative analysis of a predator-prey model with Holling type II. Functional response incorporating a prey refuge. J. Differ. Equ.. 231(2), 534–550 (2006). Publisher Full Text
Amann, H: Dynamic theory of quasilinear parabolic equations - III. Global existence. Math. Z.. 202, 219–250 (1989). Publisher Full Text
Shim, S: Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Differ. Equ.. 185, 281–305 (2002). Publisher Full Text
Shim, S: Uniform boundedness and convergence of solutions to the systems with cross-diffusion dominated by self-diffusion. Nonlinear Anal., Real World Appl.. 4, 65–86 (2003). Publisher Full Text