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Global solutions for a general predator-prey model with cross-diffusion effects
Advances in Difference Equations volume 2012, Article number: 117 (2012)
This work investigates a general Lotka-Volterra type predator-prey model with nonlinear cross-diffusion effects representing the tendency of a predator to get closer to prey. Using the energy estimate method, Sobolev embedding theorems and the bootstrap arguments, the existence of global solutions is proved when the space dimension is less than 10.
Consider the following predator-prey model with cross-diffusion effects:
where () is a bounded domain with smooth boundary ∂ Ω, ν is the outward unit normal vector of the boundary ∂ Ω, the given coefficients d, s, c, , , δ, α, γ and l are positive constants. The functions and are assumed to satisfy the following two hypotheses throughout this paper:
(H1) and for all , and there exists a positive constant K such that .
(H2) For all , for some positive constant C and a continuous function .
In the model (1), u and v represent the densities of the prey and predator respectively, the constants d, s and c refer to the predator death rate, the inter-specific competition coefficient and the conversion rate of the prey to the predator, and represents the growth rate of the prey in the absence of the predator. The function is called the functional response of the predator to the prey. The constants and are the diffusion rates of the prey and predator respectively, δ and α are referred to as self-diffusion pressures, and the term in the predator equation is cross-diffusion pressure, which expresses the population flux of the predator resulting from the presence of prey species. Biologically, this term represents the tendency of the predator to chase its prey. It is clear that such an environment of prey-predator interaction often occurs in reality. For example, in [1–5], with the similar biological interpretation, the authors also introduced the same cross-diffusion term as in (1) to the predator of various predator-prey models.
Mathematically, one of the most important problem for (1) is to establish the existence of global solutions. However, to our knowledge, there are only a few works. In , the global existence result was shown without any restrictions on space dimensions by H. Li, P. Pang and M. Wang. But their results are not valid for (1) because some restrictions are required for the term , so that the standard reaction term like Lotka-Volterra type is excluded in their works. The purpose of this paper is to establish a sufficient condition for the existence of global solutions for (1) without any restrictions on the term in the higher dimensional case. Our main result is as follows.
Theorem 1.1 Let and . Assume that , satisfy the zero Neumann boundary condition and belong to for some . Then (1) possesses a unique non-negative solution , such that and . Moreover, if or , then .
The paper is organized as follows. In Section 2, we present some known results and prove some preliminary results that are used in Section 3. In Section 3, we establish -estimates for v. In Section 4, we establish -estimates for v and give a proof of Theorem 1.1.
Theorem A Let , , where , be nonnegative functions. Then the system (1) has a unique nonnegative solution such that , where is the maximal existence time of the solution. If the solution satisfies the estimate
then . If, in addition, , then .
Lemma 2.1 (i) Let be a solution of (1) in . There exists a positive constant such that and .
(ii) For any , there exist two positive constants and such that
(iii) For any , there exists a positive constant such that .
Proof (i) Applying the maximum principle to (1), we have the result (i).
Integrating the second equation of (1) over the domain Ω and by (i) and (H2), we have
Then we have . Integrating the above inequality from 0 to T, we have
Let , then satisfies the equation
where and . Since u, , are bounded and , we can prove in the same way as Lemma 2.4 in . Moreover, satisfies
Applying Proposition 2.1 in , we obtain , which in turn implies . □
In order to establish -estimates for solutions of (1), we need the following result which can be found in .
Lemma 2.2 Let and . Suppose that w satisfies
and there exist positive constants and such that for all . Then there exists a positive constant independent of w but depending on n, Ω, q, and , such that
3 -estimates for v
Let , . In this section, and shall denote positive constants that may depend on Ω, q, n, the coefficients in (1), T (but not on t) and the following norms , and .
Lemma 3.1 There exist positive constants and such that and if , where .
Proof For any constant , multiplying the second equation of (1) by and using the integration by parts, we obtain
Integrating (2) from 0 to t, we have
The last term in (3) may be estimated by
Let and . Since and ∇u is in , using the Hölder’s inequality, we have
The substitution (4) and (5) into (3) leads to
for any . By choosing a sufficiently small ε, such that , we have
where . Setting , we define
So, from (6), we have
For any , if
then, . Therefore, by the Hölder’s inequality, we have
where . Setting , by Lemma 2.1(ii), we have
Therefore, by (10), Lemma 2.2 and the definition of E, the expression (9) yields
Then, by (7) and (11), we have
it follows from (12) that there exists a positive constant such that . By (11) and (12) we get which in turn implies that with for any q satisfying (8). Now, looking at (8), if , we have
then (8) holds for all q. So for , for all . If , then (8) is equivalent to
So, we see that v is in for all . Since (8) holds true for . So E is bounded for . It follows from (7) and (11) that is bounded for any n. Namely, there exist positive constants and such that for any n, and for . □
4 -estimates for v and a proof of Theorem 1.1
Lemma 4.1 Suppose that there are positive constants and such that . Then, for any .
Proof First, the equation of u can be written in the divergence form as
where is bounded in and is in with . Application of the Hölder continuity result (see , Theorem 10.1, p.204) yields
By similar arguments as in the proof of Lemma 2.1(iii), we see that . This implies in . Replacing by and by in the proof of Lemma 3.1, we see that either for any or else with . The latter case happens if and only if
If , applying Theorem 9.1 in  and its remark (p.351) again, we have ∇u in . Repeating the above steps, we will get the sequence of numbers
We stop and get the conclusion that for any when
It is not hard to verify by inducting that , , and
Therefore, (13) must hold for some k. We stop at this k and conclude that for any . □
Notice that for . Therefore, by Lemmas 3.1 and 4.1, we know that and for any . More precisely, there exists a positive constant depending only on T, Ω, the coefficients of (1) and initial conditions , , but not on t, such that
On the basis of the above results, we may demonstrate a proof of Theorem 1.1 analogous to that in the paper .
Proof of Theorem 1.1 Since for any if , we have . Hence it follows from the Sobolev embedding theorem that for any , which in turn implies
The second equation in (1) can be rewritten as
Since by the assumption H2, Lemmas 2.1 and 4.1, and v, and are all bounded for , the Hölder estimates  show that
Rewrite the first equation in (1) as
From (14) and (15), it follows that and are in , where . Applying the Schauder estimates, we have
In order to derive the regularity of v, it is convenient to introduce the function
Note that and are in , thus, an application of the Schauder estimates to (18) gives
Then (17) and imply that
Here, it is not hard to see that for the case or the case ,
When , . When , repeating the above bootstrap arguments by making use of (16) and (19) in place of (14) and (15), one can find that (16) and (19) are fulfilled with replaced by λ.
In the case or the above arguments are also valid if we replace (19) by (20). This completes the proof of Theorem 1.1. □
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This work was supported by the National Natural Science Foundation of China (no. 11061031, 11101334), the Fundamental Research Funds for the Gansu University, and NWNU-LKQN-11-21 Foundations.
The authors declare that they have no competing interests.
LZ directed the study and carried out the main results of this article. SF established the models and drafted the manuscript. All the authors read and approved the final manuscript.