Global behavior for a diffusive predator-prey system with Holling type II functional response

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.

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 [3]. 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 . . .

Lemma 2.1Letbe the solution of (1.1). There exists a positive constantsuch that

Proof Firstly, the existence of local solutions for system (1.1) is given in [4-6]. Roughly speaking, if , , there exists the maximum such that system (1.1) admits a unique non-negative solution

If

then .

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 onand, such that

Furthermore,

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 [[7], 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 thatis a bounded function satisfying

with the boundary conditionon, where. Then ∇Wis in.

The proof of the above lemma can be found in [[8], Proposition 2.1].

The following result can be derived from Lemma 2.3 and Lemma 2.4 of [9].

Lemma 2.4Let, . If

and there exist positive constantsandsuch 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 constantsuch 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 2.6Letbe a fixed bounded domain and. Then for allwith, one has

(1) , , ;

(2) , , ;

(3) , , ,

whereCis a positive constant dependent onq, n, Ω, T.

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 thatandsatisfy homogeneous Neumann boundary conditions and belong tofor some. Then system (1.1) has a unique non-negative solutionif the space dimension is.

Proof When , the proof is similar to the methods of [10-12]. So, we just give the proof of Theorem 3.1 for . The proof is divided into three parts.

(i) -, -estimate and -estimate for v.

Firstly, integrating the first equation of (1.1) over Ω, we have

Thus, for all , we can obtain

where .

Furthermore,

Secondly, linear combination of the second and first equations of (1.1) and integrating over Ω yields

So, we get

Further,

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

Let

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 [13] to (3.16), we need to prove the following conditions:

(1) is bounded;

(2) ;

(3) ,

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 [[5], 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 [[5], 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 [[13], 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 [13] that . Since , we have .

So,

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 [13], 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).

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