Research Article | Open Access
Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers
We consider a strongly coupled predator-prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo-Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.
The principle of competitive exclusion asserts that two or more consumer species cannot coexist indefinitely upon a single limiting resource, which dates back to the pioneering work of Volterra  in the 1920s. Subsequently, Ayala  in 1969 demonstrated experimentally that two species of Drosophila can coexist upon a single limiting resource. Ayala's experiments have received much attention (see the comprehensive survey by Cantrell and Cosner ). Schoener  in 1976 found that intraspecific interference among consumers may lead to coexistence of multiple consumer species upon a single resource. To examine more closely the implications of feeding interference among conspecific consumers on consumer-resource dynamics, Cantrell et al.  in 2004 proposed the following predator-prey system: where , and are positive constants, represents the density of the limiting resource at time and denote two consumers species. The first consumer species feeds upon the resource according to the Holling II functional response, while the second consumer species feeds upon the resource following the Beddington-DeAngelis functional response, and they compete for the common resource. For more details on the backgrounds about this system see .
The system (1.1) has a positive equilibrium under the suitable conditions, where The Jacobian matrix of the system (1.1) at can be written as where The following results were proved in :(1)the system (1.1) is dissipative;(2)the positive equilibrium of (1.1) is locally stable if and ; and(3)the positive equilibrium of (1.1) is globally stable if .
Rescaling the system (1.1) such that yields The corresponding weakly coupled reaction-diffusion system for (1.6) is as follows: where is a bounded smooth domain, is the outward unit normal vector of the boundary . The constants , and , called diffusion coefficients, are positive, and , and are nonnegative functions which are not identically zero. The system (1.7) has a constant positive steady-state solution if and only if where
In , Hei and Yu proved the following main results.
(2) Let and be fixed positive constants which satisfy and .
Then there exists a positive constant , such that (1.7) has no nonconstant positive solution if , and , where are the eigenvalues of the operator on with the homogeneous Neumann boundary condition.
(3) Let be fixed positive constants. Assume that , and (1.8) hold. Furthermore, assume that one of the following conditions is satisfied:(i), for some , and the sum is odd;(ii), , for some , and the sum is odd. Then there exists a positive constant , such that (1.7) has at least one nonconstant positive solution if , where and is the eigenspace corresponding to in .
(4) The bifurcation of nonconstant positive solutions for (1.7) was studied.
In recent years, the SKT type cross-diffusion systems have attracted the attention of a great number of investigators and have been successfully developed on the theoretical backgrounds. The above work mainly concentrate on (1) the instability and stability induced by cross-diffusion, and the existence of nonconstant positive steady-state solutions [7–14]; (2) the global existence of strong solutions [15–23]; (3) the global existence of weak solutions based on semidiscretization or finite element approximation [24–30]; and (4) the dynamical behaviors [18, 19, 31, 32], and so forth. The corresponding SKT type cross-diffusion system for (1.7) is as follows: where are positive constants, are referred as self-diffusion pressures, and are cross-diffusion pressures. The self-diffusion implies the movement of individuals from a higher to lower concentration region. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species. The value of cross-diffusion coefficient may be positive, negative, or zero. The positive cross-diffusion coefficient denotes the movement of the species in the direction of lower concentration of another species and negative cross-diffusion coefficient denotes that one species tends to diffuse in the direction of higher concentration of another species (e.g., ).
The local existence of solutions for the system (1.13) is an immediate consequence of a series of important papers [34–36] by Amann. Roughly speaking, if , and in with , then (1.13) has a unique nonnegative solution , where is the maximal existence time for the solution. If the solution satisfies the estimate then . Moreover, if , then .
For the following SKT system Yamada in  proposed four open problems:(1)the global existence of solutions of in the case and the space dimension ;(2)the global existence in the case ;(3)in order to study the asymptotic behavior of as need to establish the uniform boundedness of global solutions; and(4)the global existence of solutions for the following full SKT system: with .
Very few global existence results for (1.13) are known. The main purpose of this paper is to establish the uniform boundedness of global solutions for the system (1.13) in one space dimension. For convenience, we consider the following system: We firstly investigate the global existence and the uniform boundedness of the solutions for (1.16), then prove the global asymptotic stability of the positive equilibrium of (1.16) by an important lemma from . The proof is complete and complement to the uniform convergence theorems in papers [38–40].
For simplicity, we denote by and by . Our main results are as follows.
Theorem 1.1. Let , be the unique nonnegative solution of the system (1.16) in the maximal existence interval . Assume that Then there exist and positive constants which depend on and , such that and . Moreover, in the case that , where and are positive constants, depend on , but do not depend on .
Remark 1.2. Since the continuous embedding holds only in one space dimension, we can only establish the uniform maximum-norn estimates about time for the solution in one space dimension.
Theorem 1.3. Assume that all conditions in Theorem 1.1 are satisfied. Assume further that and (1.8) hold, where is given by (1.19). Then the unique positive equilibrium of (1.16) is globally asymptotically stable.
Examples. The following two examples satisfy all conditions of Theorem 1.3:
2. Global Solutions
In order to establish the uniform -estimates of the solutions for the system (1.16), the following Gagliardo-Nirenberg-type inequalities and the corresponding corollary play important roles (see [38, 41]).
Theorem 2.1. Let be a bounded domain with . For every function , the derivative satisfies the inequality provided one of the following three conditions is satisfied: (1) , (2) , or (3) and is not a nonnegative integer, where for all , and the positive constant depends on .
Corollary 2.2. There exists a universal constant such that
Throughout this paper, we always denote that is a Sobolev embedding constant or other kind of universal constant, are some positive constants which depend only on and are positive constants depending on and . When , depend on , but do not depend on .
Proof of Theorem 1.1. Taking integration of the three equations in (1.16) over , respectively, and combining the three integration equalities linearly, we have
It follows from the Young inequality and the Hölder inequality that
where . So there exist positive constants and depending on , and , such that
Moreover, there exists a positive constant which depends on and -norm of , such that
Multiplying the first three equations in the system (1.16) by , respectively, and integrating over , we have from which it follows that where , It is obvious that is a positive definite quadratic form of if (1.17) holds. So (1.17) implies that
Now, we proceed in the following two cases.
(i) One has . The inequality (2.2) implies that . So we have , and It follows from (2.12) and (2.13) that This means that there exist positive constants and depending on , and , such that When is independent of because the zero point of the right-hand side in (2.14) can be estimated by positive constants independent of .
(ii) One has . Repeating estimates in (i) by (2.8)′, we can obtain that there exists a positive constant depending on and the -norm of , such that When is independent of .
To estimate , we introduce the following scaling: Denoting , and using instead of , respectively, then the system (1.16) reduces to where .
We still proceed in the following two cases.(i) One has . It is not hard to verify that where .
Multiplying the first three equations in (2.17) by , integrating them over the domain , respectively, and then adding up the three integration equalities, we have where . Notice by (1.17) that there exists a positive constant depending only on , such that Thus,
Using the Young inequality, Hölder inequality, and (2.18), we can obtain the following estimates:
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of (2.21), we have Similarly, we can obtain