Global Behavior for a Strongly Coupled Predator-Prey Model with One Resource and Two Consumers

and Applied Analysis 3 Rescaling the system 1.1 such that u K – −→ u, v – −→ v, w – −→ w, rt – −→ t, a r – −→ a, bK – −→ b, A r – −→ A, BK – −→ B, D – −→ D, l r – −→ l, eK l – −→ e, L r – −→ L, EK L – −→ E 1.5


Introduction
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 1 in the 1920s. Subsequently, Ayala 2 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 3 . Schoener 4 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 The Jacobian matrix of the system 1.1 at E can be written as The following results were proved in 5 : 1 the system 1.1 is dissipative; 2 the positive equilibrium E of 1.1 is locally stable if − a 11 a 33 > 0 and a 11 a 11 a 33 − a 13 a 31 − a 12 a 21 − a 33 a 11 a 33 − a 31 a 31 < 0; and 3 the positive equilibrium E of 1.1 is globally stable if max{b, B} K − u < 1.
Abstract and Applied Analysis 3 Rescaling the system 1.1 such that The corresponding weakly coupled reaction-diffusion system for 1.6 is as follows: where Ω ⊂ R N is a bounded smooth domain, ν is the outward unit normal vector of the boundary ∂Ω, ∂ ν ∂/∂ν. The constants d 1 , d 2 , and d 3 , called diffusion coefficients, are positive, and u 0 x , v 0 x , and w 0 x are nonnegative functions which are not identically zero. The system 1.7 has a constant positive steady-state solution E * u * , v * , w * if and only if 1.9 4 Abstract and Applied Analysis In 6 , Hei and Yu proved the following main results. 1 The equilibrium u * , v * , w * of 1.7 is locally asymptotically stable if 1.8 and Then there exists a positive constant D 1 , such that 1.7 has no nonconstant positive where 0 μ 0 < μ 1 < μ 2 < · · · are the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition.
Then there exists a positive constant D 2 , such that 1.7 has at least one nonconstant positive and E μ i is the eigenspace corresponding to μ i in H 1 Ω . 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 Abstract and Applied Analysis 5 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 α ij i, j 1, 2, 3 are positive constants, α ii i 1, 2, 3 are referred as self-diffusion pressures, and α ij i, j 1, 2, 3, i / j 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., 33 .
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 u 0 x , v 0 x , and w 0 x in W 1 p Ω with p > n, then 1.13 has a unique nonnegative solution u, v, w ∈ C 0, T , W 1 p Ω ∩ C ∞ 0, T , C ∞ Ω , where T ∈ 0, ∞ is the maximal existence time for the solution. If the solution u, v, w satisfies the estimate 2 the global existence in the case γ 0; 3 in order to study the asymptotic behavior of u, v as t → ∞ need to establish the uniform boundedness of global solutions; and 4 the global existence of solutions for the following full SKT system: with α, γ, β, δ > 0. 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 u * , v * , w * of 1.16 by an important lemma from 37 . The proof is complete and complement to the uniform convergence theorems in papers 38-40 . It is obvious that u * , v * , w * is the unique positive equilibrium of the system 1.16 if 1.8 holds.
For simplicity, we denote · W k p 0,1 by | · | k,p and · L p 0,1 by | · | p . Our main results are as follows.

Abstract and Applied Analysis
Examples. The following two examples satisfy all conditions of Theorem 1.3: 1.22

Global Solutions
In order to establish the uniform W 1 2 -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 .
provided one of the following three conditions is satisfied: 1 r ≤ q, 2 0 < n r − q / mrq < 1, or 3 n r − q / mrq 1 and m − n/q is not a nonnegative integer, where 1/p j/n a 1/r − m/n 1 − a /q for all a ∈ j/m, 1 , and the positive constant C depends on n, m, j, q, r, a.

Corollary 2.2. There exists a universal constant C such that
Throughout this paper, we always denote that C is a Sobolev embedding constant or other kind of universal constant, A j , B j , C j are some positive constants which depend only on Proof of Theorem 1.1. Taking integration of the three equations in 1.16 over 0, 1 , respectively, and combining the three integration equalities linearly, we have It follows from the Young inequality and the Hölder inequality that Multiplying the first three equations in the system 1.16 by u, v, w, respectively, and integrating over 0, 1 , we have from which it follows that It is obvious that q u x , v x , w x is a positive definite quadratic form of u x , v x , w x if 1.17 holds. So 1.17 implies that

2.12
Abstract and Applied Analysis 11 Now, we proceed in the following two cases. i One has t ≥ τ 0 . The inequality 2.2 implies that |u| 6 2 It follows from 2.12 and 2.13 that When d ≥ 1, M 1 is independent of d.
i One has t ≥ τ * 1 d 1 τ 1 . It is not hard to verify that 1 0 u dx,

2.20
Abstract and Applied Analysis 13 Thus,

2.21
Using the Young inequality, Hölder inequality, and 2.18 , we can obtain the following estimates:

2.22
Applying the above estimates and Gagliardo-Nirenberg-type inequalities to the terms on the right-hand side of 2.21 , we have

2.24
Similarly, we can obtain

2.25
By the above inequalities and the condition 1.17 , we have

2.26
where λ is a constant depending only on ε α ij i, j 1, 2, 3 . Choose a small enough positive number ε which depends on α ij i, j 1, 2, 3 , a, b, e, l, A, B, D, E, and L, such that λε < C 3 . Substituting inequalities 2.24 and 2.26 into 2.21 , one can obtain

2.29
Moreover, one can obtain by 2.5 and 2.18 that

2.30
Combining 2.27 , 2.29 , and 2.30 , we have where we can transform the formulations of u x , v x , w x into fraction representations, then distribute the denominators of the absolute value of the fractions to the numerators item and enlarge the term concerning with u x , v x or w x to obtain where C is a constant depending only on ξ, η, α ij i, j 1, 2, 3 . After scaling back and contacting estimates 2.   1, 2, 3 , a, b, e, l, A, B, D 1, 2, 3 ,a, b, e, l, A, B, D, E, L and the initial functions u 0 , v 0 , w 0 , such that |u ·, t | 1,2 , |v ·, t | 1,2 , |w ·, t | 1,2 ≤ M , t ≥ 0.

2.37
Further, in the case that d 1