Research Article | Open Access
Stability Analysis of Three-Species Almost Periodic Competition Models with Grazing Rates and Diffusions
Almost periodic solution of a three-species competition system with grazing rates and diffusions is investigated. By using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained by Q. C. Lin (1999), F. D. Chen and X. X. Chen (2003), and Y. Q. Liu, S. L, Xie, and Z. D. Xie (1996).
In this paper, we study the following three-species competition system with grazing rates and diffusions: where , is the bounded open subset of with smooth boundary , which represent the habitat domain for three species. System (1.1) is supplement with boundary conditions and initial conditions: where denotes the outward normal derivation on , and represent the density of th species at point and the time of . Here, , , , , , and () denote the diffusivity rates, competition rates, and grazing rates, respectively. They are almost periodic functions in real number field . is a Laplace operator on .
System (1.1)–(1.3) describes the interaction among three species and is an important model in biomathcmatics, which has been intensively investigated, and much attention is carried to the problem [1–8]. When there is no diffusion, Jiang  and Lin  studied the existence, uniqueness, and stability on periodic solution and almost periodic solution for two-species competition system under the condition that the coefficients are the periodic function and almost periodic function, respectively; F. D. Chen and X. X. Chen  extended the results in  to n-species case. When there are no diffusion and grazing rates, Zhang and Wang [4, 5] investigated the existence of a positive periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay and the existence of multiple positive periodic solutions for a generalized delayed population model with exploited term by using the continuation theorem of coincidence degree theory; Hu and Zhang  established criteria for the existence of at least four positive periodic solutions for a discrete time-delayed predator-prey system with nonmonotonic functional response and harvesting by employing the continuation theorem of coincidence degree theory. When there are no grazing rates, Pao and Wang  proved the stability for invariable coefficient case by utilizing the method of upper and lower solutions. Liu et al.  showed the stability on the periodic solution for n-species competition system with grazing rates and diffusions. Nevertheless, generally speaking, the system does not always change strictly according to periodic laws, sometimes it changes according to almost periodic laws, and it is important to survey almost periodic solution for the multispecies competition system with grazing rates and diffusions. To sum up, we pay more attention to almost periodic solution of a three-species competition system (1.1)–(1.3) with grazing rates and diffusions; in this paper, by using the method of upper and lower solutions and Schauder fixed point theorem as well as Lyapunov stability theory, we obtain sufficient conditions which ensure the existence and globally asymptotically stable for the strictly positive space homogenous almost periodic solution, which extend and include corresponding results obtained in [2, 3, 8]. Many other results on the periodic solution and almost periodic solution can be found in [9–16].
Firstly, we give out some definitions and lemmas.
Definition 2.1. Suppose that is a continuous function in . Then is said to be almost periodic in if for every corresponds such that for any interval whose length is equal to there is at least one such that
Definition 2.2. If a smooth function satisfies (1.1) in , and every component of is the almost periodic function, we called that is a spatial homogeneity almost periodic solution for (1.1), which is denoted by .
Definition 2.3. For any nonnegative smooth initial data if there exists a unique positive solution for the system (1.1) with boundary conditions (1.2), and , , uniformly for , we called that spatial homogeneity almost periodic solution is globally asymptotically stable.
For the almost periodic function in , one denotes , , and . When is periodic function, one denotes .
3. Main Results and Proofs
Now we are in a position to state our main results and give our proofs.
Theorem 3.1. If are positive numbers, and are satisfied for , then there exists a strictly positive spatial homogeneity almost periodic solution for (1.1).
Proof. By the conditions in Theorem 3.1, we have
Let Then we have , and Therefore Furthermore, by the given conditions in Theorem 3.1, one has Thus Combining (3.5) and (3.7), we have
Let We consider the following system corresponding to (1.1): Let , ; then (3.10) becomes For any , by , , , we observe  that have almost periodic solution: By the system (3.13), we define a mapping : Combining (3.8) and (3.13), we have Therefore, , that is, . If is uniformly boundness and equicontinuous, by Ascoli-Arzela theorem, is compact mapping.
It is obvious to obtain uniformly boundedness. In fact, for any , by (3.15) we have ; that is, it satisfies
Next we prove equicontinuous. For any , we denote , and then
Let ; we obtain Recalling , we deduce that there exists a positive number such that ; then (3.18) becomes where .
Similarly, we have where , and is a positive number.
By a completely analogous argument, we obtain where , and is a positive number.
By (3.19)–(3.21), for any , we derive
Thus, is a compact mapping which maps into itself; by Schauder fixed point theorem, there exists a fixed point for ; namely, (3.11) has a solution; therefore there exists a strictly positive almost periodic solution for system (3.10). It is obvious that is also the spatial homogeneity almost periodic solution for (1.1).
Theorem 3.2. Under the conditions of Theorem 3.1, suppose that system (1.1) satisfies the following conditions: Then there exists a strictly positive spatial homogeneity almost periodic solution for (1.1), and the corresponding solution for systems (1.1)–(1.3) is globally asymptotically stable; that is, the solution satisfies
Proof. We have obtained the existence by Theorem 3.1; next we pay more attention to the stability. Concerning (3.24), we have two cases on initial data . (1) . (2)There exists a point , such that or .
For the case (1), let , , ; then . Suppose that and are the solution for (3.10) corresponding to initial datum and , respectively; then there are a pair of ordered upper and lower solutions and for (1.1)–(1.3); by Lemma 2.5, there exists a unique solution for system (1.1)–(1.3), which satisfies
If we have then (3.24) holds. Therefore, if we want to obtain (3.26), we only need to prove that the solution for (3.10) with arbitrary positive initial data satisfies
Because of the initial datum and grazing rates , by the practical meaning in biology, we know that . Now let Then one has Namely, Consider the following Lyapunov function:
Let represent the right derivation on function ; we have Integrated by the time, we have By the nonnegative of and the boundedness of , we obtain that the is bounded, and convergences, by (3.32) we get , then the limit exists, and . If , then at least one of the following three inequalities holds. Without loss of generality, we assume that . Thus there is no point of intersection between and . Suppose that ; then we have . Thus which contradicts with the convergence of . Therefore ; consequently Then we obtain (3.27).
For the case (2), firstly, choose three sufficient large positive numbers , such that and , . Let , , ; then we have Namely, , and , are a pair of ordered upper and lower solutions for systems (1.1)–(1.3). By Lemma 2.5, there exists a unique solution for systems (1.1)–(1.3), which satisfy
Secondly, we choose positive numbers such that Accordingly, we have Next, we prove in for . Firstly, we show in . If there exists one point such that , by extremum principle, we have in . However , and not being constant zero, we obtain a contradiction. Therefore we have in . Then we show in . If there exists a point such that , by the extremum principle, we have , where , which is contrary with boundary conditions (1.2). Thus we have in .
For a fixed number , by (3.41), we have Because satisfy system (1.1) in and the conditions (1.2) in , thereby is regarded as a solution for system (1.1) under initial data , nevertheless, we have in ; combining the conclusions in case (1), we have By the arbitrariness of , we obtain
If of (1.1) are periodic functions in real number field , respectively, then we have the following results.
Corollary 3.3. If are positive numbers, and