Research Article | Open Access
Existence and Uniqueness of Positive Periodic Solutions for a Delayed Predator-Prey Model with Dispersion and Impulses
An impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays is investigated, where we assume the model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches. By applying the continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system. Some known results subject to the underlying systems without impulses are improved and generalized. As an application, we also give two examples to illustrate the feasibility of our main results.
The aim of this paper is to investigate the existence and uniqueness of the positive periodic solution of the following impulsive Lotka-Volterra type predator-prey model with prey dispersal in two-patch environments and time delays: with the following initial conditions: where represents the prey population in the ith patch and represents the predator population for both patches. is the intrinsic growth rate of the prey in the ith patch and are the density-dependent coefficients of the prey at the ith patch. and are the capturing rates of the predator in patches 1 and 2, respectively, and and are the conversion rates of nutrients into the reproduction of the predator. is the death rate of the predator and denotes the dispersal rate of the prey in the ith patch . is the delay due to gestation; that is, mature adult predators can only contribute to the production of predator biomass. In addition, we have included the term in the dynamics of the predator to incorporate the negative feedback of predator crowding, where represent the population at regular harvest pulse.
As was pointed out by Xu and Chen , dispersal between patches often occurs in ecological environments, and more realistic models should include the dispersal process. During the last decade, many scholars had done excellent works on the predator-prey system with dispersal; see [2–16] and the references cited therein. In , Cui proposed the following two species predator-prey system with prey dispersal: where and represent the population density of prey species and predator species in patch 1 and is the density of prey species in patch 2. Predator species is confined to patch 1, while the prey species can diffuse between two patches. is strictly positive functions that can be viewed as the dispersal rate or inverse barrier strength. By giving a thoroughly analysis on the right hand side of the system (3), Cui obtained a sufficient and necessary condition to guarantee the predator and prey species to be permanent.
It is unlike system (3), where the predator species is confined on patch 1. In , the authors proposed a model of patches with a barrier only as far as the prey population is concerned, whereas the predator population has no barriers between patches; that is, they considered the following predator-prey system in two-patch environment: where represents the prey population in the ith patch, , at time . stands for the total predator population for both patches. The predator population is assumed to have no barriers between patches. is the specific growth rate for the prey population in the absence of predation when it is restricted to the ith patch. is the predator functional response of the predator population on the prey in the ith patch. is a positive constant that can be viewed as the dispersal rate or inverse barrier strength. is the density-dependent death rate of the predator in the absence of prey. is the conversion ratio of prey into predator. Conditions have been established in  for the existence, uniform persistence, and local and global stability of positive steady states of system (4).
The model (4), however, as was pointed out by Yang , is not perfect. Therefore, Xu et al.  had considered the following delayed periodic Lotka-Volterra type predator-prey system with prey dispersal in two-patch environments: with initial conditions: by using Gaines and Mawhins continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, they obtained a set of easily verifiable sufficient conditions to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system (5).
On the other hand, impulsive differential equations [17–19] arise frequently in the modeling of many physical systems whose states are subjects to sudden change at certain moments, for example, in population biology, the diffusion of chemicals, the spread of heat, the radiation of electromagnetic waves, the maintenance of a species through instantaneous stocking, and harvesting. There has been an increasing interest in the investigation for such equations during the past few years. There are many researchers who introduced impulsive differential equations in population dynamics [20–28]. However, to the best of the authors’ knowledge, to this day, no scholars had done works on the existence, uniqueness, and global stability of positive periodic solution of (1). Based on the idea of [10–15], we propose and study the system (1) in this paper.
For the sake of generality and convenience, we always make the following fundamental assumptions:, and are all positive periodic continuous functions with period , and ; satisfies and , are constants with and there exists a positive integer such that , . Without loss of generality, we can assume that and , then .
In what follows, we will use the notation.
Throughout this paper, we make the following notation and assumptions.
Let be a constant and , , with the norm defined by ; , , with the norm defined by ; , , if , , exists, ; , ; , , with the norm defined by ; , , with the norm defined by .Then those spaces are all Banach spaces. We also denote The aim of this paper is to obtain a set of easily verifiable sufficient conditions to guarantee the existence, uniqueness, and global stability of positive periodic solutions of the system (1) by further developing the analysis technique of [10–15]. The organization of this paper is as follows. In the next section, first, the necessary knowledge and lemmas are provided. Second, by using continuation theorem developed by Gaines and Mawhin , we establish the existence of at least one periodic solution of system (1). In Section 3, the uniqueness and global attractivity of periodic solution of system (1) are presented. Finally, we give two examples to show our results.
2. Existence of Positive Periodic Solutions
In this section, by using the continuation theorem which was proposed in  by Gaines and Mawhin, we will establish the existence conditions of at least one positive periodic solution to system (1). In doing so, we will introduce the following definitions and lemmas.
Let , be a real Banach space, let be a linear mapping, and let be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that , , it follows that is invertible; we denote the inverse of that map by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exist isomorphisms . Let denote the space of -periodic functions which are continuous for , are continuous from the left for , and have discontinuities of the first kind at point . We also denote .
Definition 1 (see ). The set is said to be quasiequicontinuous in if for any there exists such that if , , , and , then .
Lemma 2 (Gaines and Mawhin ). Let and be two Banach spaces and let be a Fredholm operator with index zero. is an open bounded set, and is L-compact on . Suppose (a)for each , every solution of is such that ;(b) for each ;(c). Then, the equation has at least one solution lying in .
Lemma 3 (see ). Assume that , are continuous nonnegative functions defined on the interval . Then there exists such that .
Lemma 5. The region , , is the positive invariable region of the system (1).
Proof. In view of biological population, we obtain , . By the system (1), we have Therefore, the conclusion is true.
Proof. Since , and is continuous on , it follows that has a unique inverse function on . Hence, it suffices to show that , . For any , by the condition , one can find that, for the equation , exists a unique solution and, for the equation , exists a unique solution ; that is and , that is, and . As it follows that . Since , thus, we have and . We can easily obtain that if , , then , , where is the unique inverse function of , which together with implies that . The proof of Lemma 6 is completed.
We denote by the inverse of , .
Theorem 7. In addition to ()(), assume the following conditions hold:, . Then, system (1) has at least one positive -periodic solution, where
Proof. We carry out the change of variable , ; then (1) can be transformed to
It is easy to see that if system (12) has one -periodic solution , then is a positive -periodic solution of system (1). Therefore, it suffices to prove system (12) has a -periodic solution. Let
where is the Euclidean norm of . Then and are Banach spaces.
Let and withIt is not difficult to show that and . So, is closed in and is a Fredholm mapping of index zero. Take It is trivial to show that , are continuous projectors such that , , and hence, the generalized inverse exists. In the following, we first devote ourselves to deriving the explicit expression of . Taking , then exists an such that Then direct integration produces that ; that is , which, together with (20), implies Then, that is Thus, for
Clearly, and are continuous. By applying Ascoli-Arzela theorem, one can easily show that are relatively compact for any open bounded set . Moreover, is obviously bounded. Thus, is -compact on for any open bounded set . Now, we reach the position to search for an appropriate open bounded set for the application of Lemma 2. Considering the operate equation , , we have Since are -periodic functions, we need only to prove the result in the interval . Integrating (26) over the interval leads to Hence, we have It follows from (26)–(28) that Multiplying the first equation of (26) by and integrating over we have Since , we obtain which yields Similarly, multiplying the second equation of (26) by and integrating over gives By using the inequalities it follows from (32)–(34) that If , then it follows from the second equation of (35) that which implies If , similarly, we obtain Set Then it follows from (37)–(39) that Note that ; then there exists such that Then it follows from (40) and (41), that Since , we can let , that is, ; then According to Lemma 6, we know . Thus, Similarly, we have On the other hand, by Lemma 6, we can see that , so we can derive