Dynamics of Mutualism-Competition-Predator System with Beddington-DeAngelis Functional Responses and Impulsive Perturbations
We explore the dynamics of a class of mutualism-competition-predator interaction models with Beddington-DeAngelis functional responses and impulsive perturbations. Sufficient conditions for existence of positive periodic solution are established by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are given for the global stability and the globally exponential stability of system by employing comparison principle and Lyapunov method.
The ecological predator-prey systems and impulsive functional differential equations have been studied extensively by many authors [1–7]. A predator's per capita feeding rate on prey, or its functional response, provides a foundation for predator-prey theory. Since 1959, Holling's prey-dependent type II functional response, a model that is a function of prey abundance only, has served as the basis for a large literature on predator-prey theory. The traditional Kolmogorov type predator-prey model with Holling's type II functional response: and its various generalized forms have received great attention from both theoretical and mathematical biologists and have been well studied [1, 8, 9]. In (1.1), and represent the densities of prey specie and predator specie at time , respectively. Predator-prey model with Holling's type II functional response assumes that predators do not interfere with one another's activities; thus competition among predators for food occurs only via the depletion of prey. However, when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be predator-dependent. The predator-prey system with the Beddington-DeAngelis functional response: was originally introduced by Beddington  and DeAngelis et al. , independently, where and represent the densities of prey specie and predator specie at time , respectively. The main difference of this functional response from Holling's type II functional response is that it contains an extra term presenting mutual interference by predators. In this model, individuals from a population of two or more predators not only allocate time to searching for and processing prey, but also spend some time engaging in encounters with other predators, the Beddington-DeAngelis type can provide better descriptions of predator feeding over a range of predator-prey abundances.
On the other hand, differential equations with impulsive effects form a wide set of different problems. During the last three decades those problems were intensively studied. Some authors devote themselves to the study of impulsive differential equation [12–22]. The main definitions and results of the theory of systems of ordinary differential equations with impulse effects were given in [12, 13, 15]. Similarity and differentiality of such problems of applied mathematics with corresponding problems of ordinary differential equations (and without the conditions of impulsive effects) were demonstrated, and general characteristics of these systems were described. Periodic and almost-periodic solutions of differential equations with impulsive effects were studied in . De La Sen investigated time-varying systems with nonnecessarily bounded everywhere continuous time-differentiable time-varying point delays . The delay-free and delayed dynamics are assumed to be time-varying and impulsive, the constructed solution trajectories of both the unforced and forced systems are obtained from different (input-state space/output space and state space to output space) operators. The system stability and the compactness of the operators describing the solution trajectories are well investigated. De La Sen and Luo also obtain sufficiency-type stability results for time-delay linear systems with constant point delays under impulsive inputs of impulses of state-dependent amplitudes occurring separately through time. They proved that the amplitudes of the impulses and the time intervals between impulses may be chosen sufficiently large if the delay-free dynamics is sufficiently stable compared to the delayed one . Xu and Sun investigated the problem of finite-time stability of linear time-varying singular systems with impulses at fixed times, they have proposed a sufficient condition for linear singular impulsive systems to be finite-time stable in terms of a set of coupled matrix inequalities . Zhang and Sun also considered the stability of impulsive linear differential equations with time delay. By using Lyapunov functions and analysis technique, they get some results for the stability of impulsive linear differential equations with time delay .
Some impulsive factors have also great impact on the growth of a population. For example, we notice that the births of many species are not continuous but happen at some regular time (e.g., the births of some wildlife are seasonal). Moreover, the human beings have been harvesting or stocking species at some time, then the species is affected by another type of impulse. If we incorporate these impulsive factors into the models of population interactions, the models must be governed by impulsive ordinary differential equations.
Population communities are embedded in periodically varying environments. Therefore, this study should take into account the biological and environmental periodicity (e.g., seasonal effects of weather, food supplies, and mating habits), we focus on the existence of periodic solution with strictly positive components. In real world, any biological or environmental parameters are naturally subject to fluctuation in time, so it is reasonable to study the corresponding nonautonomous system.
The first author investigated the existence of positive periodic solutions of a nonautonomous competitive Lotka-Volterra system with impulse and Holling type III functional response . An -dimensional Lotka-Volterra system with fixed moments of impulsive perturbations is given in  by Ahmad and Stamova. By means of piecewise continuous functions which are modifications of classical Lyapunov’ functions they give sufficient conditions for asymptotic stability of the solutions. Inspired by [23, 24], in this paper, we focus our attention on the existence of periodic solution and globally asymptotic stability of solutions for multispecies mutualism-competition-predator system with impulses in which the competition among predator species and the mutualism among prey species are simultaneously considered. In general, mutualism is relevant to two species [25, 26], thus we focus on two prey species and assume that there exists the relation of mutualism between two prey species. The primary approach is based on the coincidence degree and its related continuation theorem , which has been widely used in dealing with the existence of periodic solutions of differential equations. We will investigate the following nonautonomous system: where(i) denote the densities of prey species at time , respectively;(ii) denote the density of predator species at time , respectively;(iii) represent the sum of the birth rate and the harvesting (or stocking) rate of at time , respectively;(iv) represent the sum of the birth rate and the harvesting (or stocking) rate of at time , respectively; (v) and represent the right and left limit of at , and represent the right and left limit of at .
Let be a bounded continuous function on . Define Particularly, if are -periodic functions with respect to , we denote The range of the indices and are used in this paper unless otherwise stated.
Throughout the paper, we give the hypotheses as follows(A1) for any , , , , , , , , , , , are nonnegative continuous -periodic functions and , , ;(A2), , , , , are constants. There exists a positive integer , such that . Without loss of generality, we also suppose that and , then it follows that ;(A3) is left-continuous at , that is, the following relations are satisfied: (A4) and ;(A5).
2. Basic Concepts and Lemma
Let be a piecewise continuous function with points of discontinuity , we denote Suppose It can be easily proved that and are Banach spaces under the condition endowed the above norms.
Definition 2.1. The mapping is called a solution of system (1.1) in , if (1) is piecewise continuous in and are discontinuous points of the first kind of , and are left continuous at ;(2) satisfies system in .
Obviously, if is a solution of (1.1) satisfying in , then from the periodicity of the vector field of (1.1), we know that is an -periodic solution for (1.1). Thus, the problem which discusses the existence of solution of (1.1) will be transformed to discuss the existence of periodic solution for (1.1) in satisfying . In order to explore the existence of positive periodic solutions of (1.1), we recall some concepts and results on coincidence degree from [27, pages 39-40], borrowing notations there.
Let be normed vector spaces, a linear mapping, and a continuous mapping. If and is a closed in , then the mapping will be called a Fredholm mapping of index zero. If is a Fredholm mapping of index zero, there exists continuous projects and such that . It follows that has an inverse which is denoted by . If be an open bounded subset of , the mapping will be called -compact on provided that is bounded, and is compact. Since is isomorphic to , there exists an isomorphism .
The following continuation theorem is due to Gaines and Mawhin .
Lemma 2.3 2.3 (Continuation Theorem in ). Let a Fredholm mapping of index zero and be -compact on . Suppose that (a)for each , every solution of such that ;(b) for each and .Then the operator equation has at least one solution lying in .
Lemma 2.4. Suppose the hypotheses (A1)–(A5) hold. is a solution of (1.1), then
Proof. By integrating of (1.1) in the interval , we have
There does not exist point of discontinuity of , in the interval , it is obvious that , for , hence . We have from (1.1) that It follows from (A2) that We now integrate (1.1) in the interval , and we have for . From the above relation it follows that for .
By similar arguments, we can obtain that for . So for .
3. Existence of Positive Periodic Solution
We denote where . We also denote the matrix obtained by replacing the th column of with .
Proof. Let be the Cartesian production of -tuples, , where is -dimensional Euclidean space. It is clear that for any .
We define for any , and , where is the norm of , and is any norm of . Then it is trivial to check that are both Banach spaces when they are endowed with the above norms and , respectively.
Let Then Since is closed in is a Fredholm mapping of index zero. We respectively define and in the following: It is easy to show that , are continuous projectors such that . Furthermore, the generalized inverse exists. That follows one will calculate .
Let then that is, Note that From(3.9), we obtain and hence, that is, Thus where Clearly, and are continuous. Using the Arzela-Ascoli theorem, it is not diffcult to show that is compact for any open bounded set . Moreover, is bounded. Thus, is -compact on with any open bounded set . A isomorphism of onto can be chosen to be the identity mapping; since , there exists an isomorphism given by Corresponding to operator equation , we have
Suppose that is a solution of system (3.17) for a fixed . Integrating on both sides of (3.17) from to , we obtain Since , there exist such that On the other hand, note that there exist such that If , then . While if , we have . Thus we can obtain from (3.18) and (3.19) that It follows that From (3.17) and (3.18), we obtain By (3.22) and (3.23), we have that On the other hand, by (3.18) and (3.20), we also have and hence, By (3.18), we have where is a positive number satisfied for ; therefore, So Take . By (3.24)–(3.29), we have Here, are independent of the choice of .
Let , set , where is taken sufficiently large, such that the solution of the equation satisfies
Let . Then it is clear that satisfies the requirement (a) of Lemma 2.3.
When , is a constant vector in with , then , we check and according to mean value theorem, there exist such that For any , we have