Abstract
This paper is concerned with a discrete predator-prey model with Holling II functional response and delays. Applying Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we obtain some sufficient conditions for the existence global asymptotic stability of positive periodic solutions of the model.
1. Introduction
In recent years, numerous studies have been carried out on predator-prey interactions using Lotka-Volterra type functional response [1]. Considering the simplification of assumptions on prey searching, prey consumption, and environmental complexity, Holling suggested three different kinds of functional response to model more realistic predator-prey interactions than what is possible with the standard Lotka-Volterra type response [1, 2]. Many predator-prey systems with Holling type II functional response have been investigated. In particular, the periodic solutions are of great interest. During the past decades, a large number of excellent results have been reported for a lot of different predator-prey models with Holling type II functional response. For example, Ko and Ryu [3] investigated the qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. Zhou and Shi [4] considered the existence, bifurcation, and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional responses. Liu and Yan [5] dealt with the positive periodic solutions for a neutral delay ratio-dependent predator-prey model with a Holling type II functional response. For more related work, one can see [6–25]. Dunkel [26] pointed out that feedback control item in predator-prey models depends on the population number for certain time past and also depends on the average of the population number for a period of time past. Motivated by the viewpoint, we proposed the following predator-prey model with Holling II functional response and distributed delays: where stands for the prey and predator density at time . For the biological meaning of model (1), one can see [27].
As pointed out in [28–35], discrete time models are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have nonoverlapping generations. What is more, we can also get more accurate numerical simulation results from the discrete-time systems. Thus it is reasonable and interesting to investigate discrete-time systems governed by difference equations. Following [33, 36], we obtain the discrete form of system (1) as follows: which is a discrete time analogue of system (1), where , stands for the prey and predator density at time , , are strictly positive sequences, and is a positive constant.
In order to obtain our main results, we assume that(H1) is positive -periodic; that is, for any , where , a fixed positive integer, denotes the common period of the parameters in system (2);(H2)the following inequalities are satisfied:
The principle aim of this paper is to discuss the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model with Holling II functional response and distributed delays.
The paper is organized as follows. In Section 2, applying the coincidence degree and the related continuation theorem, a series of sufficient conditions to ensure the existence of positive solutions of difference equations are given. In Section 3, by means of the method of Lyapunov function, a set of sufficient conditions for the global asymptotic stability of the model are established. Some numerical simulations are given to illustrate the theoretical results in Section 4.
2. Existence of Positive Periodic Solutions
Throughout the paper, we always use the notations below: where is an periodic sequence of real numbers defined for . In order to explore the existence of positive periodic solutions of (2) and for the reader’s convenience, we will first summarize below a few concepts and results without proof, borrowing from [37].
Let be normed vector spaces, 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 exists an isomorphism .
Lemma 1 ([37] continuation theorem). Let be a Fredholm mapping of index zero and let be compact on . Suppose(a)for each every solution of is such that ;(b) for each , and .Then the equation has at least one solution lying in .
Lemma 2 (see [33]). Let be -periodic; that is, ; then for any fixed and any , one has
Lemma 3. is an -periodic solution of (2) with strictly positive components if and only if is an -periodic solution of
The proofs of Lemma 3 are trivial, so we omitted the details here.
Define
For , define . Let denote the subspace of all -periodic sequences equipped with the usual supremum norm , that is, , for any . It is easy to show that is a finite-dimensional Banach space.
Let and then it follows that and are both closed linear subspaces of and Next, we will be ready to establish our result.
Theorem 4. Suppose that (H1), (H2), and (H3) hold. Then system (2) has at least an -periodic solution with positive components.
Proof. Let ,
where and
Then it is trivial to see that is a bounded linear operator and
It follows that is a Fredholm mapping of index zero. Define
It is not difficult to show that and are continuous projectors such that
Furthermore, the generalized inverse (to ) exists and is given by
Obviously, and are continuous. Since is a finite-dimensional Banach space, it is not difficult to show that is compact for any open bounded set . Moreover, is bounded. Thus, is -compact on with any open bounded set .
Now we are at the point to search for an appropriate open, bounded subset for the application of the continuation theorem. Corresponding to the operator equation , we have
Suppose that is an arbitrary solution of system (16) for a certain ; summing both sides of (16) from 0 to with respect to , respectively, we obtain
It follows from (16) and (17) that
In view of the hypothesis that , there exist such that
By (17), we have
Thus
It follows from (18), (21), and Lemma 2 that
In view of (22), we derive
Obviously, are independent of . Take , where is taken sufficiently large such that , where is the unique positive solution of (6). Now we have proved that any solution of (16) in satisfies .
Let ; then it is easy to see that is an open, bounded set in and verifies requirement (a) of Lemma 1. When is a constant vector in with . Then
where
Now let us consider homotopic , where
Letting be the identity mapping and by direct calculation, we get
By now, we have proved that verifies all requirements of Lemma 1; then it follows that has at least one solution in ; that is to say, (6) has at least one -periodic solution in , say . Let ; then by Lemma 3 we know that is an -periodic solution of system (2) with strictly positive components. The proof is complete.
3. Global Asymptotic Stability
Let the delays be zero; then (2) takes the form In this section, we will present sufficient conditions for the global asymptotic stability of system (28).
Theorem 5. Assume that (H1) and (H2) are satisfied and furthermore suppose that there exist positive constants and such that Then the positive -periodic solution of system (28) is globally asymptotically stable.
Proof . Since the delays in system (2) have no effect on the periodic solution, then system (28) has a positive solution . Now we prove below that it is uniformly asymptotically stable. First, we make the change of variable
It follows from (28) that
where converges to zero as .
Define a function by
where and are all positive constants given by (34) and (35), respectively. Calculating the difference of along the solution of system (31), we get
where
It follows from the condition (29) that there exists a positive constant such that if is sufficiently large and , then
In view of Freedman [38], we can see that the trivial solutions of (31) are uniformly asymptotically stable and so is the solution of (28). Thus we can conclude that the positive periodic solution of (28) is globally asymptotically stable. The proof is complete.
4. Numerical Example
In this section, we present some numerical results of system (2) to verify the analytical predictions obtained in the previous section. Let us consider the following discrete system: where , and it is easy to see that all the conditions of Theorem 4 are fulfilled. Thus system (37) has at least a positive two-periodic solution (see Figures 1 and 2).
5. Conclusions
In this paper, a discrete predator-prey model with Holling II functional response and delays is investigated. With the aid of Gaines and Mawhin’s continuation theorem of coincidence degree theory and the method of Lyapunov function, we establish some sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the model. Since the time scales can unify the continuous and discrete situations, it is meaningful to investigate the predator-prey model with Holling II functional response and delays on time scales. We leave it for future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by State Science and Technology Support Program (Grant no. 2013BAD15B02), General Scientific Research Projects in Hunan Province, Department of Education, China (Grant no. 14C0542), Graduate Student Research Innovation Project of Hunan Province (Grant no. CX2014B306), and Guangxi Experiment Centre of Science and Technology (Grant no. LGZXKF201112).