Mathematical Analysis for a Discrete Predator-Prey Model with Time Delay and Holling II Functional Response

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 predatorprey 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 predatorprey 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:


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 predatorprey 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 predatorprey 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][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][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   () ( = 1, 2) stands for the prey and predator density at time .For the biological meaning of model (1), one can see [27].
In order to obtain our main results, we assume that (H1)   :  →  + is positive -periodic; that is,   ( + ) =   () ( = 1, 2) 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.

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  : Dom  ⊂  →  be a linear mapping, and let  :  →  be a continuous mapping.The mapping  will be called a Fredholm mapping of index zero if dimKer  = codim Im  < +∞ and Im  is closed in .If  is a Fredholm mapping of index zero and there exist continuous projectors  :  →  and  :  →  such that Im  = Ker , Im  = Ker  = Im( − ), it follows that  | Dom  ∩ Ker  : ( − ) → Im  is invertible.We denote the inverse of that map by   .If Ω is an open bounded subset of , the mapping  will be called -compact on Ω if (Ω) is bounded and   (−) : Ω →  is compact.Since Im  is isomorphic to Ker , there exists an isomorphism  : Im  → Ker .
Then the equation  =  has at least one solution lying in Dom  ⋂ Ω.
Lemma 2 (see [33]).Let  :  →  be -periodic; that is, ( + ) = (); then for any fixed  is an -periodic solution of The proofs of Lemma 3 are trivial, so we omitted the details here. Define For the subspace of all -periodic sequences equipped with the usual supremum norm ‖ ⋅ ‖, that is, ‖‖ = max ∈  |()|, for any  = {() :  ∈ } ∈   .It is easy to show that   is a finite-dimensional Banach space. Let and then it follows that   0 and    are both closed linear subspaces of   and Next, we will be ready to establish our result.
Let Ω := { = {()} ∈  : ‖‖ < }; then it is easy to see that Ω is an open, bounded set in  and verifies requirement (a) of Lemma 1.When  ∈ Ω ∩ Ker ,  = {( 1 ,  2 )  } is a constant vector in where Now let us consider homotopic ) . ( Letting  be the identity mapping and by direct calculation, we get By now, we have proved that Ω verifies all requirements of Lemma 1; then it follows that  =  has at least one solution in Dom  ∩ Ω; that is to say, (

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 ,  1 and  2 such that Then the positive -periodic solution of system (28) is globally asymptotically stable.
where  1 and  2 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.

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  1 () = 0.5 + 0.3 sin ,  2 () = 0.6 + 0.2 sin ,   () =  0.5 ( = 1, 2, 3, 4),  = 2, 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).

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.
Obviously,  and   ( − ) are continuous.Since  is a finite-dimensional Banach space, it is not difficult to show that   ( − )(Ω) is compact for any open bounded set Ω ⊂ .Moreover, (Ω) is bounded.Thus,  is -compact on Ω with any open bounded set Ω ⊂ .Now we are at the point to search for an appropriate open, bounded subset Ω for the application of the continuation theorem.Corresponding to the operator equation  = ,  ∈ (0, 1), we have