Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.


Introduction
As was pointed out by Berryman 1 , the dynamic relationship between predator and prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance.Furthermore, the study of the consequences of the hiding behavior of the prey on the dynamics of predator-prey interactions can be recognized as a major issue in both applied mathematics and theoretical ecology 2 .In general, the effects of prey refuges on the population dynamics are very complex in nature, but for modeling purposes, it can be considered as constituted by two components 2 .The first one, which affects positively the growth of prey and negatively that of predators, comprises the reduction of prey mortality due to the decrease in predation success.The second one may be the tradeoffs and by-products of the hiding behavior of prey which could be advantageous or detrimental for all the interacting populations 3 .
Sih 4 obtained a set of general conditions which ensure that the refuge use has a stabilizing effect on Lotka-Volterra-type predator-prey systems; he also examined the effect of the cost of refuge use in decreased prey feeding or reproductive rate.In 5 , González-Olivares and Ramos-Jiliberto investigated the dynamic behaviors of predator-prey system incorporating Holling type II functional response and a constant refuge: where x t , y t denote the densities of prey and predator population at any time t, respectively; c, d, k, r, β, a, and m are positive constants; here r is the intrinsic per capita growth rate of prey; K is the prey environmental carrying capacity; β is the maximal per capita consumption rate of predators; a is the amount of prey needed to achieve one-half of β; c is the conversion factor denoting the number of newly born predators for each captured prey; d is the death rate of the predator; xm is the number of prey that refuge can protect at time t.Kar 6 also studies the dynamic behaviors of system 1.1 .He obtained the conditions for the existence and stability of the equilibria and persistent criteria for the system.He also shows that the system admits a unique limit cycle when the positive equilibria is unstable.In these papers, all their finds indicate that the refuge influencing the dynamic behavior of predator-prey system greatly and increasing the amount of refuge could increase prey density and lead to population outbreaks.Kar 7 also studied the influence of harvesting on a system with prey refuge.Some scholars argued that the nonautonomous case is more realistic, because many biological or environmental parameters do subject to fluctuate with time; thus more complex equations should be introduced.Already, many scholars 8-15 studied the dynamic behaviors of nonautonomous predator-prey system incorporating prey refuge.Recently, Xu and Jia 11 proposed and studied the nonautonomous predator-prey system incorporating prey refuge and Holling type II functional response, that is, Though most dynamic behaviors of population models are based on the continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations 12 .It has been found that the dynamic behaviors of the discrete system is rather complex and contains more rich dynamics than the continuous ones 16 .Though the influence of prey refuge for continuous model has been extensively investigated, seldom did scholars investigated the influence of prey refuge for discrete predator-prey system.To the best of the authors' knowledge, to this day, only Zhuang and Wen 17 studied the local property stability of the fixed points of the discrete Leslie-Gower predator-prey systems with and without Allee effect.In this paper, we study the corresponding discrete prey-predator system of 1.2 : Here, we assume that a n , b n , c n , d n , α n , β n , and m n 0 ≤ m n < 1 , γ n are all bounded nonnegative sequences.Noting that For the point of view of biology, in the sequel, we assume that x 0 > 0, y 0 > 0, then from 1.4 , we know that the solutions of system 1.3 are positive.From now on, for any bounded sequence x n ,

Permanence
We will investigate the persistent property of the system in this section.
Lemma 2.1 see 12 .Assume that {x k } > 0 and for k ∈ N, where a k and b k are nonnegative sequences bounded above and below by positive constants.Then Theorem 2.3.Every positive solution x n , y n of system 1.3 satisfies

2.5
Here Proof.Let x n , y n be any positive solution of system 1.3 .From the first equation of system 1.3 it follows that Applying Lemma 2.1 to 2.6 leads to lim sup From the second equation of system 1.3 , similarly to the analysis of 2.6 -2.7 , we can obtain lim sup This ends the proof of Theorem 2.3.

2.9
Here

2.10
Proof.According to the first inequality of H 1 , one could choose ε > 0 small enough, such that the inequality holds.For the above ε > 0, according to Theorem 2.3, there exists an integer n * ∈ N such that for all n ≥ n * , For n ≥ n * , from 2.12 and the first equation of system 1.3 , we have

2.13
As a direct corollary of Lemma 2.2, according to 2.7 and 2.13 , one has lim inf where

2.15
Noting that and so a l − b u M 1 ≤ 0, consequently, for arbitrary ε, The above inequality leads to

2.20
According to 2.7 , 2.8 , and 2.20 , for any ε > 0, there exists n 1 ∈ N, such that for all n ≥ n 1 , Similarly to the analysis of 2.13 -2.20 , by applying 2.21 , from the second equation of system 1.3 , it follows that lim inf

2.22
Here This completes the proof of Theorem 2.4.

Global Stability
In this section, by developing the analysis technique of 18 , we obtain the conditions which guarantee the global stability of the system 1.3 .
Theorem 3.1.Assume that H 1 holds, assume further that

3.1
Then for any two positive solutions x n , y n and x n , y n of system 1.3 , one has x n exp u n , y n y n exp v n .

3.3
then system 1.3 is equivalent to

3.4
Here δ n def α n 1 − m n x n , δ n def α n 1 − m n x n .By using the mean-value theorem, it follows that

3.5
where θ i n ∈ 0, 1 i 1, 2, 3, 4 .To complete the proof, it suffices to show that In view of 3.1 , we can choose ε > 0 small enough such that

3.7
Here, p ε For the above ε > 0, according to Theorems 2.3 and 2.4, there exists a k * ∈ N, such that Noticing that θ i n ∈ 0, 1 i 1, 2, 3, 4 implies that x n exp θ 1 n u n , x n exp θ 4 n u n lie between x n and x n , y n exp θ 2 n v n , y n exp θ 3 n v n lie between y n and y n .From 3.5 , it follows that

3.10
Therefore 3.6 holds and the proof is complete.

Extinction of Predator Species and Stability of Prey Species
In this section, by developing the analysis technique of 16 , we show that under some suitable assumptions, the predator will be driven to extinction while prey will be globally attractive to a certain solution of a logistic equation.We consider a discrete logistic equation: For the above equation, we have the following lemma.
where m a l /b u exp a l − b u M 1 and M 1 is defined by Theorem 2.3.

Theorem 4.2. Assume that the inequality
holds.Let x n , y n be any positive solution of system 1.3 , then y n → 0 as n → ∞.
Proof.H 2 is equivalent to the following inequality: From 4.3 we can choose positive constant ε > 0 small enough such that inequality holds.Thus, there exists a δ > 0, Let x n , y n be any positive solution of system 1.3 .For any q ∈ N, according to the equation of system 1.3 , we obtain ln y q 1 y q − γ q − d q y q c q β q 1 − m q x q 1 α q 1 − m q x q ≤ − γ q c q β q 1 − m q x q

4.8
Theorem 2.3 implies that x n are bounded eventually, which together with the above inequality shows that y n → 0, exponentially, as n → ∞.This completes the proof of Theorem 4.2.
Theorem 4.3.Assume that a l > β u M 2 and H 2 holds, also Then for any positive solution x n , y n of system 1.3 and any positive solution x * n of system 4.1 , one has Proof.Since H 2 holds, it follows from Theorem 4.2 that then from the first equation of system 1.3 and 4.11 , Using the mean-value Theorem, one has exp u n − 1 exp θ n u n u n , θ n ∈ 0, 1 .

4.13
Then the first equation of system 1.3 is equivalent to To complete the proof, it suffices to show that We first assume that then we can choose positive constant ε > 0 small enough such that For the above ε, according to Theorems 2.3 and 2.4, Lemma 4.1, and 4.10 , there exists an integer n 2 ∈ N such that

4.18
Noting that m 1 ≤ m, then

4.19
It follows from 4.19 that

4.20
Noting that θ n ∈ 0, 1 , it implies that x * n exp θ n u n lies between x * n and x n .From 4.14 , 4.17 -4.20 , we get

4.21
This implies that
Note that

4.25
Now, we can conclude that 4.15 is satisfied as H 3 holds, and so lim n → ∞ x n − x * n 0. This completes the proof of Theorem 4.3.

Examples and Numeric Simulations
In this section, we will give two examples to show the feasibility of our results.

5.2
We could easily see that 1 − γ l /c u β u M 1 0.8457 < m l 0.85, a l − β u M 2 ≈ 0.2975 > 0, b u /b l exp a u − 1 ≈ 0.926 < 2. Clearly, conditions of Theorems 4.2 and 4.3 are satisfied.And so, lim n → ∞ x n − x * n 0 and lim n → ∞ y n 0, where x * n is any positive solution of system 4.1 .Figure 2 shows the dynamic behaviors of system 5.2 .

Discussion
We proposed a nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional responses.It is well known that prey species makes use of refuges to decrease predation risk and refuge plays an important role on the dynamic behaviors of predator-prey populations.For system 1.3 , we showed that the predator and prey will be coexistent in a globally stable state under some suitable conditions.However, in Section 4, we found that if the refuge is enough large, the predator species will be driven to extinction due to the fewer chances of predation.Obviously, increasing the amount of refuge can increase prey densities and lead to population outbreaks; such kind of finding is consistent with the continuous ones as shown by Kar 6 .In 11 , Xu and Jia studied the continuous system 1.2 .Sufficient conditions which guarantee the persistence and global stability of positive periodic solution of the system are obtained.Comparing the results of 11 with ours, we found that the conditions which guarantee the persistence of continuous system were similar to the discrete.However, for the conditions that guarantee the global stability of system, the discrete system is more complicated than that of the continuous system.Maybe the reason lies in that for the discrete population dynamics, the constructing and computing of Lyapunov function are relatively complicated than the continuous ones.Unlike the work of 11 , we argued that it is an important topic to study the extinction of the species; since more and more species are driven to extinction with the development of modern society, this motivated us to study the extinction of the predator species.At the end of the paper we would like to mention that one of the referees pointed out that "the nonautonomous character of the model is introduced to simulate the time dependent fluctuating properties of the environment.But it seems to me that a more realistic description of this dependence should be done in terms of stochastic variables rather than deterministic ones."Indeed, recently, many excellent works concerned with the continuous population model with stochastic variables had been done, see 19-21 and the references cited therein.However, to the best of the authors' knowledge, to this day, for discrete population dynamics, no similar work has been done.We leave this problem to future research.
and x N 0 > 0, where a k and b k are nonnegative sequences bounded above and below by positive constants and N 0 ∈ N. Then

Lemma 4 .1 see 17 .
For any positive solution x * n of 4.1 , one has