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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.
As was pointed out by Berryman , 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 . 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 . 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 .
Sih  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 , González-Olivares and Ramos-Jiliberto investigated the dynamic behaviors of predator-prey system incorporating Holling type II functional response and a constant refuge: where denote the densities of prey and predator population at any time , respectively; are positive constants; here is the intrinsic per capita growth rate of prey; is the prey environmental carrying capacity; is the maximal per capita consumption rate of predators; is the amount of prey needed to achieve one-half of ; is the conversion factor denoting the number of newly born predators for each captured prey; is the death rate of the predator; is the number of prey that refuge can protect at time . Kar  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  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  proposed and studied the nonautonomous predator-prey system incorporating prey refuge and Holling type II functional response, that is, where and denote the density of prey and predator populations at time , respectively; denotes the number of prey that the refuge can protect at time ; are nonnegative continuous function that have the upper and lower bounds.
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 . It has been found that the dynamic behaviors of the discrete system is rather complex and contains more rich dynamics than the continuous ones . 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  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 are all bounded nonnegative sequences. Noting that
We will investigate the persistent property of the system in this section.
Lemma 2.1 (see ). Assume that and for , where and are nonnegative sequences bounded above and below by positive constants. Then
Lemma 2.2 (see ). Assume that satisfies , and , where and are nonnegative sequences bounded above and below by positive constants and . Then
Theorem 2.3. Every positive solution of system (1.3) satisfies Here .
Proof. Let 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
From the second equation of system (1.3), similarly to the analysis of (2.6)-(2.7), we can obtain
This ends the proof of Theorem 2.3.
Theorem 2.4. Assume that inequalities hold. Let be any positive solution of system of (1.3), then Here
Proof. According to the first inequality of , one could choose small enough, such that the inequality
holds. For the above , according to Theorem 2.3, there exists an integer such that for all ,
For , from (2.12) and the first equation of system (1.3), we have
As a direct corollary of Lemma 2.2, according to (2.7) and (2.13), one has
and so , consequently, for arbitrary ,
The above inequality leads to
Letting , it follows that
According to (2.7), (2.8), and (2.20), for any , there exists , such that for all ,
Similarly to the analysis of (2.13)–(2.20), by applying (2.21), from the second equation of system (1.3), it follows that
This completes the proof of Theorem 2.4.
3. Global Stability
Theorem 3.1. Assume that holds, assume further that Then for any two positive solutions and of system (1.3), one has Here, .
then system (1.3) is equivalent to
Here . By using the mean-value theorem, it follows that
where . To complete the proof, it suffices to show that
In view of (3.1), we can choose small enough such that
For the above , according to Theorems 2.3 and 2.4, there exists a , such that for all .
Noticing that () implies that , lie between and , , lie between and . From (3.5), it follows that Let , then . In view of (3.9), we have Therefore (3.6) holds and the proof is complete.
4. Extinction of Predator Species and Stability of Prey Species
In this section, by developing the analysis technique of , 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.
Theorem 4.2. Assume that the inequality holds. Let be any positive solution of system (1.3), then as .
Proof. is equivalent to the following inequality: From (4.3) we can choose positive constant small enough such that inequality holds. Thus, there exists a , Let be any positive solution of system (1.3). For any , according to the equation of system (1.3), we obtain Summating both sides of the above inequations from 0 to , we obtain then Theorem 2.3 implies that are bounded eventually, which together with the above inequality shows that , exponentially, as . This completes the proof of Theorem 4.2.
Proof. Since holds, it follows from Theorem 4.2 that
To prove , let
then from the first equation of system (1.3) and (4.11),
Using the mean-value Theorem, one has
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 small enough such that
For the above , according to Theorems 2.3 and 2.4, Lemma 4.1, and (4.10), there exists an integer such that
Noting that , then
It follows from (4.19) that
Noting that , it implies that lies between and . From (4.14), (4.17)–(4.20), we get
This implies that
Since and is arbitrary small, we obtain ; it means that (4.15) holds when .
Note that Thus, is equivalent to or
Now, we can conclude that (4.15) is satisfied as holds, and so . This completes the proof of Theorem 4.3.
5. Examples and Numeric Simulations
In this section, we will give two examples to show the feasibility of our results.
Example 5.1. Consider the following system: One could easily see that , , then, condition is satisfied. According to Theorem 2.3, system (1.3) is permanent. Numerical simulation (see Figure 1) indicates the permanence of system (5.1).
Example 5.2. Consider the following system: We could easily see that , . Clearly, conditions of Theorems 4.2 and 4.3 are satisfied. And so, and , where is any positive solution of system (4.1). Figure 2 shows the dynamic behaviors of system (5.2).
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 . In , 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  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 , 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.
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, research was supported by the Natural Science Foundation of Fujian Province (2011J01007), the Foundation of Fujian Education Bureau (JB09001), and the Technology Innovation Platform Project of Fujian Province (2009J1007).
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