Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 247269, 11 pages
http://dx.doi.org/10.1155/2015/247269

Research Article

Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System

Liya Yang,¹ Xiangdong Xie,² and Fengde Chen²

¹College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, China
²Department of Mathematics, Ningde Normal University, Ningde, Fujian 352300, China

Received 3 June 2015; Accepted 1 September 2015

Academic Editor: Pavel Rehak

Copyright © 2015 Liya Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A nonautonomous discrete predator-prey-mutualist system is proposed and studied in this paper. Sufficient conditions which ensure the permanence and existence of a unique globally stable periodic solution are obtained. We also investigate the extinction property of the predator species; our results indicate that if the cooperative effect between the prey and mutualist species is large enough, then the predator species will be driven to extinction due to the lack of enough food. Two examples together with numerical simulations show the feasibility of the main results.

1. 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. Recently, predator-prey models have been studied widely [2–7]. It brings to our attention that all the works of [2–7] are dealing with the relationship between two species, while, in the real world, the relationship among species is very complicated and it needs to consider the three-species models. Many scholars [8–13] studied the dynamic behaviors of the three-species models.

Moreover, mutualism is one of the most important relationships in the theory of ecology. Mutualism is a symbiotic association between any two species and the interaction between the two species is beneficial to both of the species [14]. Already, many scholars [15–21] studied the dynamic behaviors of cooperative models. It brings to our attention that although predator-prey and mutualism can be recognized as major issues in both applied mathematics and theoretical ecology, few scholars have considered predator-prey system with cooperation in three species. But this phenomenon really exists in nature. For example, while aphids are preyed by natural enemies, they are protected by some natural friends like ants; there ants eat the honeydew that aphids excrete and help to overcome the resource scarcity of offspring [22, 23].

In 2009, Rai and Krawcewicz [24] proposed the following predator-prey-mutualist system:

where , , and denote the densities of prey, mutualist, and predator population at any time , respectively; they applied the equivariant degree method to study Hopf bifurcations phenomenon of the system.

Recently, Yang et al. [25] argued that, due to seasonal effects of weather, temperature, food supply, mating habits, and so forth, a more appropriate system should be a nonautonomous one, and they proposed and studied the following system:

By using the Brouwer fixed pointed theorem and constructing a suitable Lyapunov function, the authors obtained a set of sufficient conditions for the existence of a globally asymptotically stable periodic solution in system (2). It is well known that 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 are rather complex and contain more rich dynamics than the continuous ones. To the best of the authors knowledge, still no scholar proposes and studies the discrete predator-prey-mutualist system; this motivated us to study the following system:

where , , and are the population sizes of the prey, mutualist, and predator at th generation, respectively, and are the intrinsic growth rate of prey and mutualist at th generation, is the death rate of the predator at th generation, is called the conversion rate at th generation, which denotes the fraction of the prey biomass being converted to predator biomass, and is the capture rate of the prey at th generation. The sequences of , are the mutualism sequences. We mention here that, in system (3), we consider the density restriction term of predator species (); such a consideration is needed since the density of any species is restricted by the environment [10]. Here, we assume that , , and are all bounded nonnegative sequences. , are strictly positive sequences. Note that

From the point of view of biology, in the sequence, we assume that , , , and then from (4), we know that the solutions of system (3) are positive. We use the following notations for any bounded sequence :

We arrange the rest of the paper as follows. In Section 2, we establish a permanence result for (3). In Section 3, the sufficient conditions about the uniqueness and global attractivity of the periodic solution of (3) are obtained. In Section 4, the sufficient conditions about the extinction of predator species and the stability of prey-mutualist species are obtained. Finally, two suitable examples are given to illustrate that the conditions of the main theorem are feasible. We end this paper by a brief discussion.

2. Permanence

Theorem 1. Assume the inequalities , and every positive solution of system (3) satisfies

where

Proof. Since , , and , then , , and , for . We only need to prove that

Since similar result can be shown for and , then (6) follows obviously. We first assume that there exists such that . Then

Hence,

It follows that

here we used

We claim that

By way of contradiction, assume that there exists such that . Then . Let be the smallest integer such that . Then , which implies . The above argument produces that , a contradiction. This proves the claim. Now, we assume that for all . In particular, exists, denoted by . We claim that . By way of contradiction, assume that . Taking limit in the first equation in system (3) gives

which is a contradiction since

This proves the claim. Note that

It follows that (8) holds. This completes the proof of the main result.

Theorem 2. Assume the inequalities

where and are the same as in Theorem 1. Then

where

Proof. We first show that

For any , there exists such that

First, we assume that there exists such that . Note that, for ,

In particular, with , we get

which implies that . Then

Let

We claim that

By way of contradiction, assume that there exists such that . Then . Let be the smallest integer such that . Then , and clearly . The above argument produces that for all large . Then exists, denoted by . We claim that . By way of contradiction, assume that . Taking limit in the second equation in system (3) gives

which is a contradiction since

This proves the claim. Note that

Clearly, , so . We can easily see that (19) holds. The proof of the other two inequalities is similar to the above analysis and we omit the detail here. This completes the proof of the main result.

As a direct corollary of Theorems 1 and 2, from the definition of permanence, we have the following.

Theorem 3. Assume that holds. Then system (3) is permanent.
It should be noticed that, from the inequality and from the proofs of Theorems 1 and 2, one knows that where holds, the set is an invariant set of system (3).

3. Existence and Stability of a Periodic Solution

Due to seasonal effects of weather, temperature, food supply, mating habits, contact with predators, and other resources or physical environmental quantities, we can assume temporal to be cyclic or periodic [26–28]. In this section, we consider system (3) with , , , and being periodic with a common period. More precisely, we assume that there exists a positive integer such that, for ,

Let , , , be the same as in Theorems 1 and 2. Our first result concerns the existence of a periodic solution.

Theorem 4. Assume that holds; then system (3) has -periodic solution, denoted by .

Proof. As noted at the end of the last section that is an invariant set of system (3), thus we can define a mapping on by

Obviously, depends continuously on . Thus, is continuous and maps the compact set into itself. Therefore, has a fixed point . It is easy to see that the solution is -periodic solution of system (3). This completes the proof.

Now, under some additional conditions, we study the global stability of the periodic solution obtained in Theorem 4.

Theorem 5. Assume that (29) and hold, and

Then for every solution of system (3), one has

where is -periodic solution obtained in Theorem 4.

Proof. Let

Then system (3) is equivalent to

By using the mean-value theorem, it follows that

where . To complete the proof, it suffices to show that

In view of (31), we can choose small enough such that

According to Theorems 1 and 2, there exists such that

for .
Notice that implies that lies between and . Similarly, lies between and , and lies between and . From (35), we get

for . Let . Then . In view of (39), we get

This implies

Therefore (36) holds and the proof is complete.

4. Extinction of Predator Species and Stability of Prey-Mutualist Species

In this section, we also consider system (3) with , , and being periodic with a common period . By developing the analysis technique of [29], we show that, under some suitable assumption, the predator will be driven to extinction while prey-mutualist will be globally attractive to a certain solution of a logistic equation.

We consider a discrete logistic equation

Theorem 6. For any positive solution of (42), one has

where and is defined by Theorem 1. Furthermore, there exists -periodic solution for (42).

The proof of the above claim follows that of Theorems 1 and 2 with slight modification and we omit the detail here.

Theorem 7. Assume that the inequality

holds. Let be a periodic solution of (42). Then, for every positive solution of (42), one has

Proof. Let

Then system (42) is equivalent to

By using the mean-value theorem, it follows that

where . To complete the proof, it suffices to show that

we first assume that

then we can choose positive constant small enough such that

According to Theorem 6, there exists such that

Notice that implies that lies between and . From (47), we get

This implies that

Since and is arbitrarily small, we obtain , and it means that (48) holds when .
Note that

thus, is equivalent to

or

Now, we can conclude that (48) is satisfied as holds, and so

Theorem 8. Assume that the inequality

holds, where and are defined by Theorems 1 and 2. Let be any positive solution of system (3); then as .

Proof. From we can choose positive constant small enough such that inequality

holds. Thus, there exists ,

Let be any positive solution of system (3). For any , according to the equation of system (3), we obtain

Summating both sides of the above inequations from to , we obtain

and then

The above inequality shows that exponentially as . This completes the proof of Theorem 8.

Theorem 9. Assume , , and hold; also

Then for any positive solution of system (3), one has

is any positive solution of system (42) and is any positive solution of the second equation of system (3).

Proof. Since holds, it follows from Theorem 8 that

To prove , let

then from the first equation of system (3) and (66),

Using the mean-value theorem, one has

Then the first equation of system (3) is equivalent to

To complete the proof, it suffices to show that

We first assume that

and then we can choose positive constant small enough such that

For the above , according to Theorems 1, 2, and 8, there exists an integer such that

Noting that , then

It follows from (74) that

Noting that , it implies that lies between and . From (69), (72)–(75), we get

This implies that

Since and is arbitrarily small, we obtain , and it means that (70) holds when .
Note that

thus, is equivalent to

or

Now, we can conclude that (70) is satisfied as holds, and so

Next, we prove

Let

If and hold, from Theorems 1 and 2, we know that , are bounded eventually. From the second inequality of (39),

We first assume that

It follows from (70) that .
For any positive constant , there exists integer such that

Let

From (84)–(87) we can conclude that

This implies that

Since and is arbitrarily small, we obtain . Note that

thus is equivalent to

or

Now, we can conclude that . And so

We can conclude that

This completes the proof of Theorem 9.

5. Examples and Numeric Simulations

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

Example 1. Consider the following system:

One could easily see that and , and then condition is satisfied. According to Theorem 1, system (3) is permanent. Numerical simulation (see Figure 1) indicates the permanence of system (95).
Figure 1 shows the dynamic behaviors of system (95), which strongly supports our results.

Figure 1: Dynamic behavior of the solution to system (95) with the initial conditions and , respectively.

Example 2. Consider the following system:

We could easily see that , , , , and . Clearly, conditions of Theorem 9 are satisfied, and so , , and .
Figure 2 shows the dynamic behaviors of system (96), which strongly supports our results.

Figure 2: Dynamic behavior of the solution to system (96) with the initial conditions and , respectively.

6. Discussion

It is well known that prey-mutualist system can decrease predation risk; mutualism plays an important role in the dynamic behaviors of predator-prey populations. For system (3), we showed that the predator-prey-mutualist system will be coexistent in a globally stable state under some suitable conditions. We argued that it is an important topic to study the extinction of the species [29–32], since, with the development of modern society, more and more species are driven to extinction; this motivated us to study the extinction of the predator species. In Section 4, our results indicate that if the death rate of the predator species is big enough or the cooperate effect between species and is very strong, the predator species will be driven to extinction due to the fewer chances of meeting prey species. This can also be seen from ; will be driven to extinction when becomes bigger.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, the research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019) and the Foundation of Fujian Education Bureau.

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