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 usedWe 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 haswhere 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 thatfor .
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. LetThen 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 getThis 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; alsoThen 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 getThis 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 thatIt follows from (70) that .
For any positive constant , there exists integer such thatLet 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 thatThis 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.
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.
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.