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

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.


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][3][4][5][6][7].It brings to our attention that all the works of [2][3][4][5][6][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][9][10][11][12][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][16][17][18][19][20][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: ẋ =  ( 1 () −  1 ()  −  1 ()   1 () +  2 () ) , ẏ =  ( 2 () −   3 () +  4 () ) , ż =  (− 3 () +  1 ()  1 ()   1 () +  2 ()  −  2 () ) . ( 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  1 (),  2 (), and  3 () are the population sizes of the prey, mutualist, and predator at th generation, respectively,  1 () and  2 () are the intrinsic growth rate of prey and mutualist at th generation,  3 () is the death rate of the predator at th generation,  1 () is called the conversion rate at th generation, which denotes the fraction of the prey biomass being converted to predator biomass, and  1 () is the capture rate of the prey at th generation.The sequences of  4 (),  2 () are the mutualism sequences.We mention here that, in system (3), we consider the density restriction term of predator species ( 2 ()); such a consideration is needed since the density of any species is restricted by the environment [10].Here, we assume that   () ( = 1, 2, 3),   (),   () ( = 1, 2)  1 (), and   () ( = 1, 2, 3, 4) are all bounded nonnegative sequences.  () ( = 1, 2, 3),   () ( = 1, 2) are strictly positive sequences.Note that From the point of view of biology, in the sequence, we assume that  1 (0) > 0,  2 (0) > 0,  3 (0) > 0, 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.
It follows that ( 8) holds.This completes the proof of the main result.
Theorem 2. Assume the inequalities where  2 and  3 are the same as in Theorem 1. Then where Proof.We first show that lim inf For any  > 0, there exists  * ∈  such that First, we assume that there exists  0 ≥  * such that  2 ( 0 +1) ≤  2 ( 0 ).Note that, for  ≥  0 , In particular, with  =  0 , we get Let We claim that By way of contradiction, assume that there exists which is a contradiction since lim This proves the claim.Note that Clearly,   2 −  2 /  3 < 0, so  2 <   2   3 .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 ( 1 ) holds.Then system (3) is permanent.
It should be noticed that, from the inequality   1   1  1 /(  1 +   2  2 ) −   3 > 0 and from the proofs of Theorems 1 and 2, one knows that where ( 1 ) holds, the set
The proof of the above claim follows that of Theorems 1 and 2 with slight modification and we omit the detail here.
For any  ∈ , according to the equation of system (3), we obtain Summating both sides of the above inequations from 0 to −1, we obtain and then The above inequality shows that  3 () → 0 exponentially as  → +∞.This completes the proof of Theorem 8.

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