Abstract
A set of sufficient conditions is obtained for the global attractivity of the following two-species discrete mutualism model with infinite deviating arguments: and , where , , are all positive constants, , and . Our results generalize the main result of Yang et al. (2014).
1. Introduction
The aim of this paper is to investigate the stability property of the following two-species discrete mutualism model with infinite deviating arguments: together with the initial conditions
Li and Xu [1] studied the following two-species integrodifferential model of mutualism: Under the assumption , are all positive periodic functions and , by applying the coincidence degree theory, they showed that system (3) admits at least one positive -periodic solution. Chen and You [2] argued that a general nonautonomous nonperiodic system is more appropriate, and for the general nonautonomous case, by using the differential inequality theory, they showed that the system is permanent. It brings to our attention that both [1, 2] did not consider the stability property of the system, and in [3], under the assumption , are all positive constants, , we investigated the stability property of the system, and we showed that the system admits a unique globally attractive positive equilibrium. At the end of the paper, we pointed out “whether some parallel result could be established for the discrete type mutualism system is still unknown, we leave this for future investigation.”
Previously, corresponding to system (3), Li and Yang [4] and Li [5] proposed the following two-species discrete model of mutualism with infinite deviating arguments: where , is the density of mutualism species at the th generation and , and , are bounded nonnegative sequences such that They showed that, under the above assumption, system (4) is permanent. Again, none of the papers [4, 5] considered the stability property of the system. To make an intensive study on this direction, in [6], we investigated the dynamic behaviors of the following autonomous mutualism system: where are the population density of the th species at -generation. We showed that if () are all positive constants and ;()hold, system (6) admits a unique positive equilibrium , which is globally asymptotically stable. Our result shows that the dynamic behavior of the discrete type mutualism model is more complicated, and one could not expect to establish parallel result as that of continuous ones. Also, at the end of the paper, we pointed out “it seems interesting to incorporate the time delay to the system (6) and investigate the dynamic behaviors of the system, we leave this for future study.” However, to this day, we still did not study the correspondence topic on this area. For more background of system (3), (4), and (6) one could refer to [1–24] and the references cited therein. We mention here that, with , and all the coefficients being time-dependent, system (4) is a nonautonomous pure-delay system, and it is not an easy thing to investigate the stability property of the system. This motivated us to discuss the simple one, that is, the autonomous simple non-pure-delay system (1).
Concerned with the stability property of system (1)-(2), we have the following result.
Theorem 1. Assume that () and () hold, and then system (1)-(2) admits a unique positive equilibrium , which is globally attractive.
Remark 2. Obviously, Theorem 1 generalizes the main results of Yang et al. [6] to the infinite deviating arguments case. Theorem 1 can also be seen as the parallel result of the continuous one in [3]. Thus, we push on the study of the mutualism model.
2. Existence and Uniqueness of Positive Equilibrium
This section focuses on the existence and uniqueness of positive equilibrium of system (1). More precisely, we will prove the following result.
Theorem 3. Under the assumption of Theorem 1, system (1)-(2) admits a unique positive equilibrium.
Proof. The positive equilibrium of system (1) satisfies which is equivalent to where Now let us consider the function and since , it follows that , and, hence, from the continuity of , there exist two points and , such that , and since has at most two solutions, it means that admits unique positive solution . Similarly, from , one could prove admits unique positive solution . By simple computation, system (7) admits a unique positive solution , where This ends the proof of Theorem 3.
3. Proof of Theorem 1
Now we state several lemmas which will be useful in the proof of Theorem 1.
Lemma 4 (see [25]). Let , where and are positive constants, and then is nondecreasing for
Lemma 5 (see [25]). Assume that sequence satisfieswhere and are positive constants and . Then()If , then (ii)If , then
Lemma 6 (see [26]). Suppose that functions satisfy for and and is nondecreasing with respect to . If and are the nonnegative solutions of the following difference equations:respectively, and , then
Lemma 7 (see [27]). Let be nonnegative bounded sequences, and let be nonnegative sequences such that Then
Lemma 8. Let , assume that , and then are the strictly increasing function of .
Proof. Since the conclusion of Lemma 8 immediately follows.
Now we are in the position to prove the main result of this paper.
Proof of Theorem 1. Let be arbitrary solution of system (1) with initial condition (2). Denote We claim that and
From the first equation of system (1), we obtainconsidering the auxiliary equation as follows: Because of , according to (ii) of Lemma 5, we can obtain for all , where is arbitrary positive solution of (18) with initial value From Lemma 4, is nondecreasing for According to Lemma 6 we can obtain for all , where is the solution of (19) with the initial value . According to (i) of Lemma 5, we can obtain From (20) and Lemma 7 we have From the second equation of system (1), we obtain Similar to the above analysis, we have From (23) and Lemma 7 we have For enough small, without loss of generality, we may assume that , and it follows from (20)–(24) that there is an integer such that, for all , For , according to the first equation of system (1) we can obtain considering the auxiliary equation as follows: According to (ii) of Lemma 5, we can obtain for all , where is arbitrary positive solution of (28) with initial value From Lemma 4, is nondecreasing for According to Lemma 6 we can obtain for all , where is the solution of (28) with the initial value . According to (i) of Lemma 5, we have From (29) and Lemma 7 we can obtain From the second equation of system (1), we obtain Similar to the analysis of (27)–(30), we have Then, for the above , there is an integer such that, for all , Noting that, from Lemma 8, () is a strictly increasing function, then, from the first and second equations of system (1) and (26), we have From (34), similarly to the analysis of (18)–(24), we can finally obtainFor the above , it follows from (35) that there exists an integer such that, for all , It then follows from (25), (26), and (36) that For , from the strictly increasing function , and (33), we can obtain From (38), similar to the analysis of (27)–(32), we can obtain For the above , it follows from (39) that there is an integer such that, for all , Noting that Then from (33) and (40) we have Continuing the above steps, we can get four sequences , , , and such that Clearly, we have Now, we will prove is monotonically decreasing and is monotonically increasing by means of inductive method.
First of all, from (37) and (42) we have . For , we assume that and , holds, and then from the strictly increasing of function , it immediately follows that Equations of (45) show that is monotonically decreasing and is monotonically increasing. Consequently, and both exist. LetFrom (43), we have Here, (47) shows that and are all solutions of system (7). However, system (7) has unique positive solution . Therefore that is, is globally attractive. The proof of the theorem is completed.
4. Examples
In this section we shall give an example to illustrate the feasibility of the main result.
Example 1. Consider the following example: Corresponding to system (1), we have , and hence Also, Hence, all the conditions of Theorem 1 hold, and it follows from Theorem 1 that system (49) admits a unique globally attractive positive equilibrium. Figure 1 supports this assertion.
Example 2. Consider the following example: Corresponding to system (1), we have , and, obviously, However, Hence, condition () in Theorem 1 could not be satisfied, and Theorem 1 could not be applied to this example. However, numeric simulation (Figure 2) also shows that system (52) admits a unique globally attractive positive equilibrium.
5. Discussion
In [6], Yang et al. proposed system (6); under the assumption , they showed that if , then the mutualism model admits a unique globally asymptotically stable positive equilibrium.
In this paper, we try to incorporate the infinite deviating arguments, and, by developing the analysis technique of Yang et al. [6] and using the difference inequality of Chen [7], we also obtain the sufficient conditions which ensure the global attractivity of the positive equilibrium. Example 1 shows the feasibility of our main result.
Since condition () is the most important restriction on the coefficients of the system, one interesting issue is whether the result of Theorem 3 could hold if () is not satisfied. Example 2 shows that our result (Theorem 3) still have room to improve. We leave this for future investigation.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research was supported by the Natural Science Foundation of Fujian Province (2015J01019).