Abstract
Sufficient conditions are obtained for the permanence of the following discrete model of competition: ; , where , , and , , are nonnegative sequences bounded above and below by positive constants, and , .
1. Introduction
Throughout this paper, for any bounded sequence , set and
Li and Xu [1] studied the following two-species integrodifferential model of mutualism: By applying the coincidence degree theory, they showed that system (1) 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, they showed that the system is permanent. For more background and biological adjustments of system (1), one could refer to [1–6] and the references cited therein. For more work on mutualism model, one could refer to [7–34] and the references cited therein.
Li and Yang [31] and Li [32] argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations; corresponding to system (1), they 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 , , are bounded nonnegative sequences such that They showed that, under the above assumption, system (2) is permanent.
It brings to our attention the fact that the main results of [31, 32] deeply depend on the assumption , . Now, an interesting issue is proposed: is it possible for us to investigate the persistent property of system (2) under the assumption , ?
From the point of view of biology, in the sequel, we shall consider (2) together with the initial conditions: Then system (2) has a unique positive solution satisfying the initial condition (4).
From now on, we assume that the coefficients of system (2) satisfy the following.
(A) , and , , are bounded nonnegative sequences such that
We mention here that such an assumption implies that the relationship between two species is competition; indeed, under the assumption (), the first equation in system (1) can be rewritten as follows: Similar to the above analysis, the second equation in system (2) can be rewritten as follows: From (6) and (7), one could easily see that both species have negative effect on the other species; that is, the relationship between two species is competition.
Concerned with the persistent property of systems (2) and (4), we have the following result.
Theorem 1. In addition to (), assume further that holds, where then system (2) is permanent; that is, there exist positive constants , ( is defined by (9)), which are independent of the solution of system (2), such that, for any positive solution of system (2) with initial condition (4), one has
2. Proof of the Main Result
Now we state several lemmas which will be useful in proving our main result.
Lemma 2 (see [11]). Assume that satisfies and for , where and are nonnegative sequences bounded above and below by positive constants. Then
Lemma 3 (see [11]). Assume that satisfies and , where and are nonnegative sequences bounded above and below by positive constants and Then
Lemma 4 (see [34]). Let be nonnegative bounded sequences, and let be nonnegative sequences such that . Then
Now we are in the position to prove the main result of this paper.
Proof of Theorem 1. SetLet be any positive solution of system (2) with initial condition (4). From (6), it follows that By using (17), one could easily obtain that Substituting (18) into (6), it follows that Thus, as a direct corollary of Lemma 2, according to (19), one has By using (7), similar to the analysis of (17)–(20), we can obtain From (20), (21), and Lemma 4, we have Condition (6) implies that, for enough small , inequalities hold. For above , from (19)–(22), it follows that there exists such that, for all and , By using the fact that , , from (9), one could easily see that For , from (24) and (6), we have we mention here that, from (25), By using (26), we obtain Substituting (27) into (4), using (24), for , it follows that Thus, as a direct corollary of Lemma 3, according to (23) and (28), one has Letting , it follows that where Similar to the analysis of (26)–(29), by applying (24), from (7), we also have where (20), (21), (31), and (33) show that system (2) is permanent. The proof of the theorem is completed.
Competing Interests
The author declares that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Social Science Foundation of China (16BKS132), Humanities and Social Science Research Project of Ministry of Education Fund (15YJA710002), and the Natural Science Foundation of Fujian Province (2015J01283).