Permanence of a Discrete Model of Mutualism with Infinite Deviating Arguments
We propose a discrete model of mutualism with infinite deviating arguments, that is . By some Lemmas, sufficient conditions are obtained for the permanence of the system.
Chen and You  studied the following two species integro-differential model of mutualism:
where , and are continuous functions bounded above and below by positive constants: and Using the differential inequality theory, they obtained a set of sufficient conditions to ensure the permanence of system (1.1). For more background and biological adjustments of system(1.1), one could refer to [1–4] and the references cited therein.
However, many authors [5–12] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Also, since discrete time models can also provide efficient computational models of continuous models for numerical simulations, it is reasonable to study discrete time models governed by difference equations. Another permanence is one of the most important topics on the study of population dynamics. One of the most interesting questions in mathematical biology concerns the survival of species in ecological models. It is reasonable to ask for conditions under which the system is permanent.
Motivated by the above question, we consider the permanence of the following discrete model of mutualism with infinite deviating arguments:
where is the density of mutualism species at the th generation. For , and are bounded nonnegative sequences such that
Here, for any bounded sequence ,
Let we consider (1.2) together with the following initial condition:
The aim of this paper is, by applying the comparison theorem of difference equation and some lemmas, to obtain a set of sufficient conditions which guarantee the permanence of system (1.2).
In this section, we establish permanence results for system (1.2).
Lemma. Let . For any fixed is a non-decreasing function with respect to , and for , following inequalities hold: If , then for all .
Now let us consider the following single species discrete model:
where and are strictly positive sequences of real numbers defined for and . Similar to the proof of Propositions and in , we can obtain the following.
Lemma. Any solution of system (2.1) with initial condition satisfies where
Lemma 2.3 (see ). Let and be nonnegative sequences defined on , and is a constant. If then
Lemma 2.4 (see ). Let be a nonnegative bounded sequences, and let be a nonnegative sequence such that . Then
Proposition. Let be any positive solution of system (1.2), then where
Proof. Let be any positive solution of system (1.2), then from the first equation of system (1.2) we have Let , then where When is nonnegative sequence, by applying Lemma 2.3, it immediately follows that When is negative sequence, (2.12) also holds. From (2.12), we have By using the second equation of system (1.2), similar to the above analysis, we can obtain This completes the proof of Proposition 2.5.
Now we are in the position of stating the permanence of system (1.2).
Theorem. Under the assumption(1.3), system (1.2) is permanent, that is, there exist positive constants which are independent of the solutions of system (1.2) such that, for any positive solution of system(1.2) with initial condition (1.4), one has
Proof. By applying Proposition 2.5, we see that to end the proof of Theorem 2.6 it is enough to show that under the conditions of Theorem 2.6 From Proposition 2.5, For all , there exists a For all , According to Lemma 2.4, from (2.13) and (2.14) we have For above , according to (2.18), there exists a positive integer , such that, for all , Thus, for all , from the first equation of system(1.2), it follows that It follows that, for , Hence In other words, From the first equation of system (1.2) and (2.23), for all , it follows that By applying Lemmas 2.1 and 2.2 to (2.24), it immediately follows that Setting , it follows that Similar to the above analysis, from the second equation of system (1.2), we have that This completes the proof of Theorem 2.6.
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