Global Attractivity in a Discrete Mutualism Model with Infinite Deviating Arguments

A set of sufficient conditions is obtained for the global attractivity of the following two-species discrete mutualism model with infinite deviating arguments: x1(k+1) = x1(k) exp{r1[(K1+α1∑+∞ s=0 J2(s)x2(k−s))/(1+∑+∞ s=0 J2(s)x2(k−s))−x1(k)]} and x2(k+1) = x2(k) exp{r2[(K2 + α2∑+∞ s=0 J1(s)x1(k − s))/(1 + ∑+∞ s=0 J1(s)x1(k − s)) − x2(k)]}, where ri, Ki, αi, i = 1, 2, are all positive constants, ∑+∞ j=1 Ji(n) = 1, and αi > Ki. Our results generalize the main result of Yang et al. (2014).


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   () =   () ≥ 0,   (0) > 0,  = ⋅ ⋅ ⋅ , −, − + 1, . . ., −2, −1,  = 1, 2. ( Li and Xu [1] studied the following two-species integrodifferential model of mutualism: Under the assumption   (),   (),  = 1, 2, are all positive periodic functions and   >   ,  = 1, 2, 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   ,   ,   ,  = 1, 2, are all positive constants,   () ≡ 0, 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   (),  = 1, 2, is the density of mutualism species  at the th generation and {  ()}, {  ()}, {  ()}, {  ()}, and {  ()},  = 1, 2, 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   () ( = 1, 2) are the population density of the th species at -generation.We showed that if ( 1 )   ,   ,   ( = 1, 2) are all positive constants and   >   ( = 1, 2); hold, system (6) admits a unique positive equilibrium ( * 1 ,  * 2 ), 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  and the references cited therein.We mention here that, with   () ̸ = 0,  = 1, 2, 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 ( 1 ) and ( 2 ) hold, and then system (1)-( 2) admits a unique positive equilibrium ( * 1 ,  * 2 ), 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.

Proof of Theorem 1
Now we state several lemmas which will be useful in the proof of Theorem 1.
Now we are in the position to prove the main result of this paper.

Examples
In this section we shall give an example to illustrate the feasibility of the main result.
Example 2. Consider the following example: Hence, condition ( 2 ) 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.
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.