Global Stability of a Discrete Mutualism Model

and Applied Analysis 3 Lemma 2 (see [20]). Let F(λ) = λ+Bλ+C = 0, where B and C are constants. Suppose F(1) > 0 and λ 1 , λ 2 are two roots of F(λ) = 0. Then |λ 1 | < 1 and |λ 2 | < 1 if and only if F(−1) > 0 and C < 1. Let λ 1 and λ 2 be the two roots of (9), which are called eigenvalues of equilibrium E + (x 1 , x 2 ). From [20] we know that if |λ 1 | < 1 and |λ 2 | < 1, then E + (x 1 , x 2 ) is locally asymptotically stable. Theorem 3. Assume that (H 1 ) and (H 2 ) hold; then E + (x 1 , x 2 ) is locally asymptotically stable. Proof. Since (7) has two real solutions, it follows that its discriminant is positive; that is, Δ = (1 − α 1 α 2 + K 1 − K 2 ) 2 + 4 (α 1 + 1) (K 2 + α 2 K 1 ) = (1 + α 1 α 2 + K 1 + K 2 ) 2 − 4 (α 1 − K 1 ) (α 2 − K 2 ) > 0, (11) and so (1 + α 1 α 2 + K 1 + K 2 ) 2 > 4 (α 1 − K 1 ) (α 2 − K 2 ) . (12) Equation (6) combined with the above inequality implies that (1 + x 1 ) 2 (1 + x 2 ) 2 = (1 + K 1 + α 1 x 2


Introduction
There are many examples where the interaction of two or more species is to the advantage of all; we call such a situation the mutualism.For example, cellulose of white ants' gut provides nutrients for flagellates, while flagellates provide nutrients for white ants through the decomposition of cellulose to glucose.As was pointed out by Chen et al. [1] "the mutual advantage of mutualism or symbiosis can be very important.As a topic of theoretical ecology, even for two species, this area has not been as widely studied as the others even though its importance is comparable to that of predatorprey and competition interactions." Thus, it seems interesting to study some relevant topics on the symbiosis system.
The following model was proposed by Chen et al. [1] to describe the mutualism mechanism: where   refers to the intrinsic rate of population   and   >   ( = 1, 2).In the absence of other species, the carrying capacity of the species   is   .Thanks to the cooperation of the other species, the carrying capacity of the species   becomes (  +    3− )/(1 +  3− ).
Li [2] argued that the nonautonomous one is more appropriate, and he proposed the following two-species cooperative model: where   ,   ,   ,   ,   ∈ (, +),   >   ,   ,   ,   ,   ,   ( = 1, 2) are periodic functions of period  > 0.Here the author incorporates the time delays to the model, which means that the cooperation effect needs to spend some time to realize, but not immediately realize.By applying the coincidence degree theory, Li showed that the system has at least one positive periodic solution.For more works related to the system (1) and ( 2), one could refer to [1][2][3][4][5][6][7][8][9] and the references cited therein.
It is well known that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping 2 Abstract and Applied Analysis generations, and discrete time models can also provide efficient computational models of continuous models for numerical simulations.Corresponding to system (2), Li [10] proposed the following delayed discrete model of mutualism: Under the assumption that   ,   ,   ,   ,   ( = 1, 2) are positive periodic sequences with a common cycle , and   >   holds, by applying coincidence degree theory, he showed that system (3) has at least one positive -periodic solution, where  is a positive integer.Chen [11] argued that the general nonautonomous nonperiodic system is more appropriate, and he showed that the system (3) is permanent.
It brings to our attention that neither Li [10] nor Chen [11] investigated the stability property of the system (3), which is one of the most important topics on the study of population dynamics.We mention here that, with   () ̸ = 0, system (3) is a pure-delay system, and it is not an easy thing to investigate the stability property of the system.This motivated us to discuss a simple system, that is, the following autonomous cooperative system: where   () ( = 1, 2) are the population density of the th species at -generation.  ,   ,  = 1, 2, are all positive constants.
The aim of this paper is, by further developing the analysis technique of [15], to obtain a set of sufficient conditions to ensure the global asymptotical stability of the interior equilibrium of system (4).More precisely, we will prove the following result.
The rest of the paper is arranged as follows.In Section 2 we will introduce a useful lemma and investigate the local stability property of the positive equilibrium.With the help of several useful lemmas, the global attractivity of positive equilibrium of the system (4) is investigated in Section 3.An example together with its numeric simulation is presented in Section 4 to show the feasibility of our results.We end this paper by a brief discussion.
Proof.Since (7) has two real solutions, it follows that its discriminant is positive; that is, and so Equation ( 6) combined with the above inequality implies that The above inequality leads to From ( 9) and ( 14) it follows that Also, from ( 6) and ( 2 ), one has According to inequality ( 14) and ( 16), we have From Lemma 2 we can obtain that  + ( 1 ,  2 ) is locally asymptotically stable.This completes the proof of Theorem 3.

Global Stability
We will give a strict proof of Theorem 1 in this section.To achieve this objective, we introduce several useful lemmas.