Abstract
A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.
1. 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 predator-prey 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 . 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 .
Li [2] argued that the nonautonomous one is more appropriate, and he proposed the following two-species cooperative model: where , , are periodic functions of period . 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 [1–9] 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 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 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. For more work about cooperative system, we can refer to [12–19].
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 , 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 are the population density of the th species at -generation. , , are all positive constants.
Throughout this paper, we assume that the coefficients of system (4) satisfy are all positive constants and .
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.
Theorem 1. In addition to , further assume thatholds; then the unique positive equilibrium of the system (4) is globally asymptotically stable.
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.
2. Local Stability
In view of the actual ecological implications of system (4), we assume that the initial value . Obviously, any solution of system (4) with positive initial condition is well defined on , where and remains positive for all .
We determine the positive equilibrium of the system (4) through solving the following equations: which is equivalent to and so Since , , (7) admits a unique positive solution . From the first equation of system (6), one could obtain ; therefore, system (4) admits a unique positive equilibrium .
Following we will discuss the local stability of equilibrium . The Jacobian matrix of system (4) at is as follows: The characteristic equation of is where
Lemma 2 (see [20]). Let , where B and C are constants. Suppose and are two roots of . Then and if and only if and .
Let and be the two roots of (9), which are called eigenvalues of equilibrium . From [20] we know that if and , then is locally asymptotically stable.
Theorem 3. Assume that and hold; then is locally asymptotically stable.
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 , one has According to inequality (14) and (16), we have From Lemma 2 we can obtain that is locally asymptotically stable. This completes the proof of Theorem 3.
3. Global Stability
We will give a strict proof of Theorem 1 in this section. To achieve this objective, we introduce several useful lemmas.
Lemma 4 (see [20]). Let , where and are positive constants; then is nondecreasing for .
Lemma 5 (see [20]). Assume that sequence satisfies where and are positive constants and . Then(i)if , then ;(ii)if , then , .
Lemma 6 (see [21]). Suppose that functions satisfy for and and is nondecreasing with respect to . If and are the nonnegative solutions of the difference equations respectively, and , then
Proof of Theorem 1. Let be arbitrary solution of system (4) with and . Denote
We claim that and .
From the first equation of system (4), we obtain
Consider the auxiliary equation as follows:
Because of , according to (ii) of Lemma 5, we can obtain for all , where is arbitrary positive solution of (23) with initial value . From Lemma 4, is nondecreasing for . According to Lemma 6 we can obtain for all , where is the solution of (23) with the initial value . According to (i) of Lemma 5, we can obtain
From the second equation of system (4), we obtain
Similar to the above analysis, we have
Then, for sufficiently small constant , there is an integer such that
According to the first equation of system (4) we can obtain
Consider the auxiliary equation as follows:
Since , according to (ii) of Lemma 5, we can obtain for all , where is arbitrary positive solution of (23) with initial value . From Lemma 4, is nondecreasing for . According to Lemma 6 we can obtain for all , where is the solution of (23) with the initial value . According to (i) of Lemma 5, we have
From the second equation of system (4), we obtain
Similar to the above analysis, we have
Then, for sufficiently small constant , there is an integer such that
Equation (24) combined with the first equation of system (4) leads to
Similar to the analysis of (23) and (24), we have
Because of arbitrariness of , we have , where
Equation (24) combined with the second equation of system (4) leads to
Similar to the analysis of (23) and (24), we can obtain
Because of arbitrariness of , we have , where
Then, for sufficiently small constant , there is an integer such that
Equation (27) combined with the first equation of system (4) leads to
From this, we can finally obtain
Because of the arbitrariness of , we have , where
Equation (27) combined with the first equation of system (4) leads to
From the above inequality we can obtain
Because of the arbitrariness of , we have , where
Then, for sufficiently small constant , there is an integer such that
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, it is clear that , . For , we assume that and hold; then
Equations (51)–(54) show that and are monotonically decreasing and and are monotonically increasing. Consequently, and both exist. Let
From (48) and (55), we have
From (49) and (55), we get
Equations (56) and (57) show that and are all solutions of system (6). However, system (6) has unique positive solution . Therefore
That is, is globally attractive.
From Theorem 3, we get that equilibrium is locally asymptotically stable. And so, is globally asymptotically stable. This ends the proof of Theorem 1.
4. Example
In this section, we will give an example to illustrate the feasibility of the main result.
Example. Consider the following cooperative system: By calculating, we have that positive equilibrium , , , and the coefficients of system (59) satisfy and . From Theorem 1, positive equilibrium is globally asymptotically stable. Numeric simulation also supports our finding (see Figure 1).
5. Discussion
It is well known [6] that, for autonomous two-species Lotka-Volterra mutualism model, the conditions which ensure the existence of positive equilibrium are enough to ensure that the equilibrium is globally stable. However, for the two-species discrete Lotka-Volterra mutualism model, Lu and Wang [22] proved that a cooperative system cannot be permanent. That is, the dynamic behaviors of discrete Lotka-Volterra mutualism model are very different to the continuous ones.
Recently, by using the iterative method, Xie et al. [8] showed that, for a mutualism model with infinite delay, conditions which ensure the permanence of the system are enough to ensure the global stability of the system. As a corollary of their result, one could draw the conclusion that system (1) admits a unique positive equilibrium, which is globally stable. One interesting issue is proposed. For the discrete type mutualism model (4), is there any relationship between the existence of positive equilibrium and the stability property of the positive equilibrium?
In this paper, by using the linear approximation, comparison principle of difference equations, and method of iteration scheme, we showed that the conditions which ensure the local stability property of the positive equilibrium () are also enough to guarantee the global stability of the positive equilibrium .
At the end of this paper, we would like to mention here that, for the Lotka-Volterra type mutualism system with time delay, delay is one of the most important factors to influence the dynamic behaviors of the system [23–25]. It seems interesting to incorporate the time delay to the system (4) and investigate the dynamic behaviors of the system; we leave this for future study.
Conflict of 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 (2013J01011, 2013J01010) and the Foundation of Fujian Education Bureau (JA13361).