Research Article | Open Access
Global Asymptotic Stability of 3-Species Mutualism Models with Diffusion and Delay Effects
In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is investigated. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the unique positive steady-state solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition. Our approach to the problem is based on inequality skill and the method of the upper and lower solutions for a more general reaction—diffusion system. Finally, some numerical simulations are given to illustrate our results.
Modeling and analysis of the dynamics of biological populations by means of differential equations is one of the primary concerns in population growth problems. A well-known and extensively studied class of models in population dynamics is the Lotka-Volterra system which models the interaction among various species. In the earlier literature most of the discussions are devoted to coupled systems of two equations (cf. [1–11]). In the recent years, attention has been given to reaction-diffusion systems with three population species, and the main concerns in these works are the prey-predator and competition models with or without time delays (cf. [12–24]). The purpose of this paper is to investigate the asymptotic behavior of the time-dependent solution for a 3-species mutualism model, where the effect of diffusion and time delays is both taken into consideration by obtaining existence of positive solution for the corresponding inequalities. The system of equations under consideration is the Lotka-Volterra 3-species mutualism model, where the population densities do not move across the boundary of a habitat, and time delays may appear in the opposing species; the densities of three populations and are governed by the following coupled equations:
with the boundary and initial conditions
Here is a bounded domain in with boundary , denotes the outward normal derivative on and is a smooth function. The above problem (1.1)-(1.2) arises in a simple food chain describing three interacting species in a spatial habitat . For each , are positive constants. and are nonnegative constants with , and its respective diffusion rate is denoted by . The real number is the net birth rate of the th species, and is its respective intraspecific competition. The parameters and are interspecific cooperation. is a uniformly elliptic operator in the form
(cf. ). The functions , with are given either by the discrete time delay
or by the continuous time delay and the interval is given by , where is a constant representing the time delay. It is allowed that the type of time delays and the values of may be different for different . This consideration includes various combination of discrete and continuous time delays for the species and
Throughout the paper we assumed that the function is piecewise continuous in and possesses the property
The above property implies that for any constant function , we have
The same is obviously true if is given by (1.4). It is also assumed that the domain is smooth and the coefficients of are smooth functions in (cf. ). In the special case of the diffusion operator , it suffices to assume that is strictly positive on and in for some . We allow (and without the corresponding boundary condition) for some or all . In particular, if for all then the equations in (1.1)-(1.2) are reduced to the ordinary differential system (with time delays)
Problem (1.8) and various similar problems have been investigated by many investigators in the framework of ordinary differential systems (cf. [25–28] and references therein). It is to be noted that if and for every then problem (1.1) is reduced to the following 3-species mutualism models without time delays: where the domains and are defined as, respectively, and . Here is a bounded domain in with smooth boundary . is the maximal existence time of the solution. is a smooth function satisfying the compatibility condition for . The meaning of the parameters of systems (1.9) is as same as those of (1.1)-(1.2). Kim and Lin  proved that global solutions of (1.9) exist if the intraspecific competitions are strong; whereas blow-up solutions exist under certain conditions of that the intraspecific competitions are weak, and  obtained the upper bound of blow-up rate for any and the lower bound of blow-up rate for . On the other hand, if then it reduces to the 2-species cooperating models, which have been extensively investigated in the current literature (cf. [2, 10, 11] and references therein). Our conclusion about the global asymptotic stability of a positive steady-state solution to system (1.1)-(1.2) is directly applicable to the above special cases (see Corollary 3.2).
This work is motivated from the following two-prey one-predator model: and one-prey two-predator model as well as three-species food-chain model
Pao  obtained some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-steady solution for each of the above three-model problems.
It is clear that the above system has always the trivial solution (0, 0, 0) and various forms of constant semitrivial solutions, that is, constant nonnegative solutions with at least one component zero and one component positive. Our aim is to prove the existence and global asymptotic stability of a positive constant steady-state solution in the mutualism model (with respect to nonnegative initial perturbations). This global asymptotic stability result implies that system (1.1)-(1.2) is permanent, the trivial and all forms of semitrivial solutions are unstable, and the nonuniform steady-state solution does not exist.
This paper is arranged as follows. In Section 2, we give some preliminary results for a more general time-delayed reaction-diffusion system. The main results and their proofs are given in Section 3. Some numerical simulations are shown in Section 4 to illustrate our theoretical analysis.
2. Some Preliminary Results
To prove the main results in this paper we first give some preliminary results for a more general time-delayed parabolic system in the form
where and for each is, in general, a nonlinear function of and , and is a uniformly elliptic operator in the form of (1.3). The components of are given by (1.4) for and by (1.5) for where is a nonnegative integer. (The case corresponds to (1.5) for every .)
By writing and in the split forms and respectively, where and are nonnegative integers satisfying and denotes a vector with components of we write in the form
Definition 2.1. We say that the vector function possesses a mixed quasimonotone property in a subset of if for each there exist nonnegative integers and satisfying (2.2) such that is nondecreasing in and is nonincreasing in for every in (cf. [1, 12]). In particular, if for all then is said to be quasimonotone nondecreasing in . It is easily seen that the reaction functions in the models (1.1) are quasimonotone nondecreasing in .
Definition 2.2. For quasimonotone nondecreasing functions we call a pair of smooth functions , coupled upper and lower solutions of (2.1) if and if where inequality between vectors is in the componentwise sense. It is obvious that if and are constant vectors, then the above inequalities become Notice that the boundary inequalities in (2.4) are trivially satisfied. For a given pair of constant vectors , satisfying (2.5) we set where denotes the set of continuous functions on . In the following we give our basic hypotheses on with
(H) The function is quasimonotone nondecreasing for , and for each , satisfies the Lipschitz condition
where is a positive constant, and for ( is a maximum norm).
It is clear that if there exists a positive constant vector and a zero vector such that
Lemma 2.3. Let , be a pair of constant vectors satisfying (2.5) and , and let hypothesis (H) hold. Then for any in , problem (2.1) has a unique global solution for all . In particular, if condition (2.8) holds for every then problem (2.1) has a unique bounded nonnegative global solution whenever
In system (2.1) if is given in the form
then the pairs satisfy (2.8) whenever
Moreover, by the positivity lemma for the linear scalar parabolic problem
where is the solution of (2.1), the solution is positive in whenever and not identical 0 (cf. ). Since for each the solution component is also a solution of (2.11) the uniqueness property of ensures that in . This observation leads to the following.
Lemma 2.4. Let be given by (2.9) and satisfy hypothesis (H), and let be a constant vector satisfying (2.10). Then for any nontrivial nonnegative problem (2.1) has a unique bounded positive solution in .
Assume that is quasimonotone nondecreasing in . Then by using as a pair of coupled initial iterations we construct two sequences from the linear iteration process
where is the Lipschitz constant in (2.7). It is clear that the sequences are well defined. Since the initial iterations in (2.14) are the constant vectors and is independent of whenever and are constants we conclude from the uniqueness property of solution for linear boundary-value problems that the solutions of (2.14) are constants and are given by It is easy to show from the quasimonotone nondecreasing property of that the sequences possess the monotone property
It is easy to verify that the limits and are maximal and minimal solutions of systems (2.13) in respectively, furthermore, if then is the unique solution in (cf. [1, 12]). In the latter case, the time-dependent solution of (2.1) (with given by (2.12) converges to as (cf. ). To summarize the above conclusions we have the following.
Lemma 2.5. Let the conditions in Lemma 2.3 be satisfied with respect to the function given by (2.12).Then (i)the sequences given by (2.14) with are constant functions and converge monotonically to their respective constant limits and that satisfy (2.18);(ii)for any , if , then the constant pairs are also coupled upper and lower solutions of (2.1), and the problem (2.1) has a unique positive solution such that and ;(iii)if then is the unique solution of (2.13) in , and for any initial function the corresponding solution of (2.1) converges to as .
In Lemma 2.5 the convergence of the time-dependent solution to is for the class of initial functions in . For arbitrary nontrivial nonnegative initial functions we have the following result from .
Lemma 2.6. Let the conditions in Lemma 2.5 be satisfied, and let be the solution of (2.1) corresponding to an arbitrary nontrivial nonnegative . Assume that and there exists such that Then converges to as
To prove the existence and uniqueness of constant positive solution of the steady-state problem (1.13) we also need the following result.
Lemma 2.7. Let If then the inequalities have a positive solution.
Proof. With the inequalities (2.20) exists a positive solution if and only if there exist positive constants such that the following equations: have a positive solution. Let , we have From the Cramer theorem, the solutions of (2.21) are as follows: Let , , then with the equations (2.21) exists a positive solution . Therefore the inequalities (2.20) have a positive solution.
3. The Main Theorems and Proof
To prove the main results in this paper, we apply Lemmas 2.3–2.7 for the models in (1.1)-(1.2) with by constructing a suitable pair of constant upper and lower solutions and . This is equivalent to show that and satisfy condition (2.5) for the reaction functions
Theorem 3.1. Let and . If , then the following statements hold true. (i)The steady-state problem (1.9) has a unique constant positive solution.(ii) For any nontrivial nonnegative , problems (1.1)-(1.2) have a unique nonnegative global solution , and the solution is uniformly boundary in (iii)The positive constant solution of the steady-state problem (1.9) is globally asymptotic stable.
Proof. (i)Since the boundary conditions are Neumann type, we seek a pair of positive constant vectors , satisfying
It is clear that the boundary conditions are satisfied, and from the positive property of and that condition (3.1) is satisfied if
We construct the following inequalities:
Let , we have . Since and , from the Lemma 2.7 the inequalities (3.3) have a positive solution . By choosing , , where and are a sufficiently large and small positive constant, respectively, we see that all the inequalities in (3.2) are satisfied. That is, and is a pair of positive constant upper-lower solution of (1.1)-(1.2). By Lemma 2.5 the sequences governed by (2.14) with the reaction function given by (1.1) converge monotonically, respectively, to some constant limits that satisfy
Let . Then by the positivity of and , a subtraction of the above pairs of equations, leads to
Since these equations are equivalent to , where is coefficient matrix and is the column vector we conclude from the nonsingular property of that . This proves and is the unique positive solution of (1.13) in .
(ii)It is obvious that the pairs satisfy all the equalities in (3.1). By Lemma 2.3 and the arbitrariness of P, problems (1.1)-(1.2) have a unique bounded nonnegative global solution () for any nontrivial nonnegative (η_1,η_2,η_3).
(iii)Lemma 2.5 ensures that for any initial function with , where and are a sufficiently large and small positiveconstants, respectively, the corresponding solution of (1.1)-(1.2) converges to as . To show this convergence property for an arbitrary nontrivial nonnegative , we observe from Lemma 2.4, the arbitrary smallness of and the arbitrary largeness of that there exists such that on . The upper and lower bounds of show that there exists such that condition (2.19) is satisfied for The conclusion of as follows from Lemma 2.6. We next consider the stability of . We observe from Lemma 2.5 that for any there exists such that as initial function . This completes the proof.
Corollary 3.2. All the results of Theorem 3.1 hold true if for some or all . In particular, these results hold true for the ordinary differential system (1.8). They are also true if the reaction functions involve no time delays.
Proof. This follows from the argument in the proof of Theorem 3.1 by letting (and without the corresponding boundary condition) and by letting respectively.
Remark 3.3. Our model and result are different from the existence ones such as those of Pao  and Kim and Lin [22, 23]. In some sense, we enrich the results of the 3-specics Lotka-Volterra reaction-diffusion systems.
4. Numerical Simulations
In this section, we give numerical simulations supporting our theoretical analysis. As an example, we consider system (1.1) with different diffusion rates , birth rate and time delays , that is, the following systems: By using the classical implicit format solving the partial differential equations and the method of steps for differential difference equations and employing the software package MATLAB7.0, we can solve the numerical solutions of systems (4.1), (4.2), (4.3), and (4.4) which are shown respectively in Figures 1, 2, 3, and 4.
It is well known that the analysis of stability for a system of delay reaction-diffusion multispecies model is quite difficult since the reaction among multispecies is more complex. Therefore, the works on this subject are very rare. A detailed analysis on the stability for a two-prey one-predator model, one-prey two-predator model, and three-species food-chain model with delay and diffusion was given by Pao , and he obtained some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems.
In this paper, based on the ideas of Pao , we have considered a delay cooperative three-species system with Neumann boundary condition. It is shown that the system has a positive equilibrium under some certain conditions. We have obtained the similar conclusions to those of Pao . More precisely, we have obtained the following results.
(a) The steady-state problem (1.9) has a unique constant positive solution if .
(c) The positive constant solution of the steady-state problem (1.9) is globally asymptotic stable.
The condition involves only the reaction rate constants, which shows that the diffusion rates , the birth rate , and time delays do not bring effect on permanence of one species as well as contribution to its extinction. The result of global asymptotic stability implies that the three-species model system coexists, is permanent, and the trivial and all semitrivial solutions are unstable.
The authors are grateful to the referee for her/his comments. This work is supported by Science and Technology Study Project of Chongqing Municipal Education Commission (Grant no. KJ 080511) of China, Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB7415 and 2007BB2450) of China, Foundation Project of Doctor Graduate Student Innovation of Beijing University of Technology of China, the NSFC (Grant nos.10471009), and BSFC (Grant no. 1052001) of China.
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