Asymptotic behavior and numerical simulations of a Lotka-Volterra mutualism system with white noises

In the present paper, a stochastic mutualism model subject to white noises is established. We first investigate the existence and uniqueness of globally positive solution of the stochastic model. Then we study its asymptotic behavior, such as stochastic permanence and extinction, and estimate the limit of the average in time of the sample paths of every component. We also show that the stochastic system is globally attractive under some appropriate conditions. Finally, numerical simulations are presented to justify the analytical results.


Introduction
The relationship between organisms and their living environments is very close in the nature, a small change of ecological environment usually could lead to a great influence on the organisms. On the other hand, there exist many complicated interactions among various species in the biological communities, and these phenomena, such as competition, predation and mutualism, are extensive. Through the years, more and more mathematical models have been used to describe the relationship between species and their living environments in ecology. The models mainly study the population dynamic behaviors, for example, permanence, extinction, global attractivity, etc. They have long been one of the popular themes in mathematical biology due to their universal existence and importance. As far as we know, many relevant papers and monographs on population dynamics have been reported, and many important results can be found in Refs. [-]. In this paper, we are concerned with a Lotka-Volterra mutualism system expressed by where x  (t), x  (t) denote the population size, the positive coefficients r  , r  and a  , a  are the intrinsic growth rate and self-inhibition rate, respectively, positive coefficients c j measure the interspecific mutualism effects of species x j on species x i (i, j = , , i = j), and b  , b  are positive control constants.
It is well known that many factors may affect the population dynamic behaviors of biology, for instance, the fluctuating environment, delays, victuals and population density. Especially, the ecological systems in practical world are often perturbed by various types of environmental noises. Just as May and Allen show that the birth rate, the death rate and other parameters usually show random fluctuation to a certain extent due to environmental fluctuation (see Ref. []). In fact, species undergoing environmental noises is also one of the most prevalent phenomena in the nature. Mao points out that a reasonable mathematical interpretation for the noise is the so-called white noiseḂ(t), which is formally regarded as the derivative of the Brownian motion B(t), i.e.,Ḃ(t) = dB(t)/dt (see Ref. []). Hence it might happen that r i (t) are not completely known but subject to some environmental noises. In other words, r i (t) could be estimated by an average value plus an error term where σ  i (t) represent the intensities of the noises, B i (t) are the independent standard Brownian motions defined on a complete probability space ( , F, {F } t≥ , P) with a filtration {F } t≥ satisfying the usual conditions. In this sense, such systems subject to environmental white noises tend to be more suitably modeled by stochastic differential equations. Therefore, what is of most interest for our present purposes is the modification of considering the possible effects of environmental white noises for system (.) and establishing the stochastic mutualism model with the initial value x() ∈ R  + . In recent years, stochastic differential equations have been widely used in population biology because they could accurately characterize some realistic processes by means of stochastic models. Many interesting and valuable results including extinction, persistence and stability can be found in Refs. [-]. But to our best knowledge, there exist few published papers concerning system (.).
Motivated by the existing results, our contribution is as follows: • We introduce the white noise to model the evolution of a mutualism system. • We investigate the existence and uniqueness of globally positive solution, which shows that the positive solution of system (.) could not explode to infinity at any finite time. • We estimate the limit of the average in time of the sample paths of every component of the positive solution. • We derive sufficient conditions for stochastic permanence, extinction and global attractivity, and the corresponding numerical simulations are provided.

Preliminaries
In this section, we shall state some definitions and lemmas which will be useful for establishing our main results.
Definition . System (.) is said to be extinct exponentially with probability one if for any initial condition x() ∈ R  + , the solution x(t) = (x  (t), x  (t)) satisfies lim sup t→+∞ ln x i (t)/t < , i = ,  a.s. Definition . System (.) is said to be stochastically permanent if for every ε ∈ (, ), there exists a pair of positive constants α, β such that for any initial condition x() ∈ R  + , the solution x(t) satisfies , u  (t)) be any two solutions of system (.) with initial conditions x() ∈ R  + , u() ∈ R  + , respectively. If lim t→+∞ |x i (t)u i (t)| =  a.s., i = , , then system (.) is said to be globally attractive.
Lemma . Suppose that a  , a  , . . . , a n are real numbers, then the following inequality |a  + a  + · · · + a n | p ≤ C p |a  | p + |a  | p + · · · + |a n | p holds, where p >  and

Lemma . ([])
Assume that an n-dimensional stochastic process X(t) on t ≥  satisfies the condition for positive constants η, ς , μ. Then there exists a continuous versionX(t) of X(t) which has the property that for every ϑ ∈ (, ς/η), there is a positive random variable ψ(ω) such that P ω : sup <|t-s|<ψ(ω),≤s,t<∞ In other words, almost every sample path ofX(t) is locally but uniformly Hölder continuous with exponent ϑ.

Existence and uniqueness of global solution
To begin with, we first show the existence and uniqueness of global solution on system (.) which is fundamental in the present paper.
Theorem . For initial value x() = (x  (), x  ()) ∈ R  + , there is a unique solution x(t) to system (.) for all t ≥ , and x(t) will remain in R  + with probability one.
Proof The proof of this theorem is standard. It is obvious that the coefficients of system (.) satisfy the local Lipschitz condition, then for any given initial value x() = (x  (), x  ()) ∈ R  + , there exists a unique local solution (x  (t), x  (t)) on [, τ e ), where τ e is the explosion time. To show that the positive solution is global, we only need to show that τ e = +∞ a.s. Let n  be sufficiently large such that every component of x() remains in the interval [  n  , n  ]. For each integer n ≥ n  , we define the stopping time Here we set inf ∅ = +∞ (∅ denotes the empty set). Obviously, τ n is increasing as n → +∞. Assign τ +∞ = lim n→+∞ τ n , whence τ +∞ ≤ τ e a.s., if we can show that τ +∞ = +∞ a.s., then τ e = +∞ a.s. and s. for all t ≥ . To complete the proof, we only need to show that τ +∞ = +∞ a.s. By reduction to absurdity, we assume that there exists a pair of constants T >  and ε ∈ (, ) such that As a result, there is an integer n  ≥ n  such that for n ≥ n  , The nonnegativity of this function can be seen from y - -ln y ≥  for y > . Using Itô's formula one can show that Integrating both sides from  to τ k ∧ T yields Taking expectations leads to Note that for arbitrary ω ∈ n , there exist some i such that x i (τ n , ω) equals either n or  n , and thus V (x(τ n , ω)) is no less than either It then follows from (.) that where  n is the indicator function of n . Letting n → +∞ leads to the contradiction So we must have τ +∞ = +∞ a.s. This completes the proof of Theorem ..

Stochastic permanence and extinction
In this section, we prove that the pth moment of the solution of system (.) is upper bounded, and then discuss the stochastic permanence and extinction.
Lemma . Assign p > , then there exists a positive constant K(p) such that the solution x(t) of system (.) has the following property: then K * (p) is the upper bound of L(x  , x  ). Recalling (.) and integrating on both sides yield Taking expectation on both sides yields It then follows from Gronwall's inequality that This implies that This completes the proof of Lemma ..
Next we investigate the stochastic permanence.
Proof Assign  < ε <  arbitrarily, we first prove that there is a constant α >  such that It follows from r i > .σ  i that we can choose a constant m >  such that By Itô's formula, one derives that Let k be sufficiently small to satisfy We continue to define By virtue of Itô's formula again, we have Obviously, Then we only need to prove is upper bounded. In the following, we consider cases (I) and (II). Let Now let us return to (.), which leads to Integrating on both sides and taking expectations, we get For arbitrary ε ∈ (, ), let α = ( ε δ )  m . By Chebyshev's inequality, we have This gives that In the following, we turn to proving that for arbitrary fixed ε ∈ (, ), there is a constant β >  such that  p , then by Chebyshev's inequality and Lemma ., we have Consequently, This completes the proof of Theorem ..
Remark . The definition of stochastic permanence here is not a very appropriate one for stochastic population models. Many authors have introduced some more appropriate definitions of permanence for stochastic population model, for example, stochastic persistence in probability (see Refs.
[, ]) or a new definition of stochastic permanence (see Ref. []). We would like to study them in our future work.
To end with, we show that a large noise may lead to exponential extinction.
Proof An application of Itô's formula yields Integrating on both sides yields Let t → +∞ and applying the strong law of large numbers for local martingales, we obtain that This completes the proof of Theorem ..

The limit of the average in time of sample paths
In this section, we estimate the limit of the average in time of the sample paths of every component of the positive solution. To begin with, we need to introduce the following important lemma which Liu and Wang obtained and proved in Ref. [].
() If there are positive constants λ  , T and λ ≥  such that Now we are in the position to establish our threshold theorems.
Proof For arbitrarily fixed > , one has -≤ ln x i () t ≤ . Recalling (.), we have Integrating on both sides, one has It then follows from () of Lemma . that Recalling (.) again, we have Integrating on both sides, we obtain In view of () of Lemma ., we have Then the desired assertion follows from the arbitrariness of . This completes the proof of Theorem ..

Global attractivity
In this section, we will establish the sufficient criteria for global attractivity of system (.).
) be a solution of system (.) with initial value (x  (), x  ()) ∈ R  + , then almost every sample path of (x  (t), x  (t)) is uniformly continuous for t ≥ .
Proof We first prove x  (t). Let us consider the following integral equation: Thus it follows from Lemma . that almost every sample path of x  (t) is locally but uniformly Hölder continuous with exponent ϑ for ϑ ∈ (, (p -)/p), and therefore almost every sample path of x  (t) is uniformly continuous on t ≥ . By a similar procedure as above, we can demonstrate that almost every sample path of x  (t) is uniformly continuous on t ≥ . The proof of Lemma . is completed. Assign then it is easy to verify that there exists a pair of positive constants ϕ  and ϕ  such that Theorem . If (.) holds, then system (.) with initial value (x  (), x  ()) ∈ R  + is globally attractive.
Proof Let x(t) = (x  (t), x  (t)) and u(t) = (u  (t), u  (t)) be two arbitrary solutions of system (.) with initial values (x  (), x  ()) ∈ R  + , respectively. By Itô's formula, we obtain that Define a Lyapunov function by A direct calculation of the right differential d + V (t) of V (t) along the solutions yields

Conclusion
In this paper, we have considered a two-species mutualism model perturbed by white noises. We first show that the model has a unique globally positive solution, and then study the stochastic permanence and extinction of the model. The limit of the average in time of the sample paths is also estimated. Furthermore, we establish the sufficient conditions for the global attractivity of the solutions. Finally, we provide several specific examples to illustrate the analytical results. According to the numerical simulations, we find that a small noise could not disrupt the original permanence (see Figures  and ), while a large noise could make species in an equilibrium state tend to be extinct (see Figures  and ), and that the environment noises have no influence on the global attractivity (see Figures   and ).
The stability of the positive equilibrium state is one of the most interesting topics in the study of population models. For models with environmental noises, however, they could not keep the positive equilibrium state of the corresponding deterministic systems. In recent years, many authors have investigated the stability in distribution of stochastic population models (see Refs. [-]). We would like to mention that the stability in distribution could be an interesting problem associated with the study of system (.), and we leave this for future investigation.