Analysis of Stochastic Gilpin-Ayala Competition System

where x i (t) represents the population size of species i at time t, the constant b i is the growth rate of species i, and a ij represents the effect of interspecific (i ̸ = j) or intraspecific (i = j) interaction. The Lotka-Volterra models have often been severely criticized. One disadvantage of Lotka-Volterra models is that in such a model, the rate of change in the density of each species is a linear function of densities of the interacting species. In order to yield significantly more accurate results, Gilpin and Ayala proposed the the following Gilpin-Ayala models; detailed studies related to the model may be found in [1]:


Introduction
One of the most common phenomena considering ecological population is that many species which grow in the same environment compete for the limited resources or in some way inhibit others' growth.It is therefore very important to study the competition models for multispecies.It is well known that one of the famous models is the following classical Lotka-Volterra competition system: where   () represents the population size of species  at time , the constant   is the growth rate of species , and   represents the effect of interspecific ( ̸ = ) or intraspecific ( = ) interaction.The Lotka-Volterra models have often been severely criticized.One disadvantage of Lotka-Volterra models is that in such a model, the rate of change in the density of each species is a linear function of densities of the interacting species.In order to yield significantly more accurate results, Gilpin and Ayala proposed the the following Gilpin-Ayala models; detailed studies related to the model may be found in [1]: where   are the parameters to modify the classical Lotka-Volterra model.
On the other hand, population systems are inevitably affected by environmental noise.It is therefore useful to reveal how the noise affects the population systems.Recall that the parameter   in (2) represents the intrinsic growth rate of the population.In practice we usually estimate it by an average value plus an error which follows a normal distribution; then the intrinsic growth rate becomes where   () ( = 1, . . ., ) are Brown motions with   (0) = 0 and  2  represent the intensities of the noise.As a result, system (2) becomes the stochastic Gilpin-Ayala system as follows: and we impose the following condition: > 1,   > 0,   ≥ 0, 1 ≤ ,  ≤ ,  ̸ = . ( The stochastic Lotka-Volterra model has been extensively studied due to its universal existence and importance; see [2][3][4][5][6][7][8][9][10].More recently, the existence of stationary distribution and extinction of stochastic Lotka-Volterra system have received 2 Mathematical Problems in Engineering a lot of attention, which can give a good explanation of the recurring phenomena in population system.Under what conditions can a stochastic Lotka-Volterra system has a stationary distribution?It is an open topic until very recently Mao [11] gave a positive answer.Since then, this topic has received a lot of attention; the readers are referred to [11][12][13][14].In addition, the asymptotic behavior of log   ()/,  = 1, . . .,  for various stochastic Lotka-Volterra systems has been considered by many authors [4,5,10,12], which is an important and useful property on asymptotic estimation for corresponding population systems.
However, these properties for stochastic Gilpin-Ayala system (4) have not been investigated, which remain an interesting research topic.We aim to establish new results on these properties for system (4).It is well known that the stochastic Gilpin-Ayala system (4) is a highly nonlinear system; the method for classic Lotka-Volterra system cannot be directly applied to system (4).By the Lyapunov methods, and some techniques to deal with the nonquadratic item, sufficient criteria are established which ensure the existence of a stationary distribution and extinction.By using some stochastic analysis techniques, an asymptotic property for system (4) is obtained.

Notation
Throughout this paper, unless otherwise specified, let (Ω, F, {F  } ≥0 , P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets).Let () = ( 1  , . . .,    ) be a n-dimensional Brownian motion defined on the probability space.If  and  are real numbers, then  ∨  denotes the maximum of  and , and  ∧  stands for the minimum of  and .If  ∈  × is symmetric, its largest and smallest eigenvalues are denoted by  max () and  min ().Let  * = ( * 1 , . . .,  *  ) be the positive equilibrium of the deterministic Gilpin-Ayala competition system (2), that is, the solution of the following equation: In the same way as Mao et al. [8] did, we can also show the following result on the existence of global positive solution.
Lemma 1. Assume that condition (5) holds.Then, for any given initial value  0 ∈   + , there is a unique solution (,  0 ) to system (4) and the solution will remain in   + with probability 1; namely, for any  0 ∈   + .
Lemma 2. Let condition (5) hold.Then, for any  > 0 and any given initial value  0 ∈   + , there exists a constant   such that The proof of the lemma is rather standard so it is omitted.

An Asymptotic Property
The main aim of this section is to consider the large time behavior of log   ()/,  = 1, . . ., .To this end, we consider two auxiliary stochastic differential equations as follows: Then it follows from comparison principle (see [15]) that Lemma 3. Let condition (5) hold.Then the solution to system (9) has the following property: The proof is similar to Li et al. [5] and is omitted here.
Theorem 4. Let condition (5) hold and (,  0 ) be the global solution to system (4) with any positive initial value  0 .Assume moreover that Then the solution (,  0 ) of system (4) has the following property: A simple computation shows that The well-known Hölder inequality yields For  = 1, . . ., , it follows from the inequality For  = 1, . . ., , set Substituting these inequalities into (16) yields Similarly, we get Substituting ( 21) and ( 22) into ( 16) yields where   () is the solution of the following system: A simple computation shows that log Using the property of Brownian motion, we conclude that lim ( The required assertion (15) follows by letting  → ∞ on both sides of (25) and using conditions (26)-( 28).The proof is therefore completed.

Stationary Distribution
The main aim of this section is to study the existence of a unique stationary distribution of the system (4).Let us prepare a known lemma (see Hasminskii [16, pp. 106-125]).
Let () be a homogeneous Markov process in   ⊂   described by the following stochastic differential equation: The diffusion matrix is To be more precise, let   0 , denote the probability measure induced by (,  0 ), that is where B(  ) is the -algebra of all the Borel sets  ⊂   .
Lemma 5 (see [16]).We assume that there is a bounded open subset  ⊂   with a regular (i.e., smooth) boundary such that its closure  ⊂ E  , and consider the following: (i) in the domain  and some neighborhood, therefore, the smallest eigenvalue of the diffusion matrix () is bounded away from zero; (ii) if  ∈   \ , the mean time  at which a path issuing from  reaches the set  is finite, and sup ∈    < +∞ for every compact subset  ∈   .And throughout this paper one sets inf 0 = ∞.
We then have the following assertions.
(1) The Markov process () has a stationary distribution (⋅) with density in   , such that, for any borel set  ⊂   , (2) (ergodic property) Let () be a function integrable with respect to the measure (⋅).Then Remark 6.The proof is given by [16] in detail.Exactly, the existence of stationary distribution with density is referred to Theorem 4.3 on page 117 while ergodic property (33) is referred to Theorem 4.2, page 110.
Theorem 7. Let condition (5) hold and (,  0 ) be the global solution to system (4) with any positive initial value  0 .Assume that there exists  = ( 1 , . . .,   ) ≫ 0 such that Then there is a stationary distribution for system (4) and it has the ergodic property.
(43) This immediately implies condition (i) in Lemma 5.The proof is completed.Now we denote by (⋅) the stationary distribution.The mean vector of (⋅) is important and useful information on population systems, from which we can infer asymptotically the mean of   () and the size of each species.If we can show that ∫   + ||() < ∞, then the mean vector  = ( 1 , . . .,   )  is well defined.In this case, the ergodic theory stated above implies that Theorem 8. Let assumptions in Theorems 4 and 7 hold.Then Proof.The proof is composed of two parts.The first part is to show the well-definition of  by dominated control convergence theorem.The second part is to prove assertion (45).Let () = (,  0 ) for simplicity.By the ergodic property of stationary distribution, for  > 0,  > 0, we have lim The dominated convergence theorem yields that It follows from Lemma 2 that Letting  → ∞ yields That is to say, for any  > 0, the functions   are integrable with respect to the measure (⋅).The well-definition of  follows by letting  = 1 in (49) straightforward.

Extinction
One of the most basic questions one can ask in population dynamics is extinction, which means a species will be doomed.The interesting question is can the exponential extinction rate be estimated precisely?In many cases, we need to know the extinction rate of the species in order to have a suitable policy in investment and to have timely measures to protect them from the extinct disaster.
Remark 10.Theorem 9 showed that when the perturbation is large in the sense that  2  > 2  ,  = 1, . . ., , the population will be forced to expire.And the exponential extinction rate is given precisely in terms of system's coefficients.

Numerical Simulations
In this section, to illustrate the usefulness and flexibility of the theorem developed in previous section, we present a numerical example.

Conclusion
In this paper, we have investigated the asymptotic behavior for the stochastic Gilpin-Ayala competition system.Firstly, by utilizing stochastic analysis techniques and the stochastic comparison principle, the larger time behavior log   ()/,  = 1, . . ., . has been researched.Secondly, by applying some techniques to deal with the nonquadratic item, sufficient conditions are obtained under which there is a stationary distribution to the system.Based on the condition, the estimation on the mean of the stationary distribution is presented.Finally, the sufficient criteria for extinction are established.