Extinction and stationary distribution of a competition system with distributed delays and higher order coupled noises

: A stochastic two-species competition system with saturation e ﬀ ect and distributed delays is formulated, in which two coupling noise sources are incorporated and every noise source has e ﬀ ect on two species’ intrinsic growth rates in nonlinear form. By transforming the two-dimensional system with weak kernel into an equivalent four-dimensional system, su ﬃ cient conditions for extinction of two species and the existence of a stationary distribution of the positive solutions to the system are obtained. Our main results show that the two coupling noises play a signiﬁcant role on the long time behavior of system.


Introduction
In the last few decades, the mathematical models, which describe the effect of competition caused by limited resources on the growth of species, have been established and explored extensively [1][2][3][4][5]. On the other hand, time delay accompanies the whole process of species population survival and reproduction. Because of the variety of delay between different individuals of the same species or the cumulative effect of the past history of a system, it is more appropriate to establish model with distributed delay [6][7][8]. Cushing [9] and MacDonald [10] introduced the weak kernel and the strong kernel functions to describe the distributed delay, later on, these two kinds of kernels have been investigated by many scholars (see [11][12][13][14]).
With the idea of the weak kernel functions, we formulate the following competitive system with distributed delays where x and y measure the population densities of two competing species. α i e −α i t (α i > 0) are the weak kernel functions. The delays are the time passing between the intra-specific competition at a given instant and its effect, at a later time, on the dynamics of the two species. The coefficients r i , b i , c i (i = 1, 2) are positive constants, and represent the intrinsic growth rates, the intra-specific competitive rates and the inter-specific competitive rates, respectively. The term c 1 y/(1 + y) (or c 2 x/(1 + x)) is an increasing function with respect to y (or x) and has a saturation value for large enough y (or x). There are many related ecology and investigations of system (1.1). Especially, for system (1.1) without delays, Wang and Liu [15] analyzed the existence and global stability of almost periodic solutions of system, Li et al. [16] investigated the stability and Turing pattern of system with self-and cross-diffusion effect, Hu and Liu [17] took two coupling noise sources into account and obtained the sufficient conditions for survival results of system. On the other hand, for system (1.1) with discrete delays, Chen and Ho [18] discussed the persistence and global stability of system, Liu et al. [19] studied the existence of positive periodic solutions of system with impulsive perturbations. Obviously, it can be seen from the above literatures that they did not consider system with distributed delays. To reveal the effect of distributed delays we propose system (1.1). Denote By applying the linear chain technique, system (1.1) transforms into the following equivalent four- Due to environmental noise, the birth rate and other parameters involved in a system can reflect random fluctuation to some extent [20][21][22]. Some researches indicate that random interference cannot be ignored for competitive ecological models [23][24][25]. In addition, many researchers also introduced higher order perturbations into the system when the random perturbations may depend on the population's density [26,27]. Therefore, in this paper, similarly to [17,28] we further incorporate two noise sources in the system (1.3), that is, one noise source not only has influence on the intrinsic growth rate of one species but also on that of the other species. Strongly inspired by the above arguments, we assume that the white noises affect r i (i = 1, 2) mainly according to and obtain the following stochastic system which corresponds to the deterministic system (1.3) (1.5) Here, for convenience' sake, let x = x(t), y = y(t), u = u(t), v = v(t), and dB i = dB i (t)(i = 1, 2). B i are the independent standard Brownian motions defined on the complete probability space (Ω, F , P) with a filtration {F } t≥0 satisfying the usual conditions, that is B i are normally distributed with mean 0 and variance t (see [29]), and σ 2 i j (i, j = 1, 2) denote the intensities of the white noises. The rest of this paper is arranged as follows. Section 2 focuses on exploring the unique global positive solution to system (1.5). Sufficient conditions for extinction of the species and the existence of a stationary distribution of the positive solutions to system (1.5) are obtained in section 3 and section 4, respectively. In section 5, several examples are demonstrated to verify our results and a brief discussion is given.
Proof. Obviously, for any initial value ( To complete the proof, we only need to show that τ = +∞ a.s. Similarly to [29], we construct a nonnegative C 2 -function V : and n > 1 is a sufficiently large integer and K is a positive constant. Let 0 < p < 1, we assign Since − ln z → +∞ as z → 0, and z p p − ln z → +∞ as z → +∞, where p > 0, we can obtain that lim inf n→+∞,(x,y,u,v)∈R 4 + \O n V(x, y, u, v) = +∞. In the following, we will verify that LV(x, y, u, v) ≤ K. An application of Itô's formula shows that The rest proof is similar to Theorem 2.1 in [30] and hence we omit it here.
Proof. Using Itô's formula in the first equation of system (1.5), one can show that Integrating from 0 to t and dividing by t on both sides of the above equality lead to t 0 x(s)dB i (s), so that its quadratic variation is t 0 x 2 (s)ds.
Using the exponential martingale inequality and the Borel-Cantell lemma, similar to [27,31], we have that for almost all ω ∈ Ω, there is a random integer l 0 = l 0 (ω) such that for l ≥ l 0 , The strong law of local martingales implies that Taking the superior limit on both sides we get Similarly, one can obtain that The proof is completed.

Existence of stationary distribution
Consider the integral equation where σ r (s, X(s)) and B ξ (s) are vectors.
Lemma 4.1. [32] Suppose that the coefficients of (4.1) are independent of t and satisfy the following conditions for some constant N in O R ⊂ R d + for every R > 0 and there exists a nonnegative C 2 −function W(x) in R d + such that LW(x) ≤ −1 outside some compact set. Then the system (4.1) exists a solution which is a stationary distribution. Assign and for the constant 0 < p < 1, we have where M i = 2 β i max{2, J i }, J i (i = 1, 2) are positive constants which will be determined later. Since W(x, y, u, v) is a continuous function, there exists a minimum point (x,ŷ,û,v) in the interior of R 4 + . Then we can choose the following C 2 -function W : which, together with (4.4)-(4.5), leads to LW(x, y, u, v) (4.8) Let > 0 be sufficiently small such that , (4.9) Define the bounded closed set Case 1. When (x, y, u, v) ∈ O 1 , it follows from (4.8) that LW(x, y, u, v) Since M 1 = 2 β 1 max{2, J 1 }, one can see that M 1 β 1 4 ≥ 1. Then we have from (4.9) that Similarly, for any (x, y, u, v) ∈ O 3 , Since M 2 = 2 β 2 max{2, J 2 }, we have M 2 β 2 4 ≥ 1. It follows from (4.9) that Recalling (4.8), we can calculate that Case 3. When (x, y, u, v) ∈ O 5 or (x, y, u, v) ∈ O 7 , we have from (4.9) that  LW(x, y, u, v) ≤ −1.
The proof is completed.
Our theoretical results and the above numerical examples reveal that the coupling noises can change the asymptotic properties of system (1.5).
(I) For ecosystems, it is important to analyze the survival of species. It can be seen from Figure  1(a 1 )-(a 2 ) that the two competing species can survive if the competitive system is not affected by environmental noise. However, two species will go to extinction if the coupling noises are suitable large (see Figrue 1(b 1 )-(b 2 )).
(II) To analyze the statistic characteristic of the long-term behaviors of the sample trajectories, a useful approach is to study the stationary distribution. Figure 2 implies that the relatively small coupling noises can ensure the existence of a stationary distribution.