Stability of a Three-Species Cooperative System with Time Delays and Stochastic Perturbations

Considering the impacts of time delays and different kinds of stochastic perturbations, we propose two three-species delayed cooperative systems with stochastic perturbations in this paper.We establish the sufficient criteria of the asymptotical stability and stability in probability by constructing a neutral stochastic differential equation and some suitable functionals..e impacts of time delays and stochastic perturbations to the system dynamics are revealed by some numerical simulations at the end.


Introduction
In biological world, mutualism is an important interaction and many cooperative models have been proposed to describe such biological phenomena (see [1][2][3] and references cited therein).
In nature, time delays usually appear and bring some important influence to the dynamics of ecosystem models. Kuang [4] says that ignoring time delays means ignoring the reality. It is essential to take the influence of time delays into account in mathematical modelling [5][6][7][8][9]. Furthermore, in view of the complexity of natural world, single-species or two-species ecological models often cannot describe some natural phenomena accurately and many vital behaviours can only be exhibited by systems with three or more species [10][11][12][13]. Considering the effect of time delays to threespecies cooperative system, we propose the following deterministic model: dN 1 (t) � N 1 (t) r 1 − a 11 N 1 (t) + a 12 N 2 t − τ 12 + a 13 N 3 t − τ 13 dt, dN 2 (t) � N 2 (t) r 2 + a 21 N 1 t − τ 21 − a 22 N 2 (t) + a 23 N 3 t − τ 23 dt, dN 3 (t) � N 3 (t) r 3 + a 31 N 1 t − τ 31 + a 32 N 2 t − τ 32 − a 33 N 3 (t) dt.  On the contrary, the growth of populations is often subject to environmental fluctuation, so it is necessary to consider stochastic perturbation in the process of mathematical modelling [14][15][16][17][18]. Usually, there are many kinds of stochastic perturbations. e authors in [19] proposed that the stochastic perturbation of state variable around the steady-state N * 1 , N * 2 , and N * 3 was Brownian white noise, which was proportional to the distance from the equilibrium state. Consequently, we obtain the following three-species stochastic cooperative system with time delays: with initial data where N i (t) stands for the population size of the i th species at time t, r i > 0 is the growth rate of N i (t), a ii > 0 represents the intraspecific competitive coefficient of N i (t), a ij is the interspecific cooperative rate, τ ij > 0 is time delay, τ � max τ 12 , τ 13 , τ 21 , τ 23 , τ 31 , τ 32 , ϕ i (θ) is positive and continuous function defined on [−τ, 0], σ 2 i denotes the intensity of white noise, and ω i (t) t>0 is the standard independent Brownian motion defined on a complete probability Furthermore, Liu [20] proposed that the stochastic perturbation of state variable was proportional to , and then, we get the following model: For the deterministic system, whether there exists a positive equilibrium state is an important topic, which attracts many attentions of researchers, while stochastic model cannot tend to a positive fixed point; i.e., there exists no traditional positive equilibrium state, so it is popular to study the dynamics of stochastic system around the equilibrium state of its corresponding deterministic system. Furthermore, some kinds of delays are considered in biological systems, but many obtained results have no relation with delays and the effects of time delays are not revealed clearly [9,11]. Time delays are the sources of instability in population dynamics, and they can cause population fluctuations, so it is interesting to study the effects of delays to the system dynamics. Motivated by these, we propose two threespecies delayed cooperative systems with stochastic perturbation and aim to investigate how time delays affect the stability in probability around the equilibrium state and then by numerical examples to validate our theoretical results. To the best of our knowledge, this paper is the first attempt to investigate the influence of time delays on the stability in probability of a stochastic three-species cooperative system. e rest work of this paper is structured as follows. Section 2 begins with definitions and some important lemmas and notations. Section 3 focuses on the asymptotical stability and stability in probability of (4) and (6), respectively. Some numerical simulations are given in Section 4 to validate our theoretical results. Finally, a brief conclusion and future direction are given in Section 5 to conclude the paper.

Preliminaries
Let Ω, σ, P { } be a probability space and f t , t ≥ 0 be a family of σ-algebras. We consider the following neutral stochastic differential equation: where H represents the space with all f 0 -adapted functions ϕ(s) ∈ R n and x t (s) � x(t + s), s ≤ 0, ω(t) denotes the m-dimensional f t -adapted Brownian process, a(t, ϕ) and b(t, ϕ) are n-dimensional vector and n × m-dimensional matrix, respectively. Define where E denotes the mathematical expectation.
Definition 1 (see [19]). e zero solution of (7) is said to be stable in probability if for any ε, ε > 0, there exists a number δ > 0 such that the solution x(t) � x(t, ϕ) satisfies P |x(t, ϕ)| > ε < ε for any initial function ϕ ∈ H such that P ‖ϕ 0 ‖ ≤ δ � 1, in which P is the probability of an event. (6) have a unique globally positive solution on t > −τ for any initial data given above, respectively.

Stochastic Stability
In this section, we investigate the asymptotical stability and stable in probability of (4) and (6) around the equilibrium state of (1), respectively. Let x 1 � N 1 − N * 1 , x 2 � N 2 − N * 2 , and x 3 � N 3 − N * 3 , then system (4) is transformed to the following equivalent system: By the equivalent property of above transformation, the stability of (4) around the equilibrate state of (1) is equivalent to the stability of zero solution of corresponding equivalent system (11). Consequently, we only need to study the stability of zero solution of system (11).

Discrete Dynamics in Nature and Society
By use of the definition of derivative, system (11) is equivalent to the following neutral system: e linear case of (12) is as follows: For the linear and neutral system (13), the following statement is true.
Proof. First, by the characteristics of neutral functional differential system (13), we define Applying Itô formula to V 1 , we have x 2 2 (s)ds + a 13 a 12 + a 13 N * By the same way, we have 4 Discrete Dynamics in Nature and Society x 2 3 (s)ds, + a 32 a 31 + a 32 N * Adding both sides of LV 1 , LV 2 , and LV 3 yields + τ a 12 a 12 + a 13 N * Discrete Dynamics in Nature and Society 5 Define By the hypothesis (H 1 ), it is easy to get LV < 0 along all trajectories in R 3 + except N * i . Hence, by the stability theory of stochastic functional differential equations [23], the zero solution of (13) is globally asymptotically stable. is completes the proof.

Remark 2.
e globally asymptotical stability of zero solution of (13) means the globally asymptotical stability of the solution of (4) around the equilibrate state of (1), that is, lim

Remark 3.
e proof method is motivated in [19]. We apply the theory of neutral functional differential equation and define some suitable functionals V 1 , V 2 , and V 3 to obtain the sufficient conditions assuring the globally asymptotical stability of (13) around the equilibrium state N * i , which are much different from those of [20].

Theorem 2.
If (H 1 ) holds, then the zero solution of system (12) is stable in probability; that is, system (4) around the equilibrium state of (1) is stable in probability. 6 Discrete Dynamics in Nature and Society Proof. For system (12), define V 1 , V 2 , and V 3 as before and assume that there exists a number δ > 0 such that sup t≥τ |x i (s)| < δ, i � 1, 2, 3. By the Itô formula, we calculate LV 1 , LV 2 , and LV 3 along system (12), respectively, then x 2 2 (s)ds + a 13 a 12 + a 13 N * x 2 3 (s)ds, Define V 4 , V 5 , and V 6 as follows: Discrete Dynamics in Nature and Society 7 Computing the derivatives of V j (j � 4, 5, 6) and adding both sides of LV i (i � 1, 2, 3) and LV j (j � 4, 5, 6) reads By choosing sufficiently small δ > 0 such that (H 1 ) holds, then we have LV < 0. us, it follows from Lemma 2 that the zero solution of (12) is stable in probability. e proof is completed.

Remark 4.
In the process of our proof, the same method applied in linear case ( eorem 1) is generalized to nonlinear case. We define V j (j � 4, 5, 6) and compute their derivatives so as to eliminate the integral items appeared in LV i (i � 1, 2, 3). By Lemma 2, we obtain the sufficient conditions assuring the stability in probability of (4) around the equilibrium state N * i , which is relatively new in some sense.
Next, we consider system (6). By setting , and then, (6) is transformed to the following equivalent system: x 2 2 (s)ds.
(28) e rest proof is similar to proofs of eorems 1 and 2, so we omit it. e proof is completed.

t-axis
State-axis  have important impacts on the globally asymptotical stability and stability in probability around the positive equilibrium state. Conditions (H 1 ) and (H 2 ) imply that too large time delays or white noises may destroy the stability of above systems, which are also verified by numerical simulations (see Figures 1 and 2 in the next section).

Remark 6.
eorems 1-3 highlight the effects of time delays and stochastic disturbances on the stability around the positive equilibrium state, where conditions (H 1 ) and (H 2 ) seem complexity, but they are not difficult to be verified by computation software like Matlab.
An easy computation yields that neither (H 1 ) nor (H 2 ) holds; hence, (4) and (6) around the equilibrium point may be unstable (see Figures 1(a) and 1(b), respectively). e state graphs of each population in Figure 1 show that too large time delays destroy the stability of (4) and (6).

Conclusion and Future Direction
In this paper, we investigate the stability of two three-species cooperative systems with time delays and stochastic disturbances. eorem 1 gives the sufficient conditions assuring the globally asymptotical stability around the equilibrium state of the corresponding deterministic system. eorems 2 and 3 imply that systems (4) and (6) are stable in probability around the equilibrium states under some conditions, respectively. By giving Remarks 3-5, we clarify the difference of our proof from some existing methods and, particularly, show that time delays and stochastic perturbations bring some significant impacts on the globally asymptotical stability and stability in probability.
In this paper, we assume time delays are constants, whereas in practice, time delays may be time-varying (see [25][26][27]), so it is necessary to study the dynamics with timevarying delays. Furthermore, for multispecies predator-prey system or multispecies competitive system, whether some similar results can be obtained? ese are interesting and left for our future work.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.