A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.

1. Introduction

The impact of random factors cannot be neglected in the real world. Different kinds of random perturbations of stochastic models have been investigated in many pieces of literature. Bahar and Mao [1] discussed a stochastic delay Lotka-Volterra model, and they showed that environmental noise would suppress a potential population explosion and also made the solutions to be stochastically ultimately bounded. Almost at the same time, Mao [2] revealed that different types of environmental noise had different effects on delay population models. Meanwhile, Jiang and Shi [3] considered a randomized nonautonomous Logistic equation and represented the unique continuous global positive solution and positive T-periodic solution.

Recently, stochastic models with delay are paid more attention by many researchers. Shen et al. [4] studied stochastic Lotka-Volterra competitive models with variable delay, and they obtained the unique global positive solution, stochastically ultimate boundedness, and moment average in time of the solutions. Liu and Wang [5] considered stochastic Lotka-Volterra models with infinite delay; they replaced the intrinsic growth rate by a random perturbation term which was dependent on the difference between the population size and the equilibrium state in their model, and then the sufficient criteria for global asymptotic stability of the solution were established. Huang et al. [6] extended the conclusion of Liu and Wang [5] to a general case. Based on a general phase space

Xu et al. [7] and Xu [8] investigated an autonomous stochastic Lotka-Volterra model with infinite delay

()

they established the asymptotic pathwise properties of the solution to model (1), where x = (x₁, …, x_n) ^T, b = (b₁, …, b_n) ^T, A = (a_ij) _n×n, B = (b_ij) _n×n, and σ = (σ_ij) _n×n; C_r was a Banach space (see [7, 9, 10]); μ was probability measure defined on (−∞, 0] such that

In particular, when μ(θ) = e^krθ (k > 2) for θ ≤ 0, assumption (H2) was satisfied.

Motivated by the works mentioned previously, we always assume that the intensity of white noise is dependent on the difference between the population size and the equilibrium state and also is dependent on time t in this paper. Now, we consider a more general stochastic nonautonomous Lotka-Volterra model with infinite delay

()

where h_i(t) > 0, a_ij(t) (a_ii(t) < 0 if i = j), b_ij(t) and c_ij(t) are parameter functions; noise intensities σ_i(t) > 0, α_ij(t) ≥ 0, β_ij(t) ≥ 0 and γ_ij(t) ≥ 0 are continuous bounded function on [0, +∞); variable delay function δ : [0, ∞)→[0, τ] and δ^′(t) ≤ 0;

denotes an equilibrium state with respect to the deterministic part of model (2) in

(we always assume that such x^* exists in this paper). For simplicity, model (2) can be rewritten in the following form:

()

Next, we will investigate the global positive solution and its asymptotic properties of model (2). Further, the approximation distribution of solutions to model (2) is explored. Our results will extend some classical deterministic results into the stochastic cases.

2. Preliminary

Let (Ω, ℱ, {ℱ_t} _t≥0, P), throughout this paper unless otherwise specified, be a complete probability space with a filtration {ℱ_t} _t≥0 satisfying the usual conditions, and B_i(t) (i = 1,2, …, n) denote the independent 1-dimensional standard Brownian motion defined on the complete probability space. We denote the nonnegative cone and positive cone by , , respectively; that is, , . If x ∈ Rⁿ, its norm is denoted by ; if A is a vector or matrix, its transpose is denoted by A^T; if A is matrix, trace norm of matrix A is denoted by ; if A is negative matrix, we denote it by A < 0. Suppose that y(t) is a continuous bounded function on [0, +∞); we define y_u = max _t∈[0,+∞) y(t), y_l = min _t∈[0,+∞) y(t), with usual assumption inf ∅ = ∞, where ∅ denotes the empty set.

3. The Existence and Uniqueness of Global Positive Solution

In this section, we show that model (2) has a unique global solution, and the solution will remain in with probability 1.

Theorem 1. If (H1) and (H2) hold, then there is a unique solution x(t) to model (2). Moreover, x(t) remains in with probability 1, where t ∈ R.

Proof. Clearly, the coefficients of model (2) satisfy local Lipschitz continuous but do not satisfy the linear growth condition. To show that the solution x(t) is global, a.s., τ_e = ∞ is needed now, where τ_e is the explosion time.

Let k₀ > 0 be sufficiently large such that each component of initial data ξ(0) is lying in the interval (1/k₀, k₀). For each integer k ≥ k₀, define the stopping time

()

Clearly, τ_k is increasing as k → ∞. Set τ_∞ = lim _k→∞ τ_k; hence, τ_∞ ≤ τ_e a.s.. If we can prove that τ_∞ = ∞ a.s., then τ_e = ∞ a.s.. To prove this statement, let us define a C²-function

, where p ∈ (0,1). It is easy to see that V₁(x) > 0 for all

. Applying the Itô formula to model (2), it leads to

()

where

()

From the elementary inequality ab ≤ (1/2)(a² + b²) and Hölder′s inequality, we have

()

By the fact that

, u²/ρ_j − v²/(ρ_j − 1) ≤ (u + v) ² ≤ u²/l_i + v²/(1 − l_i), 0 < l_i < 1 < ρ_j, and

, it implies that

()

where 0 < K₂ < K₁ for any l_i ∈ (0,1). Then (6) yields that

()

Define

; the proof is easily checked (details can be found at the appendix).

4. Global Asymptotic Stability

Theorem 2. If (H1) and (H2) hold, there exist positive numbers such that is negative definite, and then

()

that is, x^* is globally asymptotically stable a.s., where

, η_ij = a_ij for i ≠ j, and

()

Proof. Define a C²-function by . Similar to the proof of [6] (see Theorem 2.1), we can derive

()

Since

is negative definite, LV₃(x) < 0 is valid along trajectories in

except x^*. The proof is complete.

5. Approximation Distribution of Solution

Theorem 3. If (H1), (H2), and the conditions of Theorem 2 hold, then each component x_i(t) of solution x(t) to model (2) follows asymptotically 1-dimensional normal distribution , where i = 1,2, …, n.

Proof. By the definition of equilibrium state , then . Since x^* is stable for the deterministic part to model (2), then for i = 1,2, …, n.

Linearizing the ith equation of model (2) by Taylor expansion at , then we have

()

Denote x_j(t − δ(t)) = x_jδ(t) and

, j = 1,2, ⋯, n. Since

and

, (13) can be simplified as

()

where m_i < 0, u_i ≠ 0. Then (14) implies that

()

From the definition of 1-dimensional Brownian motion, we can derive that

. According to the conditions of Theorem 2, when t → +∞, one can find that

a.s.; thus,

; in other words, x_i(t) asymptotically follows 1-dimensional normal distribution. When t tends to ∞, the mean of solution x(t) to model (2) is x^*; it is just consistent with the conclusion of Theorem 2. However, the deviation between solution x_i(t) and mean

may approach infinity; it is bad information for further analysis. Now, if the variance can be evaluated, the disadvantage will be improved.

6. Moment Estimation

First, let us prove one useful moment estimation.

Theorem 4. If (H1) and (H2) hold, then there is a positive constant G = G(p), such that limsup _t→∞E | x(t)|^p ≤ G, where p ∈ (0,1).

Proof. Define a C²-function by . For any given ɛ ∈ (0,2r), applying the Itô formula to e^ɛtV₄(x(t)) and taking expectation, it yields that

()

where

()

By fundamental inequality ab ≤ (1/2)(a² + b²),

()

Consequently,

()

where

()

Noting that H(x) is bounded in

; namely,

. Substituting this into (16), it thus follows that

()

By (H1) and (H2), we have

()

Hence, we derive

()

This implies that limsup _t→∞EV₄(x(t)) ≤ ɛ⁻¹K₃.

Again, by the fact that , it then follows that , and we have

()

the assertion follows by setting G = G(p) = n^p/2ε⁻¹K₃. The proof is complete.

Remark A. If (H1) and (H2) are valid, then limsup _t→∞ P{|x(t)| ≤ χ} ≥ 1 − ε; that is, solution x(t) to model (2) is stochastically ultimately bounded.

If we take p = 1/2, then the result is valid by using of Theorem 4 and Chebyshev′s inequality. The proof is omitted herewith.

Theorem 5. If (H1) and (H2) hold, there exists a positive constant Q, and then

()

Proof. Rewrite (20) as

()

with

()

Clearly, H₁(x) is bounded in

; namely,

. So

()

By (19), we have

()

Integrating both sides of (29) from 0 to T (T > 0 is arbitrary) and then taking expectations, we obtain that

()

Similar to (22), by the definition of δ(·), we can derive

()

Substituting (31) into (30), then we have

()

By the fact that D(x_i) = E | x_i|² − (Ex_i) ² and lim _t→∞ E(x(t)) = x^*, the conclusion is obviously valid. The proof is complete.

7. Example and Numerical Simulation

Now, we will simulate asymptotic behaviors of solutions to model (2). Let us consider the following 1-dimensional stochastic autonomous Logistic model:

()

with initial data ξ(θ) ∈ C_r, where h, a, b, c, τ, σ, α, β, and γ are all positive numbers, and the equilibrium state is x^* = h/(a + b + c). By Theorem 2, if a > 0.5x^*(σ² + Kα² + Kβ² + Kγ²), then the equilibrium state x^* of model (33) is globally asymptotically stable a.s.. According to Milstein method mentioned in Higham [11], the initial data of model (33) is given by ξ(θ) = 0.32e^0.5θ and μ(θ) = e^θ, θ ∈ (−∞, 0]; the difference equation is followed next

()

where ξ^(k) (k = 1,2, …, n) is the Gaussian random variable which follows the standard normal distribution 𝒩(0,1). If we fix the parameters h = 0.5, a = 0.6, b = 0.1, c = 0.3, τ = 1, and Δt = 0.01, then x^* = h/(a + b + c) = 0.5. When σ = α = β = γ = 0, model (33) becomes a deterministic one; then the equilibrium state x^* is globally asymptotically stable a.s. by Theorem 2 (see Figure 1). When σ = 0.1 and α = β = γ = 0, the intrinsic growth rate of model (33) is perturbed by white noise; if a > 0.5x^*σ² + b + c, the equilibrium state x^* is globally asymptotically stable a.s. (see Figure 2). When σ = 0.1 and α = β = γ = 0.01, that is, each parameter of model (33) is fluctuated by white noise, if a > 0.5x^*(σ² + Kα² + Kβ² + Kγ²) + b + c and K = (0.32e^(r+0.5)θ + x^(k)) ² ≤ (0.32 + x^(k)) ² are satisfied, then the equilibrium stable x^* is globally asymptotically stable a.s. (see Figure 3).

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Model (33) becomes a deterministic model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 11201075), the Natural Science Foundation of Fujian Province of China (no. 2010J01005), and the Technology Innovation Platform Project of Fujian Province (no. 2009J1007).

Appendix

From the definition of V₂(t), we obtain

()

where

()

It is easy to check that F(x) ≤ K^′ < ∞. Equation (A.1) becomes

()

Integrating both sides of (A.3) from 0 to τ_k∧T (T > 0 is arbitrary) and then taking expectations, it yields that

()

Noting that for every w ∈ Ω_k = {τ_k ≤ T}, by the definition of stopping time τ_k, x_i(τ_k, w) = k or 1/k for some i = 1,2, …, n, V₁(x(τ_k∧T)) ≥ min {k^p − plog k, k^−p + plog k}. It then follows that

()

Thus, lim _k→∞ P{τ_k ≤ T} = 0. Since T > 0 is arbitrary, we must have P{τ_∞ < ∞} = 0; then P{τ_∞ = ∞} = 1 as required.

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Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay

Abstract

1. Introduction

2. Preliminary

3. The Existence and Uniqueness of Global Positive Solution

4. Global Asymptotic Stability

5. Approximation Distribution of Solution

6. Moment Estimation

7. Example and Numerical Simulation

Conflict of Interests

Acknowledgments

Appendix

References

Citing Literature

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Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay

Abstract

1. Introduction

2. Preliminary

3. The Existence and Uniqueness of Global Positive Solution

4. Global Asymptotic Stability

5. Approximation Distribution of Solution

6. Moment Estimation

7. Example and Numerical Simulation

Conflict of Interests

Acknowledgments

Appendix

References

Citing Literature

Figures

References

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