Asymptotic Behaviors for Delay Lotka–Volterra Model Disturbed by G-Brownian Motion

Abstract

In this paper, we propose the stochastic Lotka–Volterra model with delay disturbed by G-Brownian motion Under a natural assumption on noise, we study existence and uniqueness of the global positive solution for the system and its asymptotic pathwise moment behavior and prove that the solution does not explode to infinity in a finite time.

1. Introduction

Since the Lotka–Volterra model (LVM in short) was provided by Lotka [1] and Volterra [2], there were extensive works concerned with the dynamics of this system and global stability and the stochastic Lotka–Volterra population model, and in here, we only mention [3, 4] (for deterministic situation) and [5–8] (for stochastic situation). The well-known two-dimensional delay Lotka–Volterra ecological population model driven by Brownian motion is

Bahar and Mao in [9] proved that the solution of (1) is almost surely nonnegative and finite. Wu and Xu in [10] investigated stochastic LVM with infinite delay. Global asymptotic stability for a stochastic delay LVM was obtained in [11].

Peng first established the stochastic analysis theory under the G-expectation framework in references [12–14]. Peng’s G-expectation space is an essential extension for probability measure space. Since then, many important theoretical results in this field are obtained, for example, SLL for sublinear expectations are obtained in [15], capacity theory results are discussed in [16] and [17–19], and other related technologies in [20–22]. Inspired by these results, we investigate a stochastic delay Lotka–Volterra model disturbed by G-Brownian motion:with , where is a n-dimensional vector, x_i(t) is the population size of species i at time t(t ≥ 0), , b_i is the species i’s growth rate, is a n × n community matrix, a_ij(I ≠ j) is the interspecific interaction effect, and a_ii is the intraspecific interaction effect. We assume that the interaction effect in this system was disturbed by a G-Brownian motion with and , where is a constant matrix, representing the total interference intensity matrix for the system; B(t) has a variance-uncertainty but not mean-uncertainty; 〈B〉(t) has a mean-uncertainty property. Therefore, (〈B〉, B) is used to characterise the disturbed growth rate, disturbed interspecific, or intraspecific interactions and interference intensity at the same time. We think the model (2) considers the stochastic interference from both mean-uncertainty and variance-uncertainty, but the traditional stochastic model cannot describe this property. Indeed, we prove the solution of (2) is quasi-surely nonnegative and finite. Some asymptotic pathwise moment estimations for the solutions of this system are presented.

2. Stochastic Delay Lotka–Volterra Model Driven by G-Brownian Motion

Definitions about sublinear expectations, G-Brownian motions, and quadratic variation process 〈B〉(t) and notations, as well as more details can also be found in [12–14]. For a matrix A, we denote and ∥A∥ = sup{|Ax|: |x| = 1}. denotes the family of continuous functions from [−τ, 0] to . We assume the matrix σ satisfies the following assumption:

The assumption (A) was first assumed by Mao et al. in [5], and it is also necessary in our framework.

Theorem 1. If the matrix σ in system (2) satisfies assumption (A), then ∀A ∈ R^n×n, b ∈ Rⁿ and {x(s): s ∈ [−τ, 0]}, then there exists a unique solution x of equation (2). Furthermore, for all s ≥ −τ quasi-surely, namely, .

Proof. Because the coefficients of equation (2) are locally Lipschitz continuous, there exist a unique local solution x(s) on s ∈ [−τ, τ_e), where τ_e is called explosion time. To see it is also global, we must show τ_∞ = ∞ q.s. Suppose k₀(k₀ > 0) is large enough s.t. x(t)(t ∈ [−τ, 0]) satisfies 1/k₀ < min |x(t)|, max |x(t)| < k₀. For any k(k ≥ k₀), set τ_k = inf{s ∈ [0, τ_e): x_i(s)∉(1/k, k), 1 < i ≤ n}, where inf ∅ = ∞. Noting that τ_k is increasing when k ⟶ ∞, let , then τ_∞ ≤ τ_e q.s. If we can prove τ_∞ = ∞ q.s., then τ_e = ∞ q.s. and q.s., t ≥ 0. If τ_∞ ≠ ∞ q.s., then ∃ a constant T > 0 s.t. V(ω: τ_∞(ω) ≤ T) ≥ ɛ for any ɛ > 0, namely, ∃ an integer k₁(k₁ ≥ k₀) s.t. V(A_k) ≔ V(ω: τ_k(ω) ≤ T) ≥ ɛ for all k ≥ k₁. Let be . Set k ≥ k₀ and T > 0. Using the G-Itô lemma for , t ∈ [0, τ_k ∧ T], we getand noting thatand, as well as by the assumption (A). Thus,Denotesince we note that there is K s.t. f(x, a, b, σ) is bounded, namely, f(x, a, b, σ) < K, thenthen . From the definition of τ_k, we know ∀ω ∈ A_k, ∃ some i s.t. x_i(τ_k, ω) ∉ (1/k, k), namely, x_i(τ_k) ≤ 1/k, or x_i(τ_k) ≥ k < ∞. Noting that the function U(x_i) is decreasing when 0 < x_i ≤ 1 and is increasing when x_i > 1, hence U(x(τ_k)) ≥ U(1/k, …, 1/k) and U(x(τ_k)) ≥ U(k, …, k), namely, . Therefore, we haveSetting k ⟶ ∞, we have the contradiction therefore, we have τ_∞ = ∞ q.s., namely, τ_e = ∞ q.s., so .

3. Asymptotic Behaviors of the Solution

Theorem 2. Under the assumption (A), if , for any β ∈ (0, 1) and δ ∈ (0, 1), ∃C₀ = C(δ) > 0 s.t. the solution x(t) of equation (2) is as follows:

Proof. LetUsing the G-Itô lemma for U(x) and notingwe haveSinceand from equation (13), we getthenWe setthen F₁(x) is bounded in , say K₁, from (16),namely,where U₀ = U(x(0)), and noting that satisfies , by (19), thentherefore,In addition, we noteso∀δ > 0, let , thenHence,

Theorem 3. Suppose the (A) is true, and there exists K(K > 0) is independent of {x(s): s ∈ [−τ, 0]}, then

Proof. Write (7) as  =  − |x|², wherethen  ≤ K − |x|², where . Taking expectation from 0 to τ_k ∧ T on both sides of equation (6), we haveLetting k ⟶ ∞ yieldsTherefore, setting T ⟶ ∞,

4. Asymptotic Moment Estimations

Theorem 4. If condition (A) is true, then ∀ {x(s): s ∈ [−τ, 0]}, x(t) in (2) satisfieswhere .

Proof. Let then by G-Itô’s lemma of Reference [13], we havewhere C₀ = log(Ṽ(x(0))). Noting that, from Lemma 3.1 in reference [19], for any integer k ≥ 1, we havesoapplying Lemma 2 in [15], we know for all but finitely many k,quasi-surely true, i.e., ∃Ω_i ⊂ Ω ( = 1) s.t. ∀ω ∈ Ω_i and k_i = k_i(ω) s.t.k ≥ k_i(ω). From equation (32) and inequality (37),t ∈ [0, k_i(ω)], k ≥ k_i(ω), in other words,where . Taking G-expectation for (39), and then , from (39), we getwhere . Set max{k_i(ω), i ∈ [1, n]} = k₀(ω), then , t ∈ [k − 1, k], k ≥  k₀(), it gets from (40):Letting ɛ tend to zero and noting that yieldThe proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest to this work.

Acknowledgments

This research was supported by the NSF-China (nos. 11761028 and 11501009), NSF-Anhui Province (1508085 JGD10), HTIT and RTFS-Yunnan Province, and Honghe University (nos. 2015HB061, 2014HB0204, and 2018JS480).