Coexistence for a kind of stochastic three-species competitive models

1 Introduction

The coexistence of species plays a vital role in protecting ecological balance, and the situation is not desirable when any species extinct. Therefore, this paper aims to discuss what the conditions for coexistence is. In term of deterministic model, Lotka-Volterra model, which was applied to study species interactions, was originally proposed by A. Lotka and V. Volterra [1, 2, 3]. Later Lotka-Volterra model is widely applied in the field of chemistry. Literature [4], based on the reversible Lotka-Volterra model, investigated the dynamic behavior of oscillatory reaction. In economy, Lotka-Volterra model is also a very useful tool to analyse enterprise competition [5]. Nowadays, Lotka-Volterra model plays a vital role in engineering field.

The classical three-dimensional competitive Lotka-Volterra model can be written as

where x(t), y(t) and z(t) denote the densities of three species at time t. For i, j = 1, 2, 3, the parameters r_i, a_ij are all positive; r_i stand for intrinsic growth rates; a_ii describe as intra-specific competition rates; a_ij(i ≠ j) represent the inter-specific competition rates. For deterministic Lotka-Volterra model, quite a few scholars contribute to investigate coexistence of species (see [6, 7, 8, 9, 10, 11, 12] and references therein).

However, as the development of stochastic analysis, deterministic model may not be suitable for reality sometimes. As a matter of fact, population systems are usually subject to environmental noise. Therefore, to describe more realistically population systems, stochastic population models have attracted considerable attention [13, 14, 15, 16, 17]. For the sake of generality, suppose that all coefficients of model(1.1) are perturbed by Brownian motions, and the model becomes

where B_i (i = 1, 2, 3, 4, 5, 6) are mutually independent Brownian motions. Although there may be an appropriate Lyapunov function to provide the condition for coexistence, it is really hard to be found in practice. Motivated this, some skillful technique should be introduced to solve this problem. To reduce unnecessary computations due to notational complexity and to make our ideas more understandable but still preserve important properties, we assume that the lowest-power terms are not affected by environment noise for simplicity, which means Δ₁ = Δ₂ = Δ₃ = 0. Thus, throughout the rest of the paper, the following model will be considered

To proceed, The rest of the paper is as follows. In Section 2, basic concept and three essential constants λ_3z, λ_3x and λ_3y are provided. Sections 3 are devoted to proving the stochastic coexistence. Then a few numerical simulations are given to manifest our results in Section 4. The last but not least, brief discussions are made to conclude the paper.

2 Basic concept and essential constants

Let (Ω, 𝓕, {𝓕_t}_t≥0, ℙ) be a complete filtered probability space with the filtration {𝓕_t}_t≥0 satisfying the usual condition, i.e., it is increasing and right continuous while 𝓕₀ contains all ℙ-null sets. In model (1.3), B_i (i = 1, 2, 3, 4, 5, 6) are 𝓕_t-adapted, mutually independent Brownian motions. Suppose that a_ij (i, j = 1, 2, 3) are positive constants. Set σ_ii ≠ 0 (i = 1, 2, 3) in order to make the diffusion be non-degenerate. For convenience, denote w = (x, y, z), w₀ = (x₀, y₀, z₀), and w(t) = (x(t), y(t), z(t)). Denote a ∨ b = max{a, b}, a ∧ b = min{a, b}, R+2,o = {(x, y) : x > 0, y > 0} and R+3,o = {(x, y, z) : x > 0, y > 0, z > 0}. Let w_w0(t) = (x_w0(t), y_w0(t), z_w0(t)) be the solution to (1.3) with initial value w₀. Previous study [15] proved that w_w0(t) remains in R+3,o with probability one if w₀ ∈ R+3,o . The definition of coexistence is given as following.

Next, a few useful properties of solutions are presented, whose proofs can be found in [15, 19, 20].

To further study, the equations on the boundary need to be considered. On the x-axis, one has

There exists a unique stationary distribution π1∗ in (0, ∞) with density

where c1∗=∫0∞x2exp2a11σ112x−r1σ112x2dx−1. Owing to the ergodicity, for any measurable function h( ⋅) : ℝ₊ → ℝ satisfying that ∫0∞|h(u)|f1∗(u)du<∞, the following equation holds,

where φ_x0(t) is the solution to (2.1) starting at x₀. Specially, for any p ∈ (−∞, 3),

Similarly, on the y-axis, one has

which in (0, ∞) has a unique stationary distribution π2∗ with density

By the ergodicity, for p ∈ (−∞, 3), one gets

where ψ_y0(t) is the solution to (2.4) starting at y₀.

On the z-axis, one has

which in (0, ∞) has a unique stationary distribution π3∗ with density

In view of the ergodicity, for p ∈ (−∞, 3), one has

where ρ_z0(t) is the solution to (2.6) starting at z₀. As to the x − y plane, Literature [18] has considered following model

which can be seen as a case on the x - y plane for (1.3). Its result provides that (2.8) has a unique invariant measure μ1∗ in R+2,o if

and

Moreover, for any μ1∗ -integrable function F(x, y) : R+2,o → ℝ, one has

Let f₁(x, y) be the density of probability measure μ1∗ in R+2,o . Then f₁(x, y) has following properties,

where f1∗ (x) and f2∗ (y) are density of π1∗ and π2∗ respectively. Define that

Similarly, on y-z plane, one gets

Using the method of [18], if

and

then (2.11) has a unique invariant measure μ2∗ in R+2,o Moreover, for any μ2∗ -integrable function F(y, z): R+2,o → ℝ, one has

Let f₂(y, z) be the density of probability measure μ1∗ in R+2,o , and then f₂(y, z) has following properties,

where f2∗ (y) and f3∗ (z) are density of π2∗ and π3∗ respectively. Define that

Analogously, on x - z plane, one has

If

and

then (2.13) has a unique invariant measure μ3∗ in R+2,o Moreover, for any μ3∗ -integrable function F(x, z) : R+2,o → ℝ, one has

Let f₃(x, z) be the density of probability measure μ3∗ in R+2,o , and then f₃(x, z) has following properties,

where f1∗ (x) and f3∗ (z) are density of π1∗ and π3∗ respectively. Define that

The reason of definition of λ_3z, λ_3x and λ_3y is as following. To determine whether z_w0(t) converges to 0 or not, it is required to consider the Lyapunov exponent of z_w0(t). The Itô’s formula manifests that

When T is large enough, the first and the third terms on the right-side of (2.15) are small. Clearly, if z_w0(t) is small in [0, T], the solution x_w0(t) is close to X_(x0,y0)(t) and y_w0(t) is close to Y_(x0,y0)(t), where X_(x0,y0)(t) and Y_(x0,y0)(t) are solutions to (2.8) starting at (x₀, y₀). Employing the ergodicity (2.9), ln⁡zw0(T)T is close to λ_3z. The definitions of λ_3x and λ_3y are also gotten in the similar way.

3 Main result

This section focuses on providing the conditions for stochastic coexistence in model (1.3).

To prove Theorem 3.1, some arguments need to be introduced.

The following formula is the well-known exponential martingale inequality. It asserts that for any a, b > 0,

if B(s) is a 𝓕_t-adapted Brownian motion while g(t) is a real-valued 𝓕_t-adapted process and ∫0tg2(s)ds < ∞, ∀ t ≥ 0 almost surely. Ordinarily, the inequality holds in finite time, but (3.1) holds since

Let any T > 1, p ∈ (1, 1.5), and 1p+1q = 1. For A ∈ 𝓕, I_A stands for the indicator function of A. Owing to part (iii) of Proposition 2.1 and Holder’s inequality, it is estimated that

for some constants θ₁, θ₂ independent of z₀, T and A. In particular, when A = Ω,

and consequently,

To satisfy the condition that z_w0(t) is small in [0, T], it requires to introduce Lemma 3.1. Let τw0σ be the stopping time that τw0σ = inf{t ≥ 0 : z_w0(t) ≥ σ}.