A stochastic ratio-dependent predator-prey model under regime switching

Abstract

This article presents an investigation of asymptotic properties of a stochastic ratio-dependent predator-prey model under regime switching. Both the white and color noises are taken into account in our model. We obtain the global existence of positive unique solution of the stochastic model. And we show the solution is bounded in mean. Moreover, the sufficient conditions for persistence in mean, extinction are obtained.

1. Introduction

The dynamic interaction between the predators and their prey has long been one of the dominant themes in mathematical biology because of its universal existence and importance. Evidences show that when predators have to search for food (and therefore have to search or compete for food), a more suitable functional response depending on the densities of both the prey and the predator should be introduced in a realistic model. Such a functional response is called a ratio-dependent functional response. Arditi and Ginzburg [1] introduced a Michaelis-Menten type ratio-dependent functional response of the form

(1.1)

where x(t) and y(t), respectively, denote population densities of prey and predator at time t. Here, g > 0 is the death rate of the predator, a, c, m, and f are positive constants that stand for prey intrinsic growth rate, capturing rate, half capturing saturation constant, and conversion rate, respectively.

As a matter of fact, population systems is often subject to environmental noise. Recently, more and more interest is focused on stochastic systems. Maiti et al. [2] considered the following stochastic model with discrete time-delay:

(1.2)

where the perturbed terms η₁(t) and η₂(t) are uncorrelated Gaussian white noises. Maiti et al. [2] assumed the Stratonovich interpretation of stochastic differential equations, and discussed the properties of SDE (1.2) by using transformations.

Guo [3] studied the stochastic model on predator-prey system of two species with ratio-dependence:

(1.3)

where a, d, c, f, g, and m are positive constants.

Taking into account the effect of randomly fluctuating environment, Ji et al. [4] considered the corresponding autonomous stochastic system described by the Itô equation

(1.4)

where B _i (t), i = 1, 2, are independent standard Brownian motions.

Now, let us consider another type of environmental noise, namely, the color noise (for example,[5–10]). The color noise can be illustrated as a switching between two or more regimes of environmental. Because population may suffer sudden-environmental changes, e.g., rain falls and changes in nutrition or food resources, etc. In general, the switching is memory-less, and the waiting time for the next switch is exponential distributed. Here, we model the regime switching by a finite-state Markov chain. We assume that there are N regimes, and the switching between these N regimes is governed by a Markov chain r(t) on the state space S = {1, 2,..., N}. Therefore, when both the white and color noises are taken into account in the system (1.1). The population system under regime switching can be described by the stochastic model as follows:

(1.5)

where B _i (t), i = 1, 2, are independent standard Brownian motions.

When both the white and color noises are taken into account in our model (1.5), we obtain the global existence of positive unique solution of the stochastic model, that is, the solution of the system is positive and not to explode to infinity in a finite time in Section 3. Section 3 also shows that the solution is bounded in mean. Moreover, the sufficient conditions for persistence in mean, extinction are obtained in Section 4.

For convenience and simplicity in the following discussion, for any sequence c(i), i ∈ S, we define

And throughout the article, we use K to denote a positive constant the exact value of which may be different in different appearances.

2. Stochastic differential equation under regime switching

Throughout this article, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all P-null sets.). Let r(t), t ≥ 0, be a right-continuous Markov chain in the probability space tasking values in a finite state space S = {1, 2,..., N} with generator Γ = (γ _ij )_N×Ngiven by

where Δ > 0. Here, γ _ij ≥ 0 is the transition rate from i to j if i ≠ j while . We assume that the Markov chain r(t) is independent of the Brownian motion. And almost every sample path of r(t) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of R₊.

We assume, as a standing hypothesis in the article, that the Markov chain is irreducible. The algebraic interpretation of irreducibility is rank (Γ) = N -1. Under this condition, the Markov chain has a unique stationary distribution π = (π₁, π₁,..., π _N ) ∈ R^1×Nwhich can be determined by solving the following linear equation

subject to

Consider a stochastic differential equation with Markovian switching

on t ≥ 0 with initial value x(0) = x₀ ∈Rⁿ, where

For the existence and uniqueness of the solution, we should suppose that the coefficients of the above equation satisfy the local Lipschitz condition and the linear growth condition. That is, for each k = 1, 2,..., there is h _k > 0 such that

for all t ≥ 0, i ∈ S and those x, y ∈ Rⁿ with , and there is an h > 0 such that

for all (x, t, i) ∈ Rⁿ × R₊ ×S.

Let C^2,1(Rⁿ × R₊ × S, R₊) denote the family of all non-negative functions V (x, t, i) on Rⁿ × R₊ × S which are continuously twice differentiable in x and once differentiable in t. If V ∈ C^2,1(Rⁿ × R₊ × S, R₊), define an operator LV from Rⁿ × R₊ × S to R by

In particular, if V is independent of i, that is V (x, t, i) = V (x, t), then

3. Positive, global and bounded solutions

As x(t), y(t) of the SDE (1.5) are sizes of the species in the system at time t, it is obvious that the positive solution are of interest. The coefficients of (1.5) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (1.5) may explode at a finite time. The following theorem shows that the solution will not explode at a finite time.

Theorem 3.1. For given initial value , there is a unique positive solution X(t) = (x(t), y(t)) to (1.5) on t ≥ 0, and the solution will remain in with probability one, namely for all t ≥ 0 almost surely.

Proof The proof is similar to [10, 11]. Since the coefficients of the equation are locally Lipschitz continuous, for given initial value , there is a unique local solution X(t) on t ∈ [0, τ _e ), where τ _e is the explosion time. To show this solution is global, we need to show that τ _e = +∞ a.s. Let k₀ > 0 be sufficiently large for every component of x(t) and y(t) all lying within the interval . For each integer k ≥ k₀, define the stopping time

where throughout this article we set inf Ø = ∞. Obviously, τ _k is increasing as k → ∞. Let τ_∞ = lim_k→∞τ _k , whence τ _∞ ≤ τ _e a.s. If we can show that τ _∞ = ∞ a.s., then τ _e = ∞ a.s. and a.s. for all t ≥ 0. So we just to prove that τ _∞ = ∞ a.s. If not, there is ε ∈ (0, 1) and T > 0 such that

Hence, there is integer k₁ ≥ k₀ such that P {τ _k ≤ T} ≥ ε for all k ≥ k₁. Define a function by V (x, y) = (x -1 - ln x) + (y -1 - ln y). The non-negativity of this function can be seen from

If , we obtain that

Making use of the generalized Itô formula yields

Set Ω _k = τ _k ≤ T for k ≥ k₁ then P (Ω _k ) ≥ ε. Note that for every ω ∈ Ω, there is x(τ _k , ω) or y(τ _k , ω) equals either k or , and hence, V (x(τ _k , ω)) is no less than either

or

Therefore,

It follows from that

where is the indicator function of Ω _k . Letting m → ∞ implies the contradiction

So, we have that τ _∞ = ∞ a.s. The proof is complete.

Theorem 3.2. For given initial value and p > 0, the solution X(t) = (x(t), y(t)) to (1.5) satisfies

Proof Define the function V (t, x) = e^tx^p, by the generalized Itô formula, we obtain

Taking expectation on both sides implies

Hence,

We show that y(t) is bounded in mean as follows. Let , by the generalized Itô formula, we have

where dropping t from x(t) and y(t). So

Then,

Obviously, the maximum value of (a(i) + g(i))Ex b(i)(Ex)² is . Therefore

By the comparison theorem, we have

It is clear that

So, we complete the proof.

4. Asymptotic behavior

4.1. Limit results

To demonstrate asymptotic properties of the stochastic system (1.5), we discuss the long time behavior of ln x(t)/t and ln y(t)/t.

Here we impose the following assumption:

(H) .

On the one hand, by the comparison theorem of stochastic equations, it is obvious that

Denote that X₂(t) as the solution to the following stochastic equation.

(4.1)

We have

On the other hand, by the comparison theorem of stochastic equations, it is obvious that we denote X₁ as the solution of stochastic differential equation

(4.2)

Consequently,

To sum up, we have

(4.3)

So we have the explicit solutions of X₁(t) and X₂(t) as follows.

and

Lemma 4.1. Under Assumption (H), for any initial value x₀ > 0, the solution X₂(t) satisfies

Proof First, we will show that

The proof is motivated by Mao and Yuan [8]. Define the Lyapunov function V (t, X₂) = e^t ln X₂, using the generalized Itô formula, we obtain

where dropping t from X₂. Thus

where

tje quadratic variation of which is

By virtue of the exponential martingale inequality, for any positive constants T, δ, β, we have

Choose T = kγ , δ= nεe ^-kδ, and , where k ∈ Z⁺, 0 < ε < 1, θ > 1 and γ > 0 above.

Hence,

Obviously, we know Applying the Borel-Cantalli lemma, we obtain that there exists some Ω _i ⊂ Ω with P (Ω _i ) = 1 such that for any ω ∈ Ω _i , an integer k _i = k _i (ω) such that for any k > k _i , we get

for all 0 ≤ t ≤ kγ. Then,

Note that t ∈ [0, kγ], s ∈ [0, t] we have

For all t ∈ [0, kγ] with k > k₀(ω), we derive

Thus, for (k -1) γ ≤ t ≤ kγ, then

Therefore,

Letting k → ∞, that is, t → ∞ we can imply

By making γ ↓ 0, ε ↑ 1, and θ ↓ 1, we get

Consequently,

as desired.

Thus, it remains to show that lim . It is clear that the quadratic variation of the stochastic integral is . Hence, the strong law of large numbers of local martingales yields that

Hence, for any ε > 0, there exists some positive T < ∞ such that

and for any t > s ≥ T, we have

Then, for any t > T

Therefore,

That is

Then,

Thus,

Since ε is arbitrary, we conclude that

as required.

Lemma 4.2. Under Assumption (H), for any initial value x₀ > 0, the solution X₁(t) satisfies

Proof Under the condition , by the same way of Lemma 4.1, we can imply the desired assertion.

Theorem 4.3. Assume the conditions (H) hold. Then for any initial value x₀ > 0, the solution x(t) to (1.5) satisfies

(4.4)

Proof By (4.3), Lemmas 4.1 and 4.2, we can conclude the assertion.

Now, let us continue to consider the asymptotic behavior of the species y(t). By the comparison theorem of stochastic equations, we have

Denote Y₂(t) as the solution to the stochastic equation as follows.

We have

On the other hand, applying the comparison theorem again, denote Y₁ as the solution of stochastic equation

Consequently,

To sum up, we have

(4.5)

Moreover, Y₁(t) and Y₂(t) have the have the following explicit solutions, respectively:

and

Lemma 4.4. Under Assumption (H), for any initial value y₀ > 0, the solutions Y₁(t) and Y₂(t) satisfy

and

Proof It is clear that the quadratic variation of the stochastic integral is . hence, the strong law of large numbers of local martingales yields that

Hence, for any ε > 0, there exists some positive T < ∞ such that

and for any t > s ≥ T, we have

It follows from Lemma 4.1 that for any ε > 0, there exists T > 0 such that

that is

Thus, for t ≥ T

So

Therefore,

Hence,

Letting t → ∞, we have

Since ε is arbitrary, we obtain

Next, we will show that

Obviously, it follows from Lemma 4.2 that for arbitrary ε > 0, there exists T > 0 such that

Hence,

Then,

we obtain

Hence,

Letting t → ∞, we have

Since ε is arbitrary, we obtain

as required.

Theorem 4.5. Under Assumption (H), for any initial value y₀ > 0, the solution y(t) to (1.5) has the property

(4.6)

Proof It follows from (4.5) and Lemma 4.4 that

Consequently,

The proof is complete.

4.2. Persistent in mean

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now, we present the definition of persistence in mean proposed in Ji et al. [4] and [12].

Definition 4.6. System (1.5) is said to be persistent in mean, if

Theorem 4.7. Assume the condition (H) hold. Then system (1.5) is persistent in mean.

Proof Define the function V = ln x, by the generalized Itô formula, we get

Thus,

Dividing both sides by t, letting t → ∞ and by the strong law of large numbers and Theorem 4.3, we obtain

Since the Markov chain r(·) is irreducible, then

Therefore,

That is

and

Moreover, define the function V = ln y, using the generalized Itô formula, we have

So, we have

Dividing both sides by t, letting t → ∞ and by the strong law of large numbers and Theorem 4.5, we have

Note that the Markov chain r(·) is irreducible, then

Obviously

Then,

And we can imply

where dropping s from x(s) and y(s). Hence

So, the system is persistent in mean.

4.3. Extinction

In Section 4.2, under the condition (H), we show that the system is persistent in mean. To a large extent, (H) as the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.

Theorem 4.7. Assume the condition , hold. Then system (1.5) will become extinct exponentially with probability one.

Proof Define the function V = ln x, by the generalized Itô formula, we get

Thus,

By the strong law of large numbers of martingales, we have

Therefore,

On the other hand, by the generalized Itô formula, we derive

So

Applying the strong law of large numbers of martingales again, then

The proof is complete.

5. Conclusions

Both the white and color noises are taken into account in our model in this article. It tells us that when the intensities of environmental noises are not too big, some nice properties such as non-explosion, boundedness, and permanence are desired. However, Theorem 4.7 reveals that a large white noise will force the population to become extinct while the population may be persistent under a relatively small noises.