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

Journal of Inequalities and Applications20112011:14

https://doi.org/10.1186/1029-242X-2011-14

Received: 21 October 2010

Accepted: 22 June 2011

Published: 22 June 2011

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.

Keywords

Itô formulaIrreducibleExponential martingale inequalityComparison theoremPersistent in meanExtinct

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,[510]). 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 ij 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 ) RNwhich 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 R n , 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 R n with , and there is an h > 0 such that

for all (x, t, i) R n × R+ ×S.

Let C2,1(R n × R+ × S, R+) denote the family of all non-negative functions V (x, t, i) on R n × R+ × S which are continuously twice differentiable in x and once differentiable in t. If V C2,1(R n × R+ × S, R+), define an operator LV from R n × 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 kk0, define the stopping time
where throughout this article we set inf Ø = ∞. Obviously, τ k is increasing as k → ∞. Let τ = limk→∞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 k1k0 such that P {τ k T} ≥ ε for all kk1. 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 kk1 then P 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 t x 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 = , δ= nεe -, 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 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. Then,
Note that t [0, ], s [0, t] we have
For all t [0, ] with k > k0(ω), we derive
Thus, for (k -1) γ ≤ t, 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 > sT, 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 > sT, we have
It follows from Lemma 4.1 that for any ε > 0, there exists T > 0 such that
that is
Thus, for tT
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.

Declarations

Acknowledgements

This research is supported by the National Natural Science Foundation of P.R. China (No. 10701020)

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, People's Republic of China
(2)
School of Mathematics and Statistics, Northeast Normal University, Changchun, People's Republic of China

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Copyright

© Lv and Wang; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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