A stochastic predator-prey system with Watt-type functional response

In this paper we consider a stochastic version of predator-prey systems with Watt-type functional response. We first prove the existence and uniqueness of the positive global solution by using the comparison theorem of stochastic equations. Then, we study the boundedness of moments of the solution. Furthermore, the growth rates, persistence and extinction of species are investigated.


Introduction
It is well known that the predator-prey system is one of the most important models in ecology.The first predatorprey system, described by a system of differential equations, was proposed by Volterra in 1920's.Since then, based on different settings, various types of predator-prey models have been proposed and studied by ecologists and mathematicians, such as considering different functional response types: Holling types [7], Hassel-Varley type [6], Beddington-DeAngelis type [3,4] and ratio-dependence type [1], adding a prey self-competition term [16], etc.We refer the reader to the book [18] for a detailed presentation.
On the other hand, it is also known that the environmental noise is an important component in an ecosystem.Indeed, for instance, in [15], Mao et al. shown that even a sufficiently small noise can suppress explosions in population dynamics.Since the real world are full of random perturbations, it is of great significance to take into account the effect of noise in the investigation of deterministic predator-prey systems.In fact, a lot of stochastic version of existing deterministic models have been introduced recently by different authors and here we only mention some of them.For example, Khasminski ǐ and Klebaner in [10] gave an analysis of Lotka-Volterra system with small random perturbations, Ji and Jiang in [9] analyzed a stochastic predator-prey system with Beddington-DeAngelis functional response, Bandyopadhyay and Chattopadhyay in [2] studied the effect of environmental fluctuations on a ratio-dependent predator-prey system, Mandal and Banerjee in [13] investigated stochastic persistence and stationary distribution of a Holling-Tanner type prey-predator model, Dung in [5] provided an explicit solution to delayed logistic equations with fractional noise, etc.

A STOCHASTIC PREDATOR-PREY SYSTEM WITH WATT-TYPE FUNCTIONAL RESPONSE
A well-known model of deterministic systems is the predator-prey model with a Watt-type functional response, proposed by Watt [23], which can be described by the following differential equations where x(t) and y(t) are functions of time representing population density of prey and predator, and all parameters are positive constants, a 1 is the intrinsic growth rate of prey, c 1 the maximum number of prey that can be eaten by a predator per unit of time, c 2 a conversion efficiency and a 2 the mortality rate of the predator.
Although some improvements to the original model ( 1) have been extensively suggested (see, for instance, [12,21,22]), the question of the effect of environmental noise on this model has not yet been addressed.The aim of this paper is to study a stochastic version of (1).More specifically, we assume that the relevant parameters a 1 , a 2 are random, then they can be modeled by where and ε 1 , ε 2 are deterministic and Ẇ1 (t), Ẇ2 (t) are independent white noises.Inserting (2) into the system (1), we get a new stochastic version in the following form where W 1 (t), W 2 (t) are independent standard Brownian motions, σ 1 , σ 2 represent the intensities of the white noises.The initial conditions x 0 , y 0 > 0.
In this paper, to investigate the nonlinear stochastic system (3), we mainly use Itô's formula, the theory of stochastic differential equations and the method of Lyapunov functionals to estimate its solution, and to analyze the long time behavior of the system.The structure of the paper is as follows.In Section 2, we establish the existence of unique positive global solution and give an estimate for moments of the solution.In Section 3, we study some features of the system, including the growth rates, persistence and extinction of predator and prey.The conclusion is given in Section 4.

The positive solution and its moments
Throughout this paper, we use the following notations.Denote R 2 + = (0, ∞) × (0, ∞), (Ω, F, P ) the complete probability space with a filtration {F t } t≥0 satisfying the usual conditions, that is, it is right continuous and increasing while F 0 contains all P -null sets.Let W 1 (t), W 2 (t) denote the independent standard Brownian motions defined on this probability space.
Since stochastic system (3) describes population dynamics, its solution has to be positive and not to explode at a finite time.By using the Itô formula, first we show that there exists a unique positive local solution of the system (3) and then by using the comparison theorem, we prove that this solution is global.We have the following theorem.

Theorem 2.1
The system (3) has a unique positive global solution on the interval [0, ∞) for any the initial condition (x 0 , y 0 ) ∈ R 2 + .

Proof
We first consider the following system with the initial (u 0 , v 0 ) = (ln x 0 , ln y 0 ).Since the coefficients of (2.1) are locally Lipschitz continuous, there is a unique local solution to (4) on the interval [0, τ ), where τ is the explosion time.Hence, by Itô's formula, t) is the unique positive local solution to (3) with the initial (x 0 , y 0 ) ∈ R 2 + .To show the solution is globally positive, we need to check that τ = ∞ almost surely.
On the interval [0, τ ), x(t) and y(t) are positive and hence, By the comparison theorem for stochastic differential equations (see, [8]) we have, for 0 ≤ t < τ : where x(t), ȳ(t) solve the following two equations, respectively The equation ( 6) is well known as a stochastic logistic equation in population dynamics, its solution is given by Similarly, the solution of ( 7) is given by ) .
Since both x(t) and ȳ(t) are defined globally, we conclude that τ = ∞.
The Theorem is proved.

Theorem 2.2
For each n > 0, the nth-moment of the prey species is bounded uniformly in time, i.e. there exists a finite constant

Proof
Applying the Itô's formula to x(t) n shows that Consequently, Put g(t) = Ex(t) n and by Lyapunov's inequality (Ex(t) n ) which can be written as From ( 9) and by the differential inequality (see, [20]), g(t) is dominated by the solution of an ordinary Bernouilli differential equation of the form: Solving the Bernouilli equation we obtain for all t ≥ 0 [g(t)] As a consequence, We therefore have The Theorem is proved.
and then We observe that lim t→∞ . Thus it is impossible to establish a similar estimate to (8) for the predator species.When n = 1, we have the following.

Theorem 2.3
The first moment of the predator species is bounded uniformly in time: ) .

Proof
Consider the following stochastic process We have and Obviously, (a . Therefore, Once again, by the differential inequality Since Ey(t) ≤ c2 c1 Ez(t), we obtain desired estimate.The Theorem is proved.

The growth rates, persistence and extinction
In this section, we give some estimates for growth rates of prey and predator.We also point out the sufficient conditions for the persistence and extinction of population in terms of its parameters.The obtained results confirm a well-known fact that if the noise is sufficiently large, the population will become extinct with probability one.
We first provide an upper bound for the growth rate of both prey and predator species.

Theorem 3.1
If 0 < m < 1 then for any p 1 > 0, p 2 ≥ 0 we have the following estimate for any initial value (x 0 , y 0 Moreover, for all m > 0, we have the following estimate for growth rate of prey species

Proof
We recall that Applying Itô's formula to e nt u(t) and e nt v(t) we have, respectively Hence, where ) .
Since 0 < m < 1, it is easy to check that there exists a positive constant K = K(n, p 1 , p 2 ) such that g(x, y) ≤ K ∀ x, y > 0.

Thus for
e nk ln k ε , and so e n ln k ε .
In the above inequality, let t → ∞ we obtain Taking the limits θ → 1 + , ε → 1 − and n → 0 + we get which deduces (10) because The inequality (11) follows from (10) by choosing p 2 = 0 and using that fact that for any m > 0 The Theorem is proved.
Remark 3.1.In the context of method of Lyapunov functionals, the condition 0 < m < 1 is inevitable one.Indeed, when m ≥ 1 we have for fixed and so, lim y→∞ g(x, y) = ∞, i.e. g(x, y) cannot be bounded by a finite constant K.
Lemma 3.1 (Strong law of large numbers) Let W (t) is a standard Brownian motion.We have Proof Refer [19].
We next study the prey species in the case without the predator species.
2 < 0, then the prey dies with a probability one even if there is no predator.
II.In the case of a 1 − σ 2 1 2 > 0, if the predator is absent, i.e., y(t) = 0 a.s.then in other words, the prey species is stable in time average.

Proof
I. From the proof of Theorem 2.1 we known that x(t) = e u(t) , where Hence, Stat., Optim.Inf.Comput.Vol.
It is clear that ] . Since ] .
Using the strong law of large numbers and the assumption a 1 − σ 2 1 2 > 0, the above inequality gives us the following estimate lim sup Similarly, with noting that max 0≤s≤t σ 1 W 1 (s) ≥ W 1 (0) = 0, we also have Combining ( 16) and (17) shows that An application of Itô's formula to ū(t) = ln x(t) yields which leads us to This, together with (18), implies that The Theorem is proved.
Definition 3.1.The population x(t) is said to be strongly persistent in mean if x(s)ds > 0, a.s.
Let us now consider the case there is the appearance of the predator species.We have the following properties for the prey species.

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On the other hand, it is easy to see that and then The Theorem is proved.
We end this paper with the properties of the predator species in the cases where our system contains or does not contain the prey species.Theorem 3.4 I.If the prey is absent, i.e., x(t) = 0 a.s.for all t ≥ 0, then the predator dies with a probability one.Furthermore, the death's rate of predator is exponential 2 > 0, i.e. the prey species is not extinction.We have Proof I.When x(t) = 0 a.s., the dynamic of predator species is expressed by  (22).
II.We recall that Then, by using the same arguments as the proof of Part I, we have From the inequality The solution of (24) can be found explicitly (for instance, see [11, page 125]).Hence, 2 )t+σ2W2(t) 2 )s+σ2W2(s)) ds )  } .
Hence, it is easy to see that max We finish the proof the Theorem by combining ( 23) and (28).
Remark 3.2.Theorem 3.4 gives us some interesting ecological interpretations.Firstly, if the system has no the prey species, then the predator population will go to extinction, which is consistent with our expectation.Secondly, although the prey population will survive, the predators die out because the diffusion coefficient σ 2 2 is too large (c 2 − a 2 − σ 2 2 2 < 0).This means that a relatively large stochastic perturbation can cause the extinction of the population.

Conclusion
In this paper, we discuss a stochastic predator-prey system with Watt-type functional response.We show that the model admits a unique global positive solution, and investigate the uniformly finite moments.Moreover, we use the Lyapunov functionals, the strong law of large numbers for Brownian motion and the theory of Itô's stochastic differential equations to study the long-term behaviors of solutions.Our obtained results partly enrich the knowledge of theory of stochastic predator-prey systems.