STOCHASTIC SPATIOTEMPORAL DIFFUSIVE PREDATOR-PREY SYSTEMS

. In this paper, a spatiotemporal diﬀusive predator-prey system with Holling type-III is considered. By using a Lyapunov-like function, it is proved that the unique local solution of the system must be a a global one if the interaction intensity is small enough. A comparison theorem is used to show that the system can be extinction or stability in mean square under some additional conditions. Finally, an unique invariant measure for the system is obtained.

1. Introduction. In the ecosystem, none of the species survives alone. The relation of two species can be described by competition, predator-prey, auspiciousness and so on. Among them, the predator-prey interaction is a significant one, which was introduced by Lotka and Volterra( [10]) and has been developing rapidly in the last decades( [4,21,5,20,8]). In this paper, we consider a stochastic homogeneous spatiotemporal diffusive predator-prey system with functional response. In 2016, J.F. Wang([20]) studied the dynamics of a deterministic homogeneous diffusive predator-prey system with Holling type-III, it takes the form: unit outer normal, we suppose the system is a closed one so that there is no flux boundary condition. But as we all know, in reality, random disturbance of various forms is everywhere. The disturbance which is described by White noise often plays an important role in the behaviour of the solution, even in the existence of the solutions. Almost all the statistical data showed that biological process has marked random fluctuation( [15]). But the deterministic systems always assume that the parameters of the models have nothing to do with the environmental disturbance. Hence the description and prediction of the deterministic systems are always less than satisfactory. Motivated by the issues, a great number of scholars introduced stochastic mathematical models instead of deterministic ones to exposit the population dynamics affected by environmental fluctuations in an ecosystem( [6,12,14,19,22,23]). Even so, as far as we know, studies about the predator-prey system with Holling type-III and the effect of a randomly fluctuating environment are relatively rare. We introduce the space-time stochastic perturbations by White noises into the equation (1) directly, and obtain the following corresponding stochastic reaction-diffusion model: where W 1 (x, t), W 2 (x, t) are independent spatially dependent Wiener fields,Ẇ (x, t) means the formal time derivative ∂ ∂t W (x, t) of W (x, t), and σ 2 1 > 0, σ 2 2 > 0 represent the intensities of the White fields. Suppose the initial values u 0 , v 0 satisfy 0 < u 0 ≤ θ u , 0 < v 0 ≤ θ v , here θ u and θ v are both positive constants.
In the corresponding deterministic system (1), the existence and boundedness of the non-negative global solution is shown in [5] by a general result. Meanwhile, there are three nonnegative constant equilibrium solutions: (0, 0), (1, 0) and (κ, v κ ) of system (1), where and the positive equilibrium solution (κ, v κ ) exists if and only if 0 < κ < 1, i.e. m > max{d, (a 2 −1)d}. [20] gives a stability result regarding the equilibrium (κ, v κ ) and (1, 0) is under some conditions respectively. But for the stochastic case, because we formulate system (2) by stochastic perturbations σ 1 uẆ 1 and σ 2 vẆ 2 directly, there is no positive equilibrium point as a solution. Hence, the solution of (2) will not tend to a point. Things are quite different from the deterministic model, even the existence of the solution. So, we begin with discussing the existence of the global solution. Comparing with the condition for the existence of the positive equilibrium solution of (1), we find if the interaction intensity m satisfies m < min{2a 2 , d − σ 2 2 2 } and d > σ 2 2 2 , (2) has a unique global positive mild solution. The difficulty here is f only satisfy local Lipschitz condition so the solution is a local one. Consequently, we apply a Lyapunov-like function and a stop time to overcome the troubles.
It is generally known that permanent is the most important property of a system, it means every species in this system can survive with other species together continuously. And closely related to that is extinction. A lot of literatures talked about permanent, extinction and stability of a ecosystem, like [6,2,1,15,7]. In Section 4, by Comparison theorem, we prove that if the strength of the White noise is large, σ 2 1 > 2, σ 2 2 > 2m, the stochastic system will not be permanent. And if where −λ ∆ be the principle eigenvalue of ∆, the system will be stability in mean square.
Finally, we prove that the global mild solution is a Markov process. Under stronger conditions σ 2 of course under the conditions for the existence too, the stable system has a unique invariant measure which is a more delicate description of the L 2 exponential stability.
Throughout this paper, we assume (Ω, F, {F t } t≥0 , P) is a complete probability space, {F t } t≥0 is a right continuous filtration and F 0 contains all P-null sets.

Preliminary.
We consider a nonlinear diffusion-reaction equation problem with White noise: where , d i > 0(i = 1, · · · , m) and α > 0. We recall some basic well known results(see, e.g. [3]). Let {W(x, t)} 0≤t≤T be a R−Wiener process in H = L 2 (D, R m ), satisfying We recall the eigenvalues and eigenvectors of the operator −A in H = L 2 (D, R m ). By the compact operator theory, the corresponding eigenvalues {λ k } and the eigen- and {e k } ⊂ H are the complete orthogonal basis functions of H.

GUANQI LIU AND YUWEN WANG
Define the Green function of linearized equation corresponding for the operator A is hence the equivalent integral equation of the equation (3) is (4) can be rewriten as here the adjoint Green operation semigroup {Γ t } t≥0 is dW t = W(·, dt).
To prove there is a local solution of (3), we introduce the following three conditions: (A3) {g(·, t)} 0≤t≤T is a H-valued process which is predictable, and
Proof. The proof is similar to that of Theorem 6.5 in [3].
Since the Itô formula is no longer valid for a mild solution, we should introduce a strong solution approximating system to which Itô's formula can be applied.
An approximating system of (3) is Lemma 2.4. Assume for any U 0 ∈ H is an given stochastic variable, E∥U 0 ∥ p < ∞(p > 2 is an integer). If f (·, ·, ·) + g(·, ·), γ(·, ·) in (3) satisfy not only the local Lipschitz condition (A2) but also conditions (A1), (A3), then Proof. Since AR(l) = AlR(l, A) = l − l 2 R(l, A) are bounded operators. By Lemma 2.2, (6) has a local mild solution, and Proposition 1.3.5 in [9] indicates the local mild solution is also a local strong solution. The remainder of the proof is similar to that of Proposition 1.3.6 in [9]. The difference is that we can deduce that

Existence of the global positive solution.
In this section, we consider the existence of the global solution for (2). The first step towards the existence is to obtain a positive local solution of (2).
For a function V (U ) ∈ C 2 (R n ; R), we define a differential operator L with (11) by Since there exists a unique local positive solution U (x, t) = (u(x, t), v(x, t)) T of (11) for t ∈ [0, τ M ], we need to consider the following approximation system of strong solution: From Lemma 2.4, we can see that (12) has a unique local strong solution U n t and lim n→∞ U n t = U (x, t), a.s. uniformly for t ∈ [0, τ M ], where U (x, t) is the unique local mild solution of (11). We need to proof τ M = ∞, then the solution is global.
In fact, Lemma 4.1 remains true if the uniformly Lipschitz condition was replaced by local Lipschitz condition. Indeed, if that would not be the case, there must exist R, T > 0, such that By Comparison Theorem Lemma 4.1, we obtain For the mild solution u 1 (x, t) of (15), where x ∈ D, Γ u is the semigroup generated by d 1 ∆ in C(D). Since Γ u is a contraction semigroup and in addition, W1 t → 0 and σ 2 1 > 2, we have ∥u 1 (x, t)∥ C(D) → 0 as t → ∞. Therefore, The interpretation of the above result is: if the intensities σ 2 1 , σ 2 2 of White noise are large, for example, σ 2 1 > 2, σ 2 2 > 2m, the prey and the predator will die out.

Markov property, uniqueness of invariant measure and ergodic.
In this section, we prove the Markov property for the solution of (11) and follow the methods in [18] and [17] to seek for an unique invariant measure µ for (11).
For a H-valued random variable X, and a probability measure P on Ω, then by L (X) we denote the law of X: Following [17], let P s,t and P (s, x, t; Π), t ≥ 0, u ∈ H, Π ∈ B(H) be the corresponding transition semigroup and transition function to U (t, s, u), here B(H) is the smallest σ−field containing all closed (or open) subsets of H. Thus where χ Π is the characteristic function of the set Π and B b (H) is the Banach space of all real bounded Borel functions, endowed with the sup norm.
First, we have the Markov property of U (t, s, U 0 ), t ≥ s.
Proof. By a series of simple but cumbersome calculations, for (11), we have Also, by Theorem 4.3, there is a positive constant C 1 , such that where C 1 is a positive constant. In this theorem, U (t, s, U 0 ) will denote the unique mild solution of        dU = AU dt + f (U, t)dt + γ(U, t)dW, t ≥ s ∂U ∂n | ∂D = 0, the corresponding approximation system of strong solution is        dU n = AU n dt + R(n)f (U n , t)dt + R(n)γ(U n , t)dW, t ≥ s ∂U ∂n | ∂D = 0, U n (s) = R(n)U 0 .

Remark 1.
If µ is the unique invariant measure for P s,t , then it is ergodic.