Stochastic two-species mutualism model with jumps

The existence and uniqueness are proved for the global positive solution to the system of stochastic differential equations describing a two-species mutualism model disturbed by the white noise, the centered and non-centered Poisson noises. We obtain sufficient conditions for stochastic ultimate boundedness, stochastic permanence, nonpersistence in the mean, strong persistence in the mean and extinction of the solution to the considered system.


Introduction
The construction of the mutualism model and its properties are presented in K. Gopalsamy [4]. Mutualism occurs when one species provides some benefit in exchange for another benefit. A deterministic two-species mutualism model is described by the system where N 1 (t) and N 2 (t) denote the population densities of each species at time t, r i (t) > 0, i = 1, 2, denotes the intrinsic growth rate of species N i , i = 1, 2, and α i (t) > K i (t) > 0, i = 1, 2. The carrying capacity of species N i (t) is K i (t), i = 1, 2, in the absence of other species. In the paper by Hong Qiu, Jingliang Lv and Ke Wang [9] the stochastic mutualism model of the form dx(t) = x(t) a 1 (t) + a 2 (t)y(t) 1 + y(t) − c 1 (t)x(t) + σ 1 (t)x(t)dw 1 (t), is considered, where a i (t), b i (t), c i (t), σ i (t), i = 1, 2, are all positive, continuous and bounded functions on [0, +∞), and w 1 (t), w 2 (t) are independent Wiener processes. The authors show that the stochastic system (1) has a unique global (no explosion in a finite time) solution for any positive initial value and that this stochastic model is stochastically ultimately bounded. The sufficient conditions for stochastic permanence and persistence in the mean of the solution to the system (1) are obtained. Population systems may suffer abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. So it is natural to introduce Poisson noises into the population model for describing such discontinuous systems. In the paper by Mei Li, Hongjun Gao and Binjun Wang [5] the authors consider the stochastic mutualism model with the white and centered Poisson noises: where x(t − ), y(t − ) are the left limit of x(t) and y(t) respectively, r i (t), b i (t), K i (t), α i (t), i = 1, 2, are all positive, continuous and bounded functions, Y is measurable subset of (0, +∞), w i (t), i = 1, 2, are independent standard one-dimensional Wiener processes,ν(t, A) = ν(t, A) − tΠ(A) is the centered Poisson measure independent on w i (t), i = 1, 2, E[ν(t, A)] = tΠ(A), Π(Y) < ∞, γ i (t, z), i = 1, 2, are random, measurable, bounded, continuous in t. The global existence and uniqueness of the positive solution to this problem are proved. The sufficient conditions of stochastic boundedness, stochastic permanence, persistence in the mean and extinction of the solution are obtained.
In this paper, we consider the stochastic mutualism model with jumps generated by centered and noncentered Poisson measures. So, we take into account not only "small" jumps, corresponding to the centered Poisson measure, but also "large" jumps, corresponding to the noncentered Poisson measure. This model is driven by the system of stochastic differential equations where w i (t), i = 1, 2, are independent standard one-dimensional Wiener processes, To the best of our knowledge, there are no papers devoted to the dynamical properties of the stochastic mutualism model (2), even in the case of the centered Poisson noise. It is worth noting that the impact of the centered and noncentered Poisson noises to the stochastic nonautonomous logistic model is studied in the papers by O.D. Borysenko and D.O. Borysenko [1,2].
In the following we will use the notations X(t) = (x 1 (t), x 2 (t)), X 0 = (x 10 , x 20 ), For the bounded, continuous functions f i (t), t ∈ [0, +∞), i = 1, 2, let us denote We will prove that system (2) has a unique, positive, global (no explosion in a finite time) solution for any positive initial value, and that this solution is stochastically ultimate bounded. The sufficient conditions for stochastic permanence, nonpersistence in the mean, strong persistence in the mean and extinction of solution are derived.
The rest of this paper is organized as follows. In Section 2, we prove the existence of the unique global positive solution to the system (2). In Section 3, we prove the stochastic ultimate boundedness of the solution to the system (2). In Section 4, we obtain conditions under which the solution to the system (2) is stochastically permanent, and in Section 5 the sufficient conditions for nonpersistence in the mean, strong persistence in the mean and extinction of the solution are obtained.
are finite measures on the Borel sets in R. On the probability space (Ω, F , P) we consider an increasing, right-continuous family of complete sub-σ- We need the following assumption.
It then follows from (7) that where 1 Ωn is the indicator function of Ω n . Letting n → ∞ leads to the contradiction ∞ > V (X 0 ) + KT = ∞. This completes the proof of the theorem.
3 Stochastically ultimate boundedness The solution X(t) to the system (2) is said to be stochastically ultimately bounded, if for any ε ∈ (0, 1) there is a positive constant χ = χ(ε) > 0 such that for any initial value X 0 ∈ R 2 + this solution has the property lim sup t→∞ P{|X(t)| > χ} < ε.

Stochastic permanence
The property of stochastic permanence is important since it means the long-time survival in a population dynamics.

Definition 2. ([5])
The solution X(t) to the system (2) is said to be stochastically permanent if for any ε > 0 there are positive constants H = H(ε), h = h(ε) such that for i = 1, 2 and for any inial value X 0 ∈ R 2 + . Theorem 3. Let Assumption 1 be fulfilled. If then the solution X(t) to the system (2) with the initial condition X 0 ∈ R 2 + is stochastically permanent.
Proof. For the process U i (t) = 1/x i (t), i = 1, 2, by the Itô formula we have Then by the Itô formula we derive for 0 < θ < 1: where I i,stoch (t), i = 1, 3, are corresponding stochastic integrals in (12). Under the Assumption 1 there exist constants |K 1 (θ)| < ∞, |K 2 (θ)| < ∞ such that for the process J(t) we have the estimate Here we used the inequality (x + y) θ ≤ x θ + θx θ−1 y, 0 < θ < 1, x, y > 0. Due to and the condition min i=1,2 inf t≥0 (a i min (t)−β i (t)) > 0 we can choose a sufficiently small 0 < θ < 1 to satisfy So from (12) and the estimate for J(t) we derive By the Itô formula and (13) we have Let us choose λ > 0 such that K 0 (θ) − λ/θ > 0. Then the function is bounded from above by some constant K > 0. So by integrating (14) and taking the expectation we obtain because the expectation of stochastic integrals are equal zero by (10) and From (11) and (16) by the Chebyshev inequality we can derive that for an arbitrary ε ∈ (0, 1) there are positive constants H = H(ε) and h = h(ε) such that

Extinction, nonpersistence and strong persistence in the mean
The property of extinction in the stochastic models of population dynamics means that every species will become extinct with probability 1.
Proof. By the Itô formula, we have where the martingale has quadratic variation Then the strong law of large numbers for local martingales ( [7]) yields lim t→∞ M (t)/t = 0 a.s. Therefore, from (17) we have So lim t→∞ x i (t) = 0, i = 1, 2, a.s.

Definition 5. ([8])
The solution X(t) = (x 1 (t), x 2 (t)), t ≥ 0, to the system (2) is said to be strongly persistence in the mean if for every initial data X 0 > 0 we have Therefore, the solution X(t) to the system (2) with the initial condition X 0 ∈ R 2 + will be strongly persistence in the mean.
Using the arbitrariness of ε > 0 we get the assertion of the theorem.