Analysis of a non-autonomous mutualism model driven by Levy jumps

This article is concerned with a mutualism ecological model with Levy noise. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded and stochastic permanence. The sufficient conditions for the system to be extinct are given and the condition for the system to be persistent are also established.


Introduction
Mutualism is an important biological interaction in nature. It occurs when one species provides some benefit in exchange for some benefit, for example, pollinators and flowering plants, the pollinators obtain floral nectar (and in some cases pollen) as a food resource while the plant obtains non-trophic reproductive benefits through pollen dispersal and seed production. Another instance is ants and aphids, in which the ants obtain honeydew food resources excreted by aphids while the aphids obtain increased survival by the non-trophic service of ant defense against natural enemies of the aphids. Lots of authors have discussed these models [1,2,5,7,10,14,13,12,24,36]. One of the simplest models is the classical Lotka-Volterra two-species mutualism model as follows: (1.1) Among various types mutualistic model, we should specially mention the following model which was proposed by May [32] in 1976: ẋ(t) = x(t) r 1 − b 1 x(t) K 1 +y(t) − ε 1 x(t) , y(t) = y(t) r 2 − b 2 y(t) K 2 +x(t) − ε 2 y(t) , where x(t), y(t) denote population densities of each species at time t, r i , K i , b i , ε i (i=1, 2) are positive constants, r 1 , r 2 denote the intrinsic growth rate of species x(t), y(t) respectively, K 1 is the capability of species x(t) being short of y(t), similarly K 2 is the capability of species y(t) being short of x(t). For (1.2), there are three trivial equilibrium points and a unique positive interior equilibrium point E * = (x * , y * ) satisfying the following equations where E * is globally asymptotically stable.
In addition, population dynamics is inevitably affected by environmental noises, May [33] pointed out the fact that due to environmental fluctuation, the birth rates, carrying capacity, and other parameters involved in the model system exhibit random fluctuation to a greater or lesser extent. Consequently the equilibrium population distribution fluctuates randomly around some average values. Therefore lots of authors introduced stochastic perturbation into deterministic models to reveal the effect of environmental variability on the population dynamics in mathematical ecology [8,11,18,17,16,21,25,26,27,34,35]. Li where W 1 (t), W 2 (t) are mutually independent Brownian motion, α i , i = 1, 2 represent the intensities of the white noise. Then the corresponding deterministic model system (1.2) may be described by the Itô problems: and Wang [28], Liu and Liang [22]. Motivated by those studies, in this paper we consider the following non-autonomous system with jumps: where x(t − ) and y(t − ) are the left limit of x(t) and y(t) respectively, r i (t), b i (t), N is a Poisson random measure with compensatorÑ and characteristic measure µ on a measurable subset Y of (0, +∞) with µ(Y) < +∞,Ñ (dt, du) = N(dt, du) − µ(du)dt, γ i : Y × Ω → R is bounded and continuous with respect to µ, and is In the next section, the global existence and uniqueness of the positive solution to problem (1.4) are proved by using comparison theorem for stochastic equations.
Sections 3 is devoted to stochastic boundedness. Section 4 deals with stochastic permanence. Section 5 discusses the persistence in mean and extinction, sufficient conditions of persistence in mean and extinction are obtained.
Throughout this paper, we let (Ω, F, {F t } t≥0 , P ) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions. For convenience, we assume that X(t) = (x(t), y(t)) and |X(t)| = x 2 (t) + y 2 (t). 1 + γ i (t, u) > 0, u ∈ Y, i = 1, 2, there exists a constant k > 0 such that We end this section by recalling three definitions which we will use in the forthcoming sections.

Definition 1.2 [29]
If for arbitrary ε ∈ (0, 1), there are two positive constants Then solution of problem (1.5) is said to be stochastically permanent.
The problem of (1.5) is said to be persistence in mean.

Existence and uniqueness of the positive solution
First, we show that there exists a unique local positive solution of (1.5).
Proof: We first set a change of variables : on t ≥ 0 with initial value u(0) = ln x 0 , v(0) = ln y 0 . Obviously, the coefficients of (2.1) satisfy the local Lipschitz condition, then making use of the theorem [9,31] about existence and uniqueness for stochastic differential equation there is a unique Itô's formula, (x(t), y(t)) is a unique positive local solution to problem (1.5) with positive initial value.
Next we need to prove solution is global, that is τ = ∞.
Proof: The reference of [17] was the main source of inspiration for its proof.
Because of (x(t), y(t)) is positive, from the first equation of (1.5), we can define the following problem is the unique solution of (2.2), and it follows from the comparison theorem for stochastic equations that On the other hand, Similarly, we can get where Combining (2.3), (2.5), (2.6) with (2.7), we obtain By Lemma 4.2 in [4], we know that Λ(t), λ(t), Θ(t), θ(t) will not be exploded in any finite time, it follows from the comparison theorem for stochastic equations [15] that (x(t), y(t)) exists globally.

Stochastically ultimate boundedness
In a population dynamical system, the nonexplosion property is often not good enough but the property of ultimate boundedness is more desired. Now, let us present a theorem about the stochastically ultimate boundedness of (1.5) for any positive initial value.
Theorem 3.1 Assume that there exists a constant L(q) > 0 such that Then for any positive initial value (x 0 , y 0 ), the solution X(t) of problem (1.5) is stochastically ultimate boundedness.

Stochastic permanence
In the study of population models, stochastic permanence is one of the most interesting and important topics. We will discuss this property by using the method as in [28] in this section. Proof: For a positive constant 0 < η < 1, we set a function Straightforward computation dV (x) by Itô , s formula shows that then whenr 1 −β 1 > 0, we can choose a sufficiently small η to satisfy Let us choose λ > 0 sufficiently small to satisfy Then, there is a positive constant L 1 satisfying where L := L 1 +ε 1 +b 1 K 1 . Integrating and then taking expectations yields Similarly, whenr 2 −β 2 > 0, we have For arbitrary ε ∈ (0, 1), choosing ζ 2 (ε) = ( λε ηL ) 1 η and using Chebyshev inequality, we yield the following inequalities, Hence, Combining Chebyshev's inequality with (3.1), (3.2), we can prove that for arbitrary ε ∈ (0, 1), there is a positive constant ζ 1 such that This completes the proof.

Persistence in mean and extinction
In the description of population dynamics, it is critical to discuss the property of persistence in mean and extinction. First, we give a Lemma using the argument as in [25,26] with suitable modifications.
(A) If there exist three positive constants T, η and η 0 such that (B) If there exist three positive constants T, η and η 0 such that u))Ñ(ds, du), then M i (t), Q i (t), i = 1, 2 are real valued local martingales vanishing at t = 0. One can see that the quadratic variations of M 1 (t) and Q 1 (t) are where M, M is Meyer's angle bracket process, and By the strong law of large numbers for local martingales [30], we have Then for arbitrary ε > 0, there exists a T 1 > 0 such that for t > T 1 Set g(t) = t 0 x(s)ds for all t > T 2 , then we have That is to say : for t ≥ T, e η 0 g dg dt ≤ e (η+ε)t , integrating this inequality from T to t, we can get Using the arbitrariness of ε we have the assertion.
The proof of (B) is similar to (A). The proof is completed.
Using Lemma 5.1, we have following theorem. with positive initial value (x 0 , y 0 ), then the problem (1.5) is persistent in mean.
Making use of Itô's formula to e t lnx, we deduce t 0 e s Y ln(1 + γ 1 (s, u))Ñ(ds, du) are martingales with the quadratic forms By the exponential martingale inequality [31], for any positive constants k, γ, δ, we can get that it follows from the Borel-Cantelli lemma that for almost all ω ∈ Ω, there is k 0 (ω) such that for each k ≥ k 0 (ω), Hence ]ds + 2δe γk lnk. Obviously, for any 0 ≤ s ≤ γk and x > 0, there is a constant A which is independent of k such that Then for 0 ≤ t ≤ γk, k > k 0 (ω), we derive That is That is For t ≥ T , we have Let ε be sufficiently small such thatr 1 −β 1 − ε > 0, then applying Lemma 5.1 to above two inequalities, we get Making use of the arbitrariness of ε we get Similarly, we yield that lim t→∞ ln y(t) t = 0.
Case (C). Similar to the arguments in Case (A) and (B), it is easy to find that: x(t) is extinction, y(t) is persistent in mean, ifȓ 1 <β 1 ,r 2 >β 2 .