KOLMOGOROV-TYPE SYSTEMS WITH REGIME-SWITCHING JUMP DIFFUSION PERTURBATIONS

. Population systems are often subject to various diﬀerent types of environmental noises. This paper considers a class of Kolmogorov-type sys- tems perturbed by three diﬀerent types of noise including Brownian motions, Markovian switching processes, and Poisson jumps, which is described by a regime-switching jump diﬀusion process. This paper examines these three dif-ferent types of noises and determines their eﬀects on the properties of the systems. The properties to be studied include existence and uniqueness of global positive solutions, boundedness of this positive solution, and asymp- totic growth property, and extinction in the senses of the almost sure and the p th moment. Finally, this paper also considers a stochastic Lotka-Volterra system with regime-switching jump diﬀusion processes as a special case.


1.
Introduction. Populations of biological species are often subject to different types of noises that have significant impact on the evolution and biodiversity; see for example, Gard [19][20][21], Lungu and Øksendal [30,31], among others. These noises include the continuous and discrete perturbations as well as random influence of the environments. They are often modeled by Brownian motions, Poisson processes, and Markovian chains. Aiming at understanding the fundamental underpinning of the different noise effects, this paper reveals their influence on the populations dynamics.
Kolmogorov-type systems of differential equations have been used to model the evolution of many biological and ecological systems because they are more general and contain many classes of the well known population models such as the Lotka-Volterra model; see for instance, [4-6, 8, 15, 16, 18, 22, 23, 25, 36, 37, 41, 42, 44, 45, 47-49, 53, 54]. The n-dimensional Kolmogorov-type system for n interacting species is described by the following n-dimensional differential equatioṅ x(t) = diag(x 1 (t), . . . , x n (t))f (x(t)), (1.1) where x = (x 1 , . . . , x n ) , diag(x 1 , . . . , x n ) represents the n × n matrix with diagonal entries x 1 , . . . , x n and elsewhere 0, and f = (f 1 , . . . , f n ) . This system can be rewritten asẋ k (t) x k (t) = f k (x(t)) for k = 1, . . . , n, which shows that f k (x(t)) represents the net growth rate of the kth species on the time t. Owing to various unpredictable factors such as continuous random processes represented by Brownian motions, discrete random process represented by jump processes taking value in a finite or countable set, and environmental changes represented by continuous-time Markov chains. We can therefore replace the net growth rate of the kth species f k (x(t)) by where w(t) is a Brownian motion, r(t) is a continuous-time Markov chain taking value in S, N (t, Ξ) is a Poisson measure counting the events happened up to time t, g k (x(t)) represents the intensity of the noise, h k (y, x(t−), r(t−)) gives the intensity of the jump, and N (1, Ξ) is a counting measure for the set Ξ in a unit time.
We therefore have the following stochastic Kolmogorov-type system with regimeswitching jump diffusion processes dx(t) = diag(x 1 (t), . . . , x n (t)) f (x(t), r(t))dt + g(x(t), r(t))dw(t) + Ξ h(y, x(t−), r(t−))N (dt, dy) , (1.2) where g = (g 1 , . . . , g n ) : R n × S → R n , h = (h 1 , . . . , h n ) : Ξ × R n × S → R n are Borel measurable. Stochastic population dynamics have received more and more attentions in recent years since they seem to be in good agreement with the reality in the nature. By introducing the Brownian noise from the interactions between the species into the classical Lotka-Volterra system, [4,32] revealed that the environmental noise may suppress the potential population explosion and guarantee the global positive solution to stochastic Lotka-Volterra system, and moreover, also shows that the stochastic Lotka-Volterra model produces many desired properties, for example, stochastically ultimate boundedness and the moment boundedness. Under the Brownian noise from the birth rate into the deterministic Lotka-Volterra system, [5,38] revealed that the stochastic Lotka-Volterra system behaves similarly to the corresponding deterministic system. [33] reviewed these two classes of the models and indicates that different structures of environmental noise may have different effects on the population dynamics. By introducing the more general stochastic perturbations including the interactions between species and the net birth rate into the system (1.2) and the corresponding Lotka-Volterra system, [45,47,48] obtained conditions under which the different noise structures have different effects on the asymptotic properties of the population dynamics. These conditions show that if the environmental noise intensity is strongly dependent on the population size (for example, the Brownian noise from the interactions between species), this noise may suppress the population explosion and guarantee the global positive solution and the asymptotic properties of the model will be determined by the noise. When the environmental noise intensity depends weakly on the population size (for example, the net birth rate noise), the stochastic system behaves similarly to the deterministic one and asymptotic properties are also independent of the noise. [49] further showed that although the interactions between the species plays the crucial roles for the existence of the global positive solution of the stochastic Lotka-Voltera systems and some asymptotic properties, the most important factor of the extinction of the species is still from the noise of the birth rate.
It is obvious that the Brownian noise can model the perturbation from continuous random processes or approximately model a sufficiently large number of iterates of independent random factors according the central limit theorem. The population may suffer sudden environmental shocks, e.g., earthquakes, hurricanes, epidemics, which cannot be modeled by the Brownian noise. To model these phenomena, [6,7] introduced the jump process into the underlying population dynamics and considered the effects of the Lévy processes on the system and examined the existence of the global solution and the asymptotic pathwise estimation and uniform boundedness. [27] was also concerned with stochastic Lotka-Volterra models perturbed by Lévy noise and examined stochastic permanence and extinction.
Besides the continuous and discrete stochastic factors mentioned above, the population growth is also different in the different environments. For instance, the growth rates of some species in the rainy season will be much different from those in the dry season. Moreover, the carrying capacities often vary according to the changes in nutrition and/or food resources. The Markov chain offers a suitable tool to describe these environmental changes. Actually, about forty years ago, the Markov chain was used to model the environmental changes in the population dynamics, for example, by using probabilistic technique, [40] examined the stationary distribution of the population system in a Markovian switching environment including two states. [42] considered the evolution of a system composed of two predator-prey deterministic systems described by Lotka-Volterra equations in a Markovian switching environment. [28] discussed a (2-dimensional) predator-prey Lotka-Volterra model in a switching diffusion environment, and then this two dimensional model was generalized as a n-dimensional Lotka-Volterra system in [29]. [53,54] were concerned with competitive Lotka-Volterra model in a switching diffusion environment and examined the certain long-run-average limits of the solution from several angles. This paper focuses on system (1.2) and examines how the three different types of noises to determine its properties including (i) existence and uniqueness of the global positive solution; (ii) boundedness including the pth moment boundedness, stochastic ultimate boundedness, and the moment average boundedness in time; (iii) asymptotic pathwise growth estimation; (iv) extinction including the almost sure extinction and the pth moment extinction.
Finally, we also apply these results to examine the Lotka-Volterra system with the regime-switching jump diffusion processes as a special case.
The rest of the paper is organized as follows. Section 2 provides the notation needed and preliminary results. Section 3 examines existence and uniqueness of the global positive solution, which shows that the Brownian noise plays a crucial role to yield the global solution. Section 4 investigates the boundedness of this global positive solution, including the pth moment boundedness, stochastic ultimate boundedness, and the moment average boundedness in time. Section 5 shows that system (1.2) grows at most polynomially. In Section 6, we analyze two classes of extinctions including the almost sure extinction and the pth moment extinction and examine the effects of these three types of noises on the extinction. Finally, we apply these results to examine the Lotka-Volterra system under regime-switching jump diffusion process formulation as a special case. 2. Preliminaries and notation. Throughout this paper, unless otherwise specified, we use the following notation. Let | · | be the Euclidean norm in R n and R + = [0, ∞), R • + = (0, ∞). If A is a vector or matrix, its transpose is denoted by A . If A is a matrix, we use the norm defined by |A| = trace(A A). The a ∨ b denotes max{a, b} and a ∧ b denotes min{a, b}. Denote by C 2 (R n ; R + ) the family of all nonnegative functions V (x) on R n that have continuous partial derivatives w.r.t. x up to the second order.
Let (Ω, F , {F t }, P) be a 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(t) be a scalar Brownian motion, Ξ be a subset of R n \ {0} that is the range space of the impulsive jumps, and N (·, ·) defined on R + × R n \ {0} is an F t -adapted Poisson random measure with compensator N (dt, dy) = N (dt, dy) − ν(dy)dt, where ν is a Lévy measure on Ξ with ν(Ξ) = λ. Denote by Ξ 0 the family of all bounded positive functions h(·) on Ξ. Let r(t) be a Markov chain taking values in S = {1, 2, . . . , m} on the space (Ω, F , P). The corresponding generator is denoted by Γ = (γ ij ) m×m , so that for sufficiently small δ > 0, Here γ ij is the transition rate from i to j and γ ij > 0 if i = j while γ ii = − j =i γ ij . As a standing hypothesis, this paper assumes that the Markov chain r(t) is irreducible, that is, the system of equations πΓ = 0 i∈S π i = 1 has a unique solution π = (π 1 , π 2 , . . . , π m ) satisfying π i > 0 for each i = 1, . . . , m.
Throughout the paper, we assume that w(t), N (t, ·), and r(t) are independent. It is well known (see e.g., [1]) that almost every sample path of the Markov chain r(·) is a right continuous step function with a finite number of sample jumps in any finite subinterval of R + . Consequently, there is a sequence of stopping times 0 = τ 0 < τ 1 < · · · < τ k → ∞ such that Let P be the set of all probability measures on the state space S. Then the Markov chain r(t) has the rate function (see [9,[11][12][13][14]) given by where p = (p 1 , . . . , p m ) ∈ P is a probability vector and u = (u 1 , . . . , u m ) ∈ R m . It is known that I(p) ≥ 0 is lower semicontinuous and I(p) = 0 if and only if p = π. Applying Theorem A.2 in Appendix A into Markov chain r(t) gives Λ(a) := lim where a = (a 1 , . . . , a m ) ∈ R m . When extinction in the sense of the pth moment is examined, Λ(·) plays an important role. It is easily seen that To examine the properties of system (1.2), we need the following assumptions: Assumption 2.1. Assume that f, g, h satisfy the local Lipschitz condition. That is, for each integer θ ≥ 1, there exists a constant K θ > 0, such that for all i ∈ S and x,x ∈ R n with |x| ∨ |x| ≤ θ.

Remark 1.
When h k = −1, if the jump takes place at t 0 , x k (t) ≡ 0 for all t ≥ t 0 , so this is a trivial case.
Remark 2. In this paper, (H3) may be replaced by the following more general condition: (H3a) h k (y, x, i) = −1 and there exist θ > 0, ς k (·, i), ϕ(·, i), ζ k (·, i), ς k (·, i) ∈ Ξ 0 such that In this paper, we often use the function for any p > 0 and x = (x 1 , . . . , x n ) ∈ R n + . We also need the following inequality about V p (x). Let us write it as the following lemma.
Lemma 2.1. For any x = (x 1 , . . . , x n ) ∈ R n + and p > 0, We also write the following Young inequality as a lemma.
Lemma 2.2. For any α, β ≥ 0 satisfying α + β > 0, for any x, y ≥ 0, 3. Global positive solutions. To examine existence and uniqueness of the global positive solutions, let us first present the following definition including the local solution and the global solution as well as the explosion time: It is called a local strong solution of Eq. (1.2) with initial value x(0) ∈ R n if for any t ≥ 0 and some positive integer k 0 , is called a maximal local strong solution on [0, ρ e ) and ρ e is called the explosion time. When ρ e = ∞, it is called a global solution. A maximal local strong solution x(t), 0 ≤ t < ρ e is said to be unique if for any other maximal local strong solutionx(t), 0 ≤ t <ρ e , we have ρ e =ρ e and x(t) =x(t) for 0 ≤ t < ρ e a.s.
. , x n )g(x, i) and H(y, x, i) = diag(x 1 , . . . , x n )h(y, x, i). By Assumption 2.1, it is easy to observe that F (x, i), G(x, i) and H(y, x, i) also satisfy the local Lipschitz condition. The definitions of F, G and H implies that F (0, i) = 0, G(0, i) = 0 and H(y, 0, i) = 0. This, together with the local Lipschitz conditions of F, G, H, implies the local linear growth condition, namely, for all i ∈ S and x ∈ R satisfying |x| ≤ θ, where K θ is a constant which may be different from K θ . By [34,Chapter 3] and [2, Chapter 6], the local Lipschitz condition, together with the local linear growth condition yields a local solution, which can be expressed as follows: Under Assumption 2.1, there exists a unique, càdlàg, and adapted, local solution x(t) on 0 ≤ t < ρ e for system (1.2), where ρ e is the explosion time.
By slightly modifying the proof of [3, Lemma 3.3], Assumption 2.1 and condition (3.1) can also guarantee that this local solution can never reach the origin if the initial value is nonzero, namely, the following result holds. By [34,Chapter 3] and [2,Chapter 6], (x(t), r(t)) is a two-component Markov process. According to the regularity of Markov processes, we can establish the existence of global solutions. Let us give the definition of regularity of Markov processes as follows: This definition implies that the regular Markov process has no finite explosion time, which is equivalent to the existence of the global solution. Let k 0 be a sufficiently large positive number such that |x 0 | < k 0 . For each integer k > k 0 , define the stopping time with the traditional setting inf{∅} = ∞, where ∅ denotes the empty set. Clearly, To examine the existence and uniqueness of global solutions for the Kolmogorovtype system (1.2), we need more notation. For any V ∈ C 2 (R n ; R + ), let us define the operator G from R n × S to R by for each i ∈ S, where L is the operator from R n × S to R for a switching diffusion process given by (also see [51]) However, we are using a function V that is independent of i, as a result, this term is 0 since j γ ij = 0. In the following, for the purpose of simplicity, we write Then for any V ∈ C 2 (R n ; R + ), applying the generalized Itô formula to system (1.2) with regime-switching jump diffusion gives (also see [50]) where V x (x)G(x, i) is the inner product between the vectors V x (x) and G(x, i). Again, normally for switching diffusions, there will be another martingale term attributes to the Markov chain (converted to a Poisson process). However, since we choose V to be independent of r(t), this term is 0. To proceed, let us present the sufficient conditions for the regularity of (x(t), r(t)) defined by system (1.2), which can be found in [50] and [51]. Lemma 3.5. Assume that f, g, h satisfy Assumption 2.1 and there is a nonnegative Applying this lemma may establish the existence and uniqueness of the global positive solutions for the Kolmogorov-type system (1.2) with the regime-switching jump diffusion process.
By the generalized Itô formula, for any t ∈ [0, τ e ), x k (t) can be expressed as Noting that the initial value x k (0) > 0, (3.7) implies that x k (t) ≥ 0 for all t ∈ [0, τ e ) for all k = 1, . . . , n. This, together with Lemma 3.3, yields x(t) > 0 for all t ∈ [0, τ e ). Now let us show that this positive local solution is actually global by using Lemma 3.5, that is, (x(t), r(t)) is regular. For any p ∈ (0, 1), letting us apply the operator (3.3) to the function V p (x) defined by (2.6) gives This, together with (H1) and (H3), yields Substituting this inequality into (3.8) yields Note that p ∈ (0, 1) and α < 2β. There must be an upper bound for the polynomial function K k (x, i) as follows: and where η max = max {i∈S} {η p (i)}. By Lemma 2.1, noting that p ∈ (0, 1), Thus by virtue of Lemma 3.5, the two component process (x(t), r(t)) is regular. This also shows that x(t) is a global positive solution.
Remark 3. According to this proof, the bounded property of the polynomial function K k,p (x, i) plays a crucial role to guarantee the regularity of (x(t), r(t)), in which condition 2β > α guarantees this bounded property. This implies that this regularity is induced by the Brownian motion w(t). In our previous papers, for example, [46], it is shown that the Brownian motion may suppress the potential explosion. In [52], we also shows that although the Markov switching may change the explosion time, it cannot change the trend of the explosion for the explosion system. In (3.9), although Ξ [ς p k (y, i) − 1]ν(dy) < ∞ can change the value of η k,p (i), it has no effect on the bounded property of K k,p (x, i). This implies that similar to the Markov chain, the jump process under condition (H3) may change the explosion time, but it cannot change the trend of the explosion.
which can be added into the polynomial function K k (x, i) and have no effect on the bounded property for any θ > 0 since we can choose sufficiently small p > 0. In the following, we can deal with (H3a) similar to this technique.
4. Boundedness. Theorems 3.6 shows that the solutions of Eq. (1.2) will remain in the positive cone R n + • . This property enables us to further examine how the solution varies in this cone in more detail. Comparing with the existence and uniqueness of the global positive solutions, boundedness is more interesting from the biological point of view. In the following, we will discuss the pth moment boundedness, stochastic ultimate boundedness and the moment average boundedness in time. Proof. By Theorem 3.6, the solution x(t) of (1.2) remains in R n + • almost surely for all t ≥ 0. Applying the generalized Itô formula to e t V p (x(t)) gives where G V p is given and estimated by (3.8). According to (3.9), we therefore have a j (i) Note that p ∈ (0, 1), 2β > α and Substituting this estimation into (4.2) gives which implies that Choosing K p = n k=1 max {i∈S} {K k,p (i)} gives the desired result. By the pth moment boundedness, the stochastically ultimate boundedness will follow directly. We describe it as a theorem below. This proof is from the Chebyshev inequality. We omit it (please see [45]).  Proof. Applying the Itô formula to V p (x(t)) gives where G V is given and estimated by (refsolueq1). According to (3.9), we therefore have

FUKE WU, GEORGE YIN AND ZHUO JIN
where By the similar techniques to Theorems 3.6 and 4.1, there exists a constant K k,p (i) such that max {x>0} { K k,p (x, i)} ≤ K k,p (i). This, together with (4.6) gives which implies that )∨0 ] gives the desired result.
Remark 5. In Theorems 4.1 and 4.3, the boundedness properties of the functions K k,p (x, i) and K k,p (x, i) play crucial roles, in which, similar to Theorem 3.6, 2β > α determines their bounded properties. This shows that the Brownian motion plays the most important role in these bounded properties. According to K p and K p , the Markov chain r(t) and the jump process N (t, ·) can have effect on the boundedness, but they cannot determine the bounded properties of K k,p (x, i) and K k,p (x, i).
This theorem shows that the trajectory of system (1.2) grows at most polynomially. By virtue of this result, for any > 0, there is a positive random time T > 0 such that for any t ≥ T , |x(t)| ≤ t 2+ with probability 1. In other words, with probability one, the solution will not grow faster than t 2+ . Remark 6. According to the proof of this theorem, the bounded property of the polynomial function Γ(x, i) plays an important role. It can be observed that condition 2β > α determine this bounded property. This shows that the Brownian motion mainly contributes to this asymptotic pathwise estimation.
6. Extinction. This section reveals the role of the stochastic factors that lead to extinction of the population. We discuss two classes of extinction including the pth moment extinction and the almost sure extinction. Let us give the definition of these two classes of extinction. where Then the regular solution of system (1.2) satisfies lim sup Thus, if γ < 0, system (1.2) becomes extinct almost surely.

Remark 7.
In this proof, the boundedness property of the polynomial Γ 0 (x, i, p, ε) plays an important role. It can be observed that the condition 2β > α determines this bounded property, so high order term of the diffusion term g contributes to this almost sure extinction. Moreover, γ < 0 determines the almost sure extinction of the population. We can easily observe that −ξ 2 min (i) contributes to γ < 0. This shows that the sufficiently large ξ will lead to the almost sure extinction of the population. This implies the constant term of the diffusion term g can play an important role for the almost sure extinction of the population. Note that if ς max (y, i) ∈ (0, 1), Ξ log ς max (y, i)ν(dy) contributes to γ < 0. This shows that the jump process can also lead to almost sure extinction of the population. The definition of γ shows that when some γ(i) = Γ(i) − ζ 2 min (i)/2 + Ξ log ς max (y, i)ν(dy) > 0, but some γ(i) < 0, the whole population can become extinct almost surely. This shows that the Markov chain r(t) also contributes the almost sure extinction of the population. 6.2. pth-moment extinction. Let us now explore the effects of the Brownian motion w(t), the jump process N (t, ·), and the Markovian chain r(t) on the pth moment extinction. where η p = (η p (1), . . . , η p (m)) is defined by (3.12). If Λ(η p ) < 0, system (1.2) is pth-moment extinct.
Remark 8. Since the definition of η p is from the boundedness of the polynomial function K k,p (x, i), in which the condition 2β > α plays a crucial role, so the high order term of the diffusion term contributes to the pth moment extinction of the population. By the definition of η p and Λ(η p ), if a factor can contribute to η p < 0 contributing to Λ(η p ) < 0, it will induce the pth moment extinction of the population. According to this standard, for p ∈ (0, 1), the constant term of the diffusion term contributes to the pth moment extinction since η p (i) contains −p(1 − p)ξ 2 k (i)/2. According to (3.11), if ς k (y, i) ∈ (0, 1), the jump process also contributes to η p < 0 contributing to Λ(η p ) < 0, so it will also induce the pth moment extinction of the population.
Because Λ(η p ) is a functional, we cannot examine the contribution of r(t) for Λ(η p ) < 0 directly. To examine the role of the Markov chain in the pth moment extinction of the population, let us introduce the following lemma. Lemma 6.4. Let η p = (η p (1), . . . , η p (m)) and Λ(η p ) be defined by (3.12) and (2.4), respectively. Then whereη = i∈S π i η(i) and Proof. By the definition of Λ(η p ) in (2.4), for any p ∈ (0, 1), there exists ap(p) ∈ P dependent on p such that Λ(η p ) = i∈S η p (i)p i (p) − I(p(p)). (6.10) Remark 9. This theorem shows that for sufficiently small p > 0, the Brownian motion and the jump process contribute to the pth moment extinction similar to the case with Theorem 6.3, but it can show that the Markov chain r(t) plays the same role as the almost sure extinction, namely, the Markov chain can contribute to the pth moment extinction.
According to the stability theory (for example, [35]), under the linear growth condition, the pth moment stability implies the almost sure stability. However, the Kolmogorov-type system (1.2) doesn't satisfy the linear growth condition, we cannot employ this existing result. In this paper, since γ andη have the similar expressions, we can look for conditions which leads to these two classes of extinctions. For example, let us define Clearly, µ ≥ γ and µ ≥η, so if µ < 0, system (1.2) becomes extinct almost surely and pth moment.
Applying the results of Kolmogorov-type system (1.2) to the Lotka-Volterra system (7.1) gives the following theorem.
In this theorem, it is obvious thatη ≤ γ, so γ < 0 implies that Lotka-Volterra system (7.1) becomes extinct almost surely and pth moment for the sufficiently small p > 0.
Appendix A. Appendix. We recall some large deviations results, which can be founded in [9] and [10]. Let (X, B, · ) be a polish space, and p(t, x, dy) be the transition probability of X-valued a continuous time Markov process X(t) with X(0) = x. Denote by T t the strong continuous Markovian semigroup with respect to X(t) such that T t : C(X) → C(X) (Feller semigroup). Let L be the infinitesimal generator of the semigroup T t having domain D ⊂ C(X). Then D is dense in C(X).
Let P be the space of all probability measures on the state space X. For any p ∈ P, define the rate function by Then the rate function is non-negative and holds the following property (see [14]): Theorem A.1. I(p) = 0 if and only if p is the invariant measure of the transition probability function p(t, x, dy).
Let Ω x be the space of X-valued càdlàg X(t), 0 ≤ t < ∞, with X(0) = x. For each t > 0 ω ∈ Ω x and Borel set A ⊂ X, define the occupation time measure by which is the proportion of time up to time t that a particular sample path ω = X(·) spends in the set A. Note that for each t > 0 and each ω, L t (ω, ·) is a probability measure.
Remark A.1. For the Markov chain r(t) defined in this paper, we have the following assertions (see [10]) L t (ω, i) = 1 t