Asymptotic properties of a stochastic n-species Gilpin–Ayala competitive model with Lévy jumps and Markovian switching

Accepted Manuscript Asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model with Lévy jumps and Markovian switching Qun Liu PII: DOI: Reference:

S1007-5704(15)00016-7 http://dx.doi.org/10.1016/j.cnsns.2015.01.007 CNSNS 3452

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Communications in Nonlinear Science and Numerical Simulation

Received Date: Revised Date: Accepted Date:

9 November 2014 15 January 2015 16 January 2015

Please cite this article as: Liu, Q., Asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model with Lévy jumps and Markovian switching, Communications in Nonlinear Science and Numerical Simulation (2015), doi: http://dx.doi.org/10.1016/j.cnsns.2015.01.007

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Asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model with L´ evy jumps and Markovian switching Qun Liu1 School of Mathematics and Information Science, Guangxi Universities Key Lab of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, P.R. China

Abstract In this paper, a stochastic n-species Gilpin-Ayala competitive model with L´evy jumps and Markovian switching is proposed and studied. Some asymptotic properties are investigated and sufficient conditions for extinction, nonpersistence in the mean and weak persistence are established. The threshold between extinction and weak persistence is obtained. The results illustrate that the asymptotic properties of the considered system have close relationships with L´evy jumps and the stationary distribution of the Markovian chain. Moreover, some simulation figures are presented to confirm our main results. Keywords: Stochastic Gilpin-Ayala competitive model; Persistence; L´evy jumps; Markovian chain; Extinction.

1 Corresponding author. 15878046432.

E-mail address:

[email protected];

Tel.:

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Preprint submitted to Communications in Nonlinear Science and Numerical SimulationJanuary 21, 2015

1. Introduction The deterministic Gilpin-Ayala competitive model with n interacting species can be expressed as follows n h i X dxi (t) λ = xi (t) ri − aij xj ij (t) , dt j=1

1 ≤ i ≤ n,

(1)

where xi (t) is the population size of the ith species at time t, ri > 0 denotes the intrinsic growth rate of the ith species, aij > 0 stands for the effect of interspecific (for i 6= j) interaction, λij ≥ 0 represents the nonlinear measure of interspecific interference. However, in the real world, population systems are always affected by stochastic environmental noise, which is an important component in an ecosystem (see e.g. Gard [1, 2]). Introducing the environmental noise into system (1) and imposing the impacts of the environmental perturbations on the parameters ri and aij , 1 ≤ i, j ≤ n, system (1) will become the following stochastic Itˆo’s equation n n n h  i X X X λij λ dxi (t) = xi (t) ri − aij xj (t) dt+ σ1j dB1j (t)+ σ2j xj ij (t)dB2j (t) , j=1

j=1

j=1

(2) 2 where σpj (p = 1, 2; j = 1, . . . , n) represents the intensity of the white noise, B(t) = (Bpj (t))T2×n is a given Brownian motion defined on a complete probability space (Ω, F, P) with a filtration {Ft }t≥0 satisfying the usual conditions. Recently, stochastic Gilpin-Ayala competition system has been widely investigated, here we only mention a few [3, 4, 5, 6, 7]. On the other hand, population systems may be subjected to sudden and abrupt environmental perturbations, such as earthquakes, planting, epidemics, harvesting, tsunami and so on (see e.g. [8, 9, 10, 11, 12]). These events are so severe and strong that they can change the population size greatly in a short time. So these phenomena can not be described accurately by Brownian motion. Introducing L´evy jumps into the underlying population systems may be a good method to describe these phenomena (see e.g. [8, 9, 10, 11, 12, 13, 14, 15, 16]). There has been an increasing interest literature concerned with Stochastic Differential Equations (SDEs) with jumps. Here we only mention Bao et al. [8, 9], Liu and Wang [10, 11], Liu and Bai [12], Wu et al. [13], Liu and Liang [14], Liu and Chen [15], Liu et al. [16], 2

Applebaum [17, 18, 19], Situ [20] and Kunita [21]. Taking the influences of L´evy jumps into account, then system (2) will become n n h  X X λ dxi (t) = xi (t− ) ri − aij xj ij (t− ) dt + σ1j dB1j (t) j=1

+

n X

λ

σ2j xj ij (t− )dB2j (t) +

Z

Y

j=1

j=1

i e (dt, dy) , γi (y)N

(3)

where xi (t− ) denotes the left limit of xi (t), 1 ≤ i ≤ n. N represents a Poisson counting measure on (0, ∞) × Y with characteristic measure ν on a measurable subset Y of (0, ∞) satisfying ν(Y) < ∞, it is assumed that ν is e (dt, dy) = N (dt, dy) − ν(dy)dt, γi : Y × Ω → R a L´evy measure such that N is bounded and continuous with respect to ν and is B(Y) × Ft -measurable. Moreover, in an ecosystem there exist several types of environmental noise. Besides the white noise, in this paper, we consider a classical colored noise, i.e., telegraph noise. The telegraph noise can be illustrated as a switching between two or more regimes of environment. Frequently, the switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed. Therefore, we can replace the stochastic factors in system (3) by a continuous-time Markovian chain ζ(t), t ≥ 0 (see e.g. [22, 23]). Inspired by the above discussions, system (3) has Markovian switching of the following form n n h  X X λij − − dxi (t) = xi (t ) ri (ζ(t)) − aij (ζ(t))xj (t ) dt + σ1j (ζ(t))dB1j (t) j=1

+

n X

λ

σ2j (ζ(t))xj ij (t− )dB2j (t) +

Z

Y

j=1

j=1

i e (dt, dy) . γi (ζ(t), y)N

(4)

To make system (4) more realistic, we shall consider the following stochastic Gilpin-Ayala competitive model with L´evy jumps and Markovian switching n n h  X X λij − − dxi (t) = xi (t ) ri (ζ(t)) − aij (ζ(t))xj (t ) dt + σ1j (ζ(t))dB1j (t) j=1

+

n X

θ

σ2j (ζ(t))xj ij (t− )dB2j (t) +

Z

Y

j=1

3

j=1

i e (dt, dy) , γi (ζ(t), y)N

(5)

where θij ≥ 0, 1 ≤ i, j ≤ n. System (5) will switch from one mode to another according to the law of the Markovian chain. If the initial state ζ(0) = l ∈ S, then system (5) satisfies n n h  X X λ dxi (t) = xi (t− ) ri (l) − aij (l)xj ij (t− ) dt + σ1j (l)dB1j (t) j=1

+

n X

θ

σ2j (l)xj ij (t− )dB2j (t) +

Z

Y

j=1

j=1

i e (dt, dy) γi (l, y)N

till time ς when the Markovian chain switches to ζ(ς) = p ∈ S from ζ(0), then the equation satisfies n n  h X X λij − σ1j (p)dB1j (t) aij (p)xj (t ) dt + dxi (t) = xi (t ) ri (p) − −

j=1

+

n X

θ

σ2j (p)xj ij (t− )dB2j (t) +

Z

Y

j=1

j=1

e (dt, dy) γi (p, y)N

i

until the next switching. The system will continue to switch as long as the Markovian chain switches. Some recent results about the stochastic Gilpin-Ayala system have appeared (see e.g. [5, 14]). Liu and Liu [5] considered the delay Gilpin-Ayala competition system under regime switching and they investigated the boundedness and asymptotic moment estimation of the solution. In [14], Liu and Liang studied a stochastic non-autonomous Gilpin-Ayala system driven by L´evy noise, they studied the persistence and extinction of the system. Nevertheless, in [5], the authors didn’t take the L´evy jumps into account, while in [14], the authors didn’t consider the effects of Markovian switching. This suggests that the proposed system is more realistic and meaningful. It is well known that persistence and extinction are two important topics in mathematical ecology due to their theoretical and practical significance. From the viewpoint of applications, investigating persistence and extinction is a most interesting topic in population dynamics. Moreover, the critical value of persistence and extinction is also significant in practice. Thus, in this paper, we aim to study the impacts of the Markovian switching and L´evy jumps on the persistence and extinction of Eq. (5). The paper is organized as follows. In Section 2, we prove that there exists a unique global 4

positive solution to system (5) for any given positive initial value. In Section 3, we establish sufficient criteria for extinction and persistence of system (5). Moreover, the threshold between extinction and weak persistence is also obtained. In Section 4, some numerical simulations are introduced to illustrate our main results. Finally, some conclusions and discussions are given to close this paper. 2. Global positive solutions Throughout this paper, we suppose that mink∈S aii (k) > 0 and 1 + γi (k, y) > 0. When γi (k, y) > 0, the perturbation denotes increasing of the species (e.g. planting), while −1 < γi (k, y) < 0 represents decreasing (e.g. harvesting and epidemics), k ∈ S, y ∈ Y. Let ζ(t), t ≥ 0 be a right-continuous Markovian chain taking values in a finite state space S = {1, . . . , m} with generator Q = (qpl )m×m given by  qpl ∆t + o(∆t), if l 6= p; P = {ζ(t + ∆t) = l|ζ(t) = p} = 1 + qpp ∆t + o(∆t), if l = p, where Pm ∆t > 0, qpl ≥ 0 is the transition rate from p to l if p 6= l while l=1 qpl = 0. Throughout this paper, we suppose that B, ζ and N are independent. Furthermore, as the standard hypothesis, we assume that ζ(t) has a unique stationary distribution π = (π1 , . . . , πm ) which P can be obtained by solving the following linear equation πQ = 0 subject to m i=1 πi = 1 and πi > 0 for ∀i ∈ S. In the sequel, for simplicity, let me define the following symbols: Z t f (s)ds, f ∗ = lim sup f (t), f∗ = lim inf f (t), fˆ = min f (k), hf (t)i = t−1 t→∞

t→∞

0

k∈S

f˘ = max f (k), γˆi (y) = min γi (k, y), γ˘i (y) = max γi (k, y), i = 1, . . . , n. k∈S

k∈S

k∈S

For the jump-diffusion coefficient, we impose the following assumption Assumption 1. There is a constant c > 0 such that Z [ln(1 + γi (k, y))]2 ν(dy) ≤ c, y ∈ Y, k ∈ S, i = 1, . . . , n. Y

When γi (k, y) > 0, the perturbation denotes increasing of the species (e.g. planting), while γi (k, y) < 0 represents decreasing (e.g. harvesting and epidemics), k ∈ S, y ∈ Y. For more detail info, please refer to [24]. 5

Since xi (t) in system (5) represents the population size of the ith species at time t, it should be nonnegative. So for further investigation, we should firstly give some conditions under which system (5) has a unique global positive solution. Theorem 1. Let Assumption 1 hold, then for any given initial value ζ(0) ∈ S and x(0) = x0 ∈ Rn+ , system (5) has a unique global positive solution x(t) ∈ Rn+ on t ≥ 0 almost surely (a.s.), where Rn+ = {x ∈ Rn : xi > 0, 1 ≤ i ≤ n}. Proof. The proof of this theorem is similar to Theorem 2.1 in [14] and so we omit it. We have verified that system (5) has a unique global positive solution under certain conditions. However, from the biology viewpoint, the nonexplosion property and positivity in a population system are not enough. So in the following section, we shall study the asymptotic properties of the positive solutions. 3. The threshold between extinction and persistence At first, we should cite a strong law of large numbers for local martingales. Lemma 1 ([25]). Suppose that M (t), t ≥ 0, is a local martingale with M (0) = 0. Then lim ρM (t) < ∞

⇒ lim

t→∞

where ρM (t) =

Z

t→∞

t 0

M (t) = 0 a.s., t

dhM i(s) , (1 + s)2

t≥0

and hM i(t) := hM, M i(t) is Meyer’s angle bracket process (see e.g. [8]). In the following, we study the long term behaviors of the positive solutions. Firstly, let me present some concepts which will be used later. Definition 1 (see e.g. [26]). Let x(t) be the positive solution of Eq. (5), (a) if limt→∞ x(t) = 0 a.s., then species x(t) is said to be extinctive almost surely (a.s.). Rt (b) if limt→∞ hxλ (t)i = limt→∞ t−1 0 xλ (s)ds = 0 a.s., then species x(t) is said to be non-persistent in the mean a.s. (c) if x∗ = lim supt→∞ x(t) > 0 a.s., then species x(t) is said to be weakly persistent a.s. Theorem 2. Let Assumption 1 hold, then for any given initial value ζ(0) ∈ S 6

and x(0) = x0 ∈ Rn+ , the solution x(t) = (x1 (t), . . . , xn (t))T of system (5) satisfies m ln xi (t) X ≤ πp hi (k) a.s. lim sup t t→∞ p=1 Pm Particularly, if by sysp=1 πp hi (k) < 0, then the species xi (t) modeled Pn 2 tem (5) will become extinctive a.s., where hi (k) = ri (k) − 0.5 j=1 σ1j (k) − R (γ (k, y) − ln(1 + γ (k, y)))ν(dy). i Y i Proof. Making use of the generalized Itˆo’s formula with jumps to ln xi (t) leads to d ln xi (t) =

h

n X

hi (ζ(t)) −

λ aij (ζ(t))xj ij (t− )

j=1

+

n X

σ1j (ζ(t))dB1j (t) +

j=1

+

Z

Y

Then we obtain

n X

n i 1X 2 2θ σ2j (ζ(t))xj ij (t− ) dt − 2 j=1 θ

σ2j (ζ(t))xj ij (t− )dB2j (t)

j=1

e (dt, dy). ln(1 + γi (ζ(t), y))N

Z th n X λ ln xi (t) = ln xi (0) + hi (ζ(s)) − aij (ζ(s))xj ij (s− ) 0

−

1 2

n X j=1

j=1

i 2θ 2 σ2j (ζ(s))xj ij (s− ) ds + M1 (t) + M2 (t) + M3 (t),

(6)

where M1 (t) =

Z tX n

σ1j (ζ(s))dB1j (s), M2 (t) =

0 j=1

Z tX n

θ

σ2j (ζ(s))xj ij (s− )dB2j (s),

0 j=1

M3 (t) =

Z tZ 0

Y

e (ds, dy). ln(1 + γi (ζ(s), y))N

The quadratic variations of M1 (t), M2 (t) and M3 (t) are hM1 i(t) =

Z tX n 0 j=1

2 (ζ(s))ds σ1j

≤

n X j=1

7

(˘ σ1j )2 t,

hM2 i(t) =

Z tX n 0 j=1

and hM3 i(t) =

Z tZ 0

2θ

2 σ2j (ζ(s))xj ij (s− )ds

[ln(1 + γi (ζ(s), y))]2 ϑ(dy)ds ≤ ct a.s.

Y

It then follows from Lemma 1 that lim

t→∞

Mi (t) = 0 a.s., t

i = 1, 3.

(7)

In view of the exponential martingale inequality (see e.g. [27]), for any positive numbers T , α and β, one can obtain that i o n h α (8) P sup M2 (t) − hM2 i(t) > β ≤ e−αβ . 2 0≤t≤T Choose T = n, α = 1 and β = 2 ln n, then we derive n h i o 1 1 P sup M2 (t) − hM2 i(t) > 2 ln n ≤ 2 . 2 n 0≤t≤n Using the Borel-Cantelli’s Lemma (see e.g. [27]), we have that for almost all ω ∈ Ω, there exists a random integer n0 = n0 (ω) such that for n ≥ n0 , i h 1 sup M2 (t) − hM2 i(t) ≤ 2 ln n a.s. 2 0≤t≤n That is to say, M2 (t) ≤ 2 ln n +

1 2

Z tX n 0 j=1

2θ

2 σ2j (ζ(s))xj ij (s− )ds

for all 0 ≤ t ≤ n, n ≥ n0 a.s. When the above inequality and (7) are used in (6), we can derive Z t Z tX n λ ln xi (t) ≤ ln xi (0) + hi (ζ(s))ds − aij (ζ(s))xj ij (s− )ds + 2 ln n 0

0 j=1

+M1 (t) + M3 (t) Z t hi (ζ(s))ds + 2 ln n + M1 (t) + M3 (t) ≤ ln xi (0) + 0

8

(9)

for all 0 ≤ t ≤ n, n ≥ n0 a.s. Moreover, for n − 1 ≤ t ≤ n, n ≥ n0 , one can see that Z 2 ln n M1 (t) M3 (t) ln xi (0) 1 t ln xi (t) hi (ζ(s))ds + ≤ + + + . t t t 0 n−1 t t Taking the superior limit and applying (7) and the ergodic property of the Markovian chain, one can easily obtain the desired assertion. Thus the proof of Theorem 2 is completed. Remark 1. Obviously, x(t) ≡ 0 is the trivial Pm solution of Eq. (5). By virtue of Theorem 2, one can easily see that if p=1 πp hi (k) < 0, then the trivial solution of Eq. (5) is almost surely exponentially stable, it shows that the species xi (t) will go to extinction exponentially with probability one. P Theorem 3. Let Assumption 1 hold. If m π p=1 p hi (k) = 0, then the species xi (t) represented by Eq. (5) is non-persistent a.s. R in the mean P −1 t Proof. Due to the fact that limt→∞ t h (ζ(s))ds = m p=1 πp hi (k) and 0 i (7), for any given  > 0, there is a positive constant T1 such that t−1

Z

t

hi (ζ(s))ds ≤

0

m X

πp hi (k) + /2 = /2,

t ≥ T1 .

p=1

Substituting the above inequality into (6) gives Z t Z t ln xi (t) ≤ ln xi (0) + hi (ζ(s))ds − aii (ζ(s))xλi ii (s− )ds + M1 (t) 0

0

+M3 (t) + 2 ln n Z t xλi ii (s− )ds + M1 (t) + M3 (t) + 2 ln n ≤ t/2 − a ˆii 0

for all T1 ≤ t ≤ n, n ≥ n0 a.s. Note that there is a T > T1 such that for all T ≤ n − 1 ≤ t ≤ n and n ≥ n0 , one can see that M1 (t)/t ≤ /8, M3 (t)/t ≤ /8, (ln n)/t ≤ /8. In other words, we have verified Z t xλi ii (s− )ds ln xi (t) ≤ ln xi (0) + t − a ˆii 0

for sufficiently large t > T . Let gi (t) =

Rt 0

xλi ii (s− )ds, then one gets

ln(dgi /dt) ≤ λii t − λii a ˆii gi (t) + ln xλi ii (0), t > T, 9

which shows that eλii aˆii gi (t) (dgi /dt) ≤ xλi ii (0)eλii t ,

t > T.

Integrating the above inequality from T to t results in λii a ˆii gi (t) a ˆ−1 − eλii aˆii gi (T ) ] ≤ xλi ii (0)−1 [eλii t − eλii T ]. ii [e

Rewriting the above inequality leads to eλii aˆii gi (t) ≤ eλii aˆii gi (T ) + xλi ii (0)ˆ aii −1 eλii t − xλi ii (0)ˆ aii −1 eλii T . Taking the logarithm on both sides yields λii a ˆii gi (T ) gi (t) ≤ λ−1 ˆ−1 + xλi ii (0)ˆ aii −1 eλii t − xλi ii (0)ˆ aii −1 eλii T }. ii a ii ln{e

Thus, it is shown that n Z t o∗ t−1 xλi ii (s− )ds 0 n o∗ λii λii −1 −1 λii a ˆii gi (T ) −1 λii t −1 λii T ≤ λ−1 a ˆ . t ln[e + x (0)ˆ a  e − x (0)ˆ a  e ] ii ii ii ii i i Making use of the L’Hospital’s rule results in −1 aii −1 eλii t ]}∗ = ˆ−1 ln[xλi ii (0)ˆ hxλi ii i∗ ≤ λ−1 ii a ii {t

 a.s. a ˆii

By the arbitrariness of , we can get hxλi ii i∗ ≤ 0, which is the required statement. Thus the proof of Theorem 3 P is completed. Theorem 4. Let Assumption 1 hold. If m p=1 πp hi (k) > 0, then the species xi (t) denoted by Eq. (5) will be weakly persistent a.s. Proof. At first, we shall prove that [t−1 ln xi (t)]∗ ≤ 0 a.s.

(10)

Applying the generalized Itˆo’s formula with jumps to et ln xi leads to Z t h n X λ t e ln xi (t) − ln xi (0) = es ln xi (s) + hi (ζ(s)) − aij (ζ(s))xj ij (s− ) 0

j=1

1 − 2 +

n X

i

2θ 2 σ2j (ζ(s))xj ij (s− )

j=1 n X

N2j (t) + N3 (t),

j=1

10

ds +

n X

N1j (t)

j=1

(11)

Rt Rt θ where N1j (t) = 0 es σ1j (ζ(s))dB1j (s), N2j (t) = 0 es σ2j (ζ(s))xj ij (s− )dB2j (s) Rt R e (ds, dy) are martingales with the and N3 (t) = 0 es Y ln(1 + γi (ζ(s), y))N quadratic forms Z t Z t 2θ 2s 2 2 hN1j i(t) = e σ1j (ζ(s))ds, hN2j i(t) = e2s σ2j (ζ(s))xj ij (s− )ds, 0

hN3 i(t) =

0

Z

t

e

2s

0

Z

2

[ln(1 + γi (ζ(s), y))] ν(dy)ds ≤ c

Z

t

e2s ds.

0

Y

In view of the exponential martingale inequality (8), by choosing T = γn, α = e−γn and β = θeγn ln n, one can get that i o n h 1 1 P sup Nlj (t) − e−γn hNlj i(t) > θeγn ln n ≤ θ , l = 1, 2 2 n 0≤t≤γn and P

n

sup 0≤t≤γn

h

i o 1 1 N3 (t) − e−γn hN3 i(t) > θeγn ln n ≤ θ , 2 n

where θ > 1 and γ > 1. By the Borel-Cantelli’s Lemma, one can obtain that for almost all ω ∈ Ω, there exists a random number n0 = n0 (ω) such that for every n ≥ n0 , 1 Nlj (t) ≤ e−γn hNlj i(t) + θeγn ln n, 0 ≤ t ≤ γn, l = 1, 2 2 and

1 N3 (t) ≤ e−γn hN3 i(t) + θeγn ln n, 0 ≤ t ≤ γn. 2 Substituting the above inequalities into (11) yields Z t h n X λ t s e ln xi (t) − ln xi (0) ≤ e ln xi (s) + hi (ζ(s)) − aij (ζ(s))xj ij (s− ) 0

j=1

Z n i 1 1 −γn t 2s X 2 2θij − 2 e σ1j (ζ(s))ds σ (ζ(s))xj (s ) ds + e − 2 j=1 2j 2 0 j=1 Z t Z t n X 1 1 2θ 2 + e−γn e2s e2s ds + 3θeγn ln n σ2j (ζ(s))xj ij (s− )ds + ce−γn 2 2 0 0 j=1 Z t h Z es ln xi (s) + ri (ζ(s)) − (γi (ζ(s), y) − ln(1 + γi (ζ(s), y)))ν(dy) = n X

0

Y

11

−

n X

n

λ aij (ζ(s))xj ij (s− )

j=1 n X

1X 2 − σ (ζ(s))[1 − es−γn ] 2 j=1 1j

i 1 1 2θ 2 σ2j (ζ(s))xj ij (s− )[1 − es−γn ] − c[1 − es−γn ] ds + 3θeγn ln n 2 j=1 2 Z t h Z ≤ es ln xi (s) + ri (ζ(s)) + (|γi (ζ(s), y)| + | ln(1 + γi (ζ(s), y))|)ν(dy) −

0

Y

n

n X

1X 2 λ σ1j (ζ(s))[1 − es−γn ] − aij (ζ(s))xj ij (s− ) − 2 j=1 j=1

n i 1X 2 1 2θ − σ2j (ζ(s))xj ij (s− )[1 − es−γn ] − c[1 − es−γn ] ds + 3θeγn ln n. 2 j=1 2

Since a ˆii > 0, then it is easy to see that for any 0 ≤ s ≤ γn and xi > 0, i = 1, 2, . . . , n, there exists a constant K which is independent of n such that Z ln xi + ri (ζ(s)) + (|γi (ζ(s), y)| + | ln(1 + γi (ζ(s), y))|)ν(dy) Y

− −

n X

n

λ aij (ζ(s))xj ij

j=1 n X

1 2

1X 2 σ (ζ(s))[1 − es−γn ] − 2 j=1 1j

1 2θ 2 (ζ(s))xj ij [1 − es−γn ] − c[1 − es−γn ] ≤ K. σ2j 2 j=1

That is to say, for any 0 ≤ t ≤ γn, we have already obtained et ln xi (t) − ln xi (0) ≤ K(et − 1) + 3θeγn ln n. That is to say, ln xi (t) ≤ e−t ln xi (0) + K(1 − e−t ) + 3θe−t eγn ln n. If γ(n − 1) ≤ t ≤ γn and n ≥ n0 , one has e−t ln xi (0) K(1 − e−t ) 3θe−γ(n−1) eγn ln n ln xi (t) ≤ + + . t t t t Letting t → ∞, then we get the desired assertion (10). 12

P Now suppose that m p=1 πp hi (k) > 0, we shall prove that lim supt→∞ xi (t) > 0 a.s. If this assertion is not true, let E be E = {lim supt→∞ xi (t) = 0}, then P(E) > 0. It follows from (6) that Z t n E DX λ −1 −1 t [ln xi (t) − ln xi (0)] = t aij (ζ(t))xj ij (t− ) hi (ζ(s))ds − 0

−

n 1D X

2

j=1

j=1

E 2θ 2 σ2j (ζ(t))xj ij (t− ) + t−1 M1 (t) + t−1 M2 (t) + t−1 M3 (t).

(12)

λ

On the other hand, for all ω ∈ E, we can obtain limt→∞ xj ij (t, ω) = 0, 1 ≤ i, j ≤ n, then in view of the law of large numbers for local martingales (see e.g. [27]), we can get limt→∞ Mit(t) = 0, i = 1, 2, 3. Substituting the above inequalities into (12) gives [t−1 ln xi (t, ω)]∗ =

m X

πp hi (k) > 0.

p=1

Then P{[t−1 ln xi (t)]∗ > 0} > 0, which contradicts (10). This completes the proof of Theorem 4. Remark 2. From Theorems 2-4, one can easily see that persistence and extinction the species xi (t) represented by system (5) only depend on the Pof m value of p=1 πp hi (k), which means that the long term behaviors of system (5) have close relationships with L´evy jumps P and the stationary distribution of the Markovian switching. Moreover, if m 0, then the species p=1 πp hi (k) < P xi (t) denoted by system (5) will go to extinction a.s.; if m p=1 πp hi (k) = 0, then the species x (t) modeled by system (5) will be non-persistent in the Pm i mean a.s.; if p=1 πp hi (k) > 0, then the species P xi (t) represented by Eq. (5) will be weakly persistent a.s. Obviously, m p=1 πp hi (k) is the threshold between extinction and weak persistence. 4. Numerical simulations In this section, we shall introduce an example and some simulation figures to demonstrate our main results. In the following discussion, we firstly give the stationary distribution π = (π1 , . . . , πm ) of the Markovian chain ζ(t) directly because π can be obtained by solving the following linear equations πQ = 0,

m X i=1

13

πi = 1,

where Q is the generator of the Markovian chain ζ(t). Example 1. Consider the following stochastic 2-species Gilpin-Ayala competitive model with L´evy jumps and Markovian switching h   − 0.4 − 0.3 −  dx (t) = x (t ) r (ζ(t)) − a (ζ(t))x (t ) − a (ζ(t))x (t ) dt  1 1 1 11 12 1 2    0.6 −  + σ11 (ζ(t))dB11 (t) + σ12 (ζ(t))dB12 (t) + σ21 (ζ(t))x1i (t )dB21 (t)    R  − e  + σ22 (ζ(t))x1.2 2 (t )dB22 (t) + Y γ1 (ζ(t), y)N (dt, du) , h  − 0.6 − 1.5 −   dx (t) = x (t ) r (ζ(t)) − a (ζ(t))x (t ) − a (ζ(t))x (t ) dt 2 2 2 21 22 1 2    0.9 −   + σ11 (ζ(t))dB11 (t) + σ12 (ζ(t))dB12 (t) + σ21 (ζ(t))x1i (t )dB21 (t)   R   − e + σ22 (ζ(t))x1.8 2 (t )dB22 (t) + Y γ2 (ζ(t), y)N (dt, du) , (13) where ζ = ζ(t) is a Markovian chain with state space S = {1, 2}. Let r1 (1) = e − 1.96, r1 (2) = e − 1.9, a11 (ζ(t)) = a12 (ζ(t)) ≡ 0.2, 2 2 2 2 σ11 (ζ(t)) = σ12 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, 2 2 γ (u) = e − 1, i = 1, 2. That is to say, h (1) = r (1) − 0.5(σ11 (1) + σ12 (1)) − 1 1 Ri 2 2 RY (γ1 (u) − ln(1 + γ1 (u))) = 0.08, h1 (2) = r1 (2) − 0.5(σ11 (2) + σ12 (2)) − (γ (u) − ln(1 + γ1 (u))) = −0.02. By solving the equations Y 1  π11 + π12 = 1, 0.08π11 − 0.02π12 = 0 yields π11 = 0.2, π12 = 0.8. Now let us choose r2 (1) = e − 1.9, r2 (2) = e − 1.8, a21 (ζ(t)) = a22 (ζ(t)) ≡ 2 2 2 2 0.4, σ11 (ζ(t)) = σ12 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, γi (u) = e − 1, i = 1, 2. By a simple calculation, we can obtain π21 = 0.8, π22 = 0.2. Let us simulate the above example. There exist two kinds of random processes in system (13). One is the Markovian switching, the other is the Brownian motion. As for Brownian motion, we shall use the Milstein method (see e.g. [28]). In Fig. 1, we choose r1 (1) = e − 1.96, r1 (2) = e − 1.9, r2 (1) = e − 1.9, r2 (2) = e − 1.8, a11 (ζ(t)) = a12 (ζ(t)) ≡ 0.2, a21 (ζ(t)) = a22 (ζ(t)) ≡ 0.4, 2 2 2 2 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, γi (u) = σ11 (ζ(t)) = σ12 e − 1, i = 1, 2. The only difference between conditions of Fig. 1(a)-(c) is that the values of π11 and π21 are different. In Fig. 1(a), we choose π11 = 0.18 and π21 = 0.75. In view of Theorem 2, one can obtain that both species x1 and x2 represented by system (13) will be extinctive a.s. In Fig. 1(b), we choose 14

π11 = 0.2, π21 = 0.8. By Theorem 3, one can see that both species x1 and x2 will be non-persistent in the mean a.s. In Fig. 1(c), we choose π11 = 0.4 and π21 = 0.96. It follows from Theorem 4 that both species x1 and x2 will be weakly persistent a.s. 5. Conclusions and discussions This paper is devoted to the study of a stochastic n-species Gilpin-Ayala competitive model with L´evy jumps and Markovian switching. Some asymptotic properties are investigated and sufficient conditions for extinction, nonpersistence in the mean and weak persistence are obtained. The threshold between extinction and weak persistence is also obtained. The results demonstrate that the asymptotic properties of the considered system have close relationships with L´evy jumps and the stationary distribution of the Markovian switching. To the best of our knowledge, the present paper is the first attempt to study the asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model with L´evy jumps. Some interesting topics deserve further investigations, on the one hand, one may propose some more realistic but complex models, such as considering the effects of time delays and impulsive perturbations on system (5). On the other hand, in the present paper, we only consider the function γi (i = 1, . . . , n) depends on t, one may introduce a more general form, that is, it should depend on x(t) (see e.g. [29]). Furthermore, it will be also interesting to investigate the stochastic permanence, global asymptotic stability and other important properties of system (5). We leave these for further investigations and look forward to solving them in future work. Acknowledgments The author wishes to express his gratitude to the editors and the reviewers for the helpful comments. This work is supported by NNSF of China Grant No.11271087, No.61263006 and 2014YJZD02. References [1] T.C. Gard, Persistence in stochastic food web models, Bull. Math. Biol. 46 (1984) 357-370. [2] T.C. Gard, Stability for multispecies population models in random environments, Nonlinear Anal. 10 (1986) 1411-1419. 15

[3] B. Lian, S. Hu, Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J. Math. Anal. Appl. 339 (2008) 419-428. [4] B. Lian, S. Hu, Stochastic delay Gilpin-Ayala competition models, Stoch. Dyn. 6 (2006) 561-576. [5] Y. Liu, Q. Liu, A stochastic delay Gilpin-Ayala competition system under regime switching, Filomat 27 (2013) 955-964. [6] M. Vasilova, M. Jovanovi´c, Dynamics of Gilpin-Ayala competition model with random perturbation, Filomat 24 (2010) 101-113. [7] M. Vasilova, M. Jovanovi´c, Stochastic Gilpin-Ayala competition model with infinite delay, Appl. Math. Comput. 217 (2011) 4944-4959. [8] J. Bao, X. Mao, G. Yin, C. Yuan, Competitive Lotka-Volterra population dynamics with jumps, Nonlinear Anal. 74 (2011) 6601-6616. [9] J. Bao, C. Yuan, Stochastic population dynamics driven by L´evy noise, J. Math. Anal. Appl. 391 (2012) 363-375. [10] M. Liu, K. Wang, Stochastic Lotka-Volterra systems with L´evy noise, J. Math. Anal. Appl. 410 (2014) 750-763. [11] M. Liu, K. Wang, Dynamics of a Leslie-Gower Holling-type II predatorprey system with L´evy jumps, Nonlinear Anal. 85 (2013) 204-213. [12] M. Liu, C.Z. Bai, On a stochastic delayed predator-prey model with L´evy jumps, Appl. Math. Comput. 228 (2014) 563-570. [13] R.H. Wu, X.L. Zou, K. Wang, Asymptotic properties of stochastic hybrid Gilpin-Ayala system with jumps, Appl. Math. Comput. 249 (2014) 53-66. [14] Q. Liu, Y.L. Liang, Persistence and extinction of a stochastic nonautonomous Gilpin-Ayala system driven by L´evy noise, Commun. Nonlinear Sci. Numer. Simulat. 19 (2014) 3745-3752. [15] Q. Liu, Q.M. Chen, Analysis of a stochastic delay predator-prey system with jumps in a polluted environment, Appl. Math. Comput. 242 (2014) 90-100. 16

[16] Q. Liu, Q.M. Chen, Z.H. Liu, Analysis on stochastic delay LotkaVolterra systems driven by L´evy noise, Appl. Math. Comput. 235 (2014) 261-271. [17] D. Applebaum, L´evy Processes and Stochastic Calculus, 2nd ed., Cambridge University Press, 2009. [18] D. Applebaum, M. Siakalli, Asymptotic stability properties of stochastic differential equations driven by L´evy noise, J. Appl. Probab. 46 (2009) 1116-1129. [19] D. Applebaum, M. Siakalli, Stochastic stabilization of dynamical systems using L´evy noise, Stoch. Dyn. 10 (2010) 497-508. [20] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Springer-Verlag, New York, 2012. [21] H. Kunita, Itˆo’s stochastic calculus: its surprising power for applications, Stochastic Processes Appl. 120 (2010) 622-652. [22] N.H. Du, V.H. Sam, Dynamics of a stochastic Lotka-Volterra model perturbed by white noise, J. Math. Anal. Appl. 324 (2006) 82-97. [23] M. Slatkin, The dynamics of a population in a Markovian environment, Ecology 59 (1978) 249-256. [24] M. Liu, C.Z. Bai, A remark on a stochastic logistic model with L´evy jumps, Appl. Math. Comput. 251 (2015) 521-526. [25] R. Lipster, A strong law of large numbers for local martingales, Stochastics, 3 (1980) 217-228. [26] M. Liu, K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model. 54 (2011) 2139-2154. [27] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, 2006. [28] D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM. Rev. 43 (2001) 525-546. 17

[29] Y.C. Zang, J.P. Li, J.G. Liu, Dynamics of nonautonomous stochastic Gilpin-Ayala competition model with jumps, Abst. Appl. Anal. 2013 (2013), ID 978151, 12 pages.

18

a 0.9

the population x1 the population x2

0.8 0.7 0.6 0.5 0.4 0.3

0.2 0.1 0

0

5

10

15

20

25 Time

30

35

40

45

50

Fig. 1. Solutions of system (13) for r1 (1) = e − 1.96, r1 (2) = e − 1.9, r2 (1) = e − 1.9, r2 (2) = e − 1.8, a11 (ζ(t)) = a12 (ζ(t)) ≡ 0.2, a21 (ζ(t)) = 2 2 2 2 a22 (ζ(t)) ≡ 0.4, σ11 (ζ(t)) = σ12 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, γi (u) = e − 1, i = 1, 2, initial value (x1 (0), x2 (0)) = (0.45, 0.45) and step size ∆t = 0.01. (a) is with π11 = 0.18, π21 = 0.75.

1

b

2.5

the population x1 the population x2

2

1.5

1

0.5

0

0

5

10

20

15

25 Time

30

35

40

45

50

Fig. 1. Solutions of system (13) for r1 (1) = e − 1.96, r1 (2) = e − 1.9, r2 (1) = e − 1.9, r2 (2) = e − 1.8, a11 (ζ(t)) = a12 (ζ(t)) ≡ 0.2, a21 (ζ(t)) = 2 2 2 2 a22 (ζ(t)) ≡ 0.4, σ11 (ζ(t)) = σ12 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, γi (u) = e − 1, i = 1, 2, initial value (x1 (0), x2 (0)) = (0.45, 0.45) and step size ∆t = 0.01. (b) is with π11 = 0.2, π21 = 0.8.

2

c

3

the population x1 the population x2

2.5

2

1.5

1

0.5

0

0

5

10

15

25 Time

20

30

35

40

45

50

Fig. 1. Solutions of system (13) for r1 (1) = e − 1.96, r1 (2) = e − 1.9, r2 (1) = e − 1.9, r2 (2) = e − 1.8, a11 (ζ(t)) = a12 (ζ(t)) ≡ 0.2, a21 (ζ(t)) = 2 2 2 2 a22 (ζ(t)) ≡ 0.4, σ11 (ζ(t)) = σ12 (ζ(t)) ≡ 0.12, σ21 (ζ(t)) = σ22 (ζ(t)) ≡ 0.02, ν(Y) = 1, γi (u) = e − 1, i = 1, 2, initial value (x1 (0), x2 (0)) = (0.45, 0.45) and step size ∆t = 0.01. (c) is with π11 = 0.4, π21 = 0.96.

3

● We consider a stochastic n-species Gilpin-Ayala competitive model with Levy jumps and Markovian switching. ● We study some asymptotic properties and establish sufficient criteria for extinction, non-persistence in the mean and weak persistence of the considered system. ● We obtain the threshold between extinction and weak persistence. ● We introduce some numerical simulations to demonstrate our main results.