体制切换下随机时滞捕食者－食饵模型的持久性与灭绝

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59 (2014) APPLICATIONS OF MATHEMATICS No. 3, 331–343

PERSISTENCE AND EXTINCTION OF A STOCHASTIC DELAY

PREDATOR-PREY MODEL UNDER REGIME SWITCHING

Zhen Hai Liu, Qun Liu, Nanning

(Received July 17, 2012)

Abstract. The paper is concerned with a stochastic delay predator-prey model under

regime switching. Suﬃcient conditions for extinction and non-persistence in the mean

of the system are established. The threshold between persistence and extinction is also

obtained for each population. Some numerical simulations are introduced to support our

main results.

Keywords: persistence; extinction; Markov switching; delay; stochastic perturbations

MSC 2010 : 34B16, 34C25

1. Introduction

The deter ministic delay predator-prey model can be expr e ssed as follows:

dx(t)/dt = x(t)[r

(t) − a

(t)x(t) − a

(t)x

(t − τ

) −a

(t)y

(t)],(1)

dy(t)/dt = y(t)[−r

(t) + a

(t)x(t) − a

(t)y(t) − a

(t)y

(t − τ

)],

where x(t) and y(t) are the prey population density and the predator p opulation

density at time t, respectively; r

(t) and r

(t) represent the intrinsic growth rates

of the prey and the preda tor at time t, respe c tively; a

(t) and a

(t) denote the

density-dependent coeﬃcients of the prey and the predator, respectively; a

(t) is the

capturing r ate of the predator and a

(t) denotes the rate of c onversion of nutrients

into the reproduction of the pr e dator; a

(t) provides a measure of intra-speciﬁc

interference and a

(t) pr ovides a measure of inter-speciﬁc interference; τ

and τ

are

two positive constants which stand for the time delays; θ

, θ

> 0 and θ

, θ

The research has been supported by NNSF of China Grant Nos. 11271087, 61263006.

331

provide nonlinear measures of intra-sp e c iﬁc interference, θ

provides a nonlinear

measure of inter-speciﬁc interference; r

(t) and a

(t) are positive continuous bounded

functions on R

= [0 , +∞), i = 1, 2; j = 1, 2, 3.

On the other hand, in the real world, population system is inevitably aﬀected

by the environmental noise (see e.g. [2], [13], [11], [12]). As we all know, there are

various types of environmental nois es. First, we sha ll consider a classical colored

noise, say the telegraph no ise. May [10 ] pointed out that due to the environmental

noise, the birth rate, carrying capacity, competition c oeﬃcient and other parameters

involved in the sys tem are often aﬀected by the telegraph noise. Several authors

(see e.g. [9 ], [7], [14], [4]) have revealed that we can model the telegraph noise by a

continuous-time Markov chain γ(t), t > 0 with a ﬁnite-state space S = {1, 2, . . . , m}.

Let γ(t) be generated by Q = (q

), that is,

P{γ(t + ∆t) = j|γ(t) = i} =

(

∆t + o(∆t) if j 6= i;

1 + q

∆t + o(∆t) if j = i,

where q

> 0 for i, j = 1, 2, . . . , m with j 6= i and

j=1

= 0 for i = 1, 2, . . . , m.

Then model (1) will become

(2)

dx(t)/dt = x(t)[r

(γ(t)) − a

(γ(t))x(t) − a

(γ(t))x

(t − τ

) − a

(γ(t))y

(t)],

dy(t)/dt = y(t)[−r

(γ(t)) + a

(γ(t))x(t) − a

(γ(t))y(t) − a

(γ(t))y

(t − τ

)].

The mechanism of system (2) is explained as follows. Assume that γ(0) = κ ∈ S ,

then (2) satisﬁes

dx(t)/dt = x(t)[r

(κ) − a

(κ)x(t) − a

(κ)x

(t − τ

) − a

(κ)y

(t)],

dy(t)/dt = y(t)[−r

(κ) + a

(κ)x(t) − a

(κ)y(t) − a

(κ)y

(t − τ

)]

for a random amount of time until γ(t) jumps to another state, say ς ∈ S . Then

the system obeys

dx(t)/dt = x(t)[r

(ς) − a

(ς)x(t) − a

(ς)x

(t − τ

) − a

(ς)y

(t)],

dy(t)/dt = y(t)[−r

(ς) + a

(ς)x(t) − a

(ς)y(t) − a

(ς)y

(t − τ

)]

for a random amount of time until γ(t) jumps to a new state again.

Further , let us consider the white nois e. Recall that r

(i) r e presents the intrinsic

growth rate in regime i (i ∈ S ). We estimate it by an error term plus an average

332

value. Sometimes, the error term follows a normal dis tribution. Consequently, we

can replace r

(i) by r

(i) + σ

(i)

(t) (see e.g. [7], [14], [4]), where

(t) is a

white noise and σ

(i) stands for the intensity of the white noise. In the same way,

−a

(i), −a

(i), −r

(i), a

(i), −a

(i) and −a

(i) will become −a

(i) +

(i)

(t), −a

(i) + σ

(i)

(t), −a

(i) + σ

(i)

(t), −r

(i) + σ

(i)

(t),

−a

(i)+σ

(i)

(t), −a

(i)+σ

(i)

(t) and −a

(i)+σ

(i)

(t) (see e.g. [8]).

Then we obtain the following stochastic delay predator-prey model under regime

switching:

dx(t) = x(t)[r

(γ(t)) − a

(γ(t))x(t) − a

(γ(t))x

(t − τ

)(3)

− a

(γ(t))y

(t)] dt + σ

(γ(t))x(t) dB

(t) + σ

(γ(t))x

(t) dB

(t)

+ σ

(γ(t))x(t)x

(t − τ

) dB

(t) + σ

(γ(t))x(t)y

(t) dB

(t),

dy(t) = y(t)[−r

(γ(t)) + a

(γ(t))x(t) − a

(γ(t))y(t) − a

(γ(t))y

(t − τ

)] dt

+ σ

(γ(t))y(t) dB

(t) + σ

(γ(t))x(t)y(t) dB

(t)

+ σ

(γ(t))y

(t) dB

(t) + σ

(γ(t))y(t)y

(t − τ

) dB

(t),

where B(t) =



(t) B

(t)

(t) B

(t)



is a given 2×4 dimensional Brow-

nian motion deﬁned on a complete probability space (Ω, F , P) with a ﬁltration

}

t∈R

satisfying the usual conditions. Suppose that the Markov chain γ(·) is

independent of B(t). As the standing hypothesis, we a ssume that γ(·) has a unique

stationary distribution π = (π

, π

, . . . , π

) which can b e o bta ined by solving the

linear equation πQ = 0 subject to

i=1

= 1 and π

> 0, i ∈ S . Throughout this

article, we assume that min

i∈S

(i) > 0, min

i∈S

(i) > 0, min

i∈S

(i) > 0, min

i∈S

(i) > 0,

j 6= k, j = 1, 2; k = 1, 2, 3; l = 1, 2, 3, 4 and deﬁne ˆν = max

i∈S

ν(i), ˘ν = min

i∈S

ν(i).

To begin with, we give the following useful deﬁnition.

Deﬁnition 1. 1. The population x(t) is said to go to extinction if lim

t→+∞

x(t) = 0.

2. The population x(t) is said to be nonpersis tent in the mean if hx(t)i

∗

= 0,

where hf(t)i =

f(s) ds/t, f

∗

= lim sup

t→+∞

f(t), f

∗

= lim inf

t→+∞

f(t).

The organization of this paper is as follows: In Section 2, we analyze the persistence

and extinction of a stochastic delay predator-prey model under regime switching.

Some simulation ﬁgures are provided to illustra te our main results in Section 3.

Finally we give some conclusions and discussio n.

333

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