Acta Mechanica Sinica (2014) 30(5):739–745

DOI 10.1007/s10409-014-0069-y

RESEARCH PAPER

Dynamics of a prey-predator system under Poisson white noise excitation

Shan-Shan Pan · Wei-Qiu Zhu

Received: 11 December 2013 / Revised: 8 February 2014 / Accepted: 29 April 2014 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2014

Abstract The classical Lotka–Volterra (LV) model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is investigated by using the stochastic averaging method. The averaged generalized Itoˆ stochastic differential equation and

Fokker–Planck–Kolmogorov (FPK) equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized

FPK equation is obtained by using the perturbation method.

The effect of prey self-competition parameter ε2s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo (MC) simulation.

Keywords Prey-predator ecosystem · Poisson white noise ·

Stochastic averaging · Approximate stationary solution · Perturbation method 1 Introduction

Recently, the interest in prey-predator system has experienced a rapid growth due to their intrinsic relevance to modern population biology and ecology [1–5]. For some decades, much effort has been devoted to mathematical modeling of the dynamics of interacting species through a set of coupled nonlinear differential equations [3]. A well-known model is the Lotka–Volterra (LV) model, which describes the time evolution of two interacting species, referred to as prey and predator [6–8]. This model includes the prey selfThe project was supported by the National Natural Science Foundation of China (10932009, 11072212, 11272279, and 11321202).

S.-S. Pan ·W.-Q. Zhu ( )

Department of Mechanics,

State Key Lab of Fluid Power Transmission and Control,

Zhejiang University, 310027 Hangzhou, China e-mail: wqzhu@yahoo.com competition term and the coupled term which provide a balance between the two populations. Without these two terms, the prey population would grow with a constant birth rate in absence of the predator, while the predator population would decrease exponentially without the prey. Then more researches on the model have been made, such as taking the predator-saturation term into account [9, 10], and considering the effect of time delay of prey population on predator population [1, 11–13].

All the above mentioned models are deterministic.

However, it is widely believed that interaction between species and with natural environment greatly influences the global features of ecosystems [14–16]. Therefore, changes in the environment (e.g., temperature, rainfall, pollution, disaster, etc.) inspire intense discussion [13, 17]. Gaussian white noise variations are usually added to the predator death rate and prey birth rate to model the continuous environmental fluctuations [1, 14]. In fact, the Gaussian white noise always assumes the presence of continuous perturbations, while in real systems there are some unavoidable sparse, yet drastic, impulses which may qualitatively change the system behavior and even completely invalidate the deterministic predictions. The impact of sudden natural disasters such as earthquakes, forest fires, floods, on the biological population, looking from the macroscopic time scale, is discrete pulse-type perturbation, which cannot be appropriately described by using Gaussian white noise. So, Wu and Zhu [18] proposed a pulse-type LV model to describe the behaviors of two interacting species under discrete environmental fluctuations, which were modeled as Poisson white noises. In

Ref. [18], the generalized cell mapping method was applied to obtain the probability distributions of the predator and prey populations. However, the generalized cell mapping method is a numerical one. It is not convenient to use this method for parametric analysis.

Stochastic averaging method is a powerful technique for predicting the response of nonlinear systems to random excitations. Stochastic version of the classical LV model and its improved model, in which the random variability was 740 S.-S. Pan, W.-Q. Zhu modeled as Gaussian white noise, has been investigated using the stochastic averaging method [14, 15, 19, 20]. Xu et al. [21] gave an averaging principle for stochastic dynamical systems with Le´vy noise and studied stochastic bifurcations in a bistable Duffing–van der Pol oscillator with colored noise through averaging method [22]. In recent years, the stochastic averaging method has been developed for quasilinear or nonlinear dynamic systems driven by Poisson white noise excitation [23, 24].

In the present paper, the stochastic averaging method [23, 24] is applied to obtain the probability distributions of the predator and prey populations in a state of statistical quasistationarity. Averaged generalized Itoˆ stochastic differential equation and FPK equation of the system are derived. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of the prey self-competition parameter ε2s on the ecosystem behavior is evaluated. It is shown that the analytical and MC simulation results agree very well. 2 The model

The pulse-type stochastic model of the prey-predator ecosystem is given as follows

X˙1 = X1(a1 − ε2sX1 − bX2 + εξ1(t)),

X˙2 = X2(−c + f X1 + εξ2(t)), (1) where X1(t) and X2(t) are two stochastic processes representing population densities of the prey and the predator, respectively; a1, b, c, and f are positive constants; ε > 0 is a small parameter; a1 is the birth rate of the prey; c is the death rate of the predator. Nonlinear term −ε2sX21 denotes the prey self-competition, which makes prey population not grow infinitely without predator (For prey, in the environment of the habitat, because the consumption of resources such as food, water, space is limited, competing for resources is bound to form the competition between individuals within populations, which to a certain extent, restrict the expansion of the prey population). Nonlinear terms −bX1X2 and f X1X2 provide a balance between the two populations, and lead to relative reduction of prey population and relative growth of predator population. ξ1(t) and ξ2(t) are two independent Poisson white noises, which can be treated as the formal derivative of the following compound Poisson processes