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Population dynamical behaviors of stochastic logistic system with jumps

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This paper is concerned with a stochastic logistic model driven by martingales with jumps. In the model, generalized noise and jump noise are taken into account. This model is new and more feasible. The explicit global positive solution of the system is presented, and then sufficient conditions for extinction and persistence are established. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 935 – 948 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1307-25 Population dynamical behaviors of stochastic logistic system with jumps Ruihua WU1,2,∗, Ke WANG1 Department of Mathematics, Harbin Institute of Technology, Weihai, P.R. China 2 College of Science, China University of Petroleum (East China), Qingdao, P.R. China 1 Received: 10.07.2013 • Accepted: 22.02.2014 • Published Online: 01.07.2014 • Printed: 31.07.2014 Abstract: This paper is concerned with a stochastic logistic model driven by martingales with jumps. In the model, generalized noise and jump noise are taken into account. This model is new and more feasible. The explicit global positive solution of the system is presented, and then sufficient conditions for extinction and persistence are established. The critical value of extinction, nonpersistence in the mean, and weak persistence in the mean are obtained. The pathwise and moment properties are also investigated. Finally, some simulation figures are introduced to illustrate the main results. Key words: Logistic equation, martingale, extinction, persistence 1. Introduction The classical nonautonomous logistic equation is [ ] dx(t) = x(t) a(t) − b(t)x(t) dt (1.1) for t ≥ 0 with initial value x(0) > 0. In this model, x(t) denotes the population size at time t, a(t) is the intrinsic growth rate, and a(t)/b(t) is the carrying capacity at time t . Both a(t) and b(t) are positive continuous functions. System (1.1) models the population density of a single species whose members compete among themselves for limited resources such as food or living space. For the detailed model construction, readers can refer to [24]. Because of its importance in theory and practice, many authors have studied model (1.1) and its generalization. Many good results on the dynamical behavior of solutions have been reported; see, e.g., Freedman and Wu [5], Lisena [16], .