Stochastic Logistic Systems with Jumps

This paper is concerned with a stochastic nonautonomous logistic model with jumps. In the model, the martingale and jump noise are taken into account. This model is new and more feasible and applicable. Sufficient criteria for the existence of global positive solutions are obtained; then asymptotic boundedness in pth moment, stochastically ultimate boundedness, and asymptotic pathwise behavior are to be considered. 1. Introduction The well-known logistic system is an important and applicable system in both ecology and mathematical ecology. The classical nonautonomous logistic equation can be described by for with initial value . In this model, is the population size at time , so we are only concerned with positive solutions. The coefficient is the intrinsic growth rate and stands for the carrying capacity at time . Both and are continuous bounded functions on . System (1) models the population density of a single species whose members compete among themselves for a limited amount of food and living space. About the detailed model construction, readers can refer to [1]. Because of the importance in theory and practice, many authors have studied deterministic model (1) and its generalization. Many good results on the dynamical behavior have been reported; see, for example, Freedman and Wu [2], Lisena [3], Golpalsamy [4], Kuang [5], and the references therein. Among them, the books [4, 5] are good references in this area. However, in the real world, the population systems are inevitably subject to much stochastic environmental noise which is important in ecosystem (see, e.g., Gard [6, 7]). In model (1), the parameters are all deterministic and irrespective of the environmental fluctuations; therefore, they have limitations in applications and it is difficult to fit data and predict the future accurately [8]. May [1] proposed the fact that because of the environmental noise, the birth rate, carrying capacity, and other parameters involved in the system exhibit random fluctuation to a greater or lesser extent. So it is necessary to find a more practical model. According to the well-known central limit theorem, the sum of all stochastic environmental noise follows a normal distribution, and we usually call the white noise and denote it by . We impose the stochastic perturbation on and then get the following It？'s equation: where is a standard Brownian motion defined on a complete probability space with a filtration satisfying the usual conditions (i.e, is right continuous and contains all -null sets) and denotes the intensity of the noise. The white noise has