Existence and exponential stability of piecewise mean-square almost periodic solutions for impulsive stochastic Nicholson's blowflies model on time scales

Keywords: Time scales Nicholson's blowflies model Patch structure Piecewise mean-square almost periodic solution Impulsive stochastic a b s t r a c t In this paper, a class of impulsive stochastic Nicholson's blowflies model with patch structure and nonlinear harvesting terms is introduced and studied on time scales. By using contraction mapping principal and Gronwall–Bellman inequality technique, some sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions for the model with infinite delays are established on time scales. Finally, an example is given to demonstrate the validity of the conditions of the main theorem. On the applied aspect of dynamic systems, as every one knows, one of the most popular dynamic population models is Nicholson's blowflies model N 0 ðtÞ ¼ ÀdNðtÞ þ pNðt À sÞe ÀaNðtÀsÞ ; which was proposed by Gurney et al. [1] to describe the population of the Australian sheep-blowfly and to agree with Nich-olson's experimental data [2]. Here, NðtÞ is the size of the population at time t; p is the maximum per capita daily egg production , 1=a is the size at which the population reproduces at its maximum rate, d is the per capita daily adult death rate, and s is the generation time. The model and its modifications have been extensively and intensively studied and numerous results about its stability, persistence, attractivity, periodic solutions, almost periodic solutions and so on (see [3–7]) have been obtained. In recent years, the theory of time scale, which was introduced by Hilger in his PhD thesis [8], has been established in order to unify continuous and discrete analysis. In fact, the progressive field of dynamic equations on time scales contains, links and extends the classical theory of differential and difference equations. This theory represents a powerful tool for applications to economics, biological models, quantum physics among others. See, for instance, Ref. [9]. Because of this fact, it has been attracting the attention of many mathematicians (see Refs. [10–14]). On the other hand, recently, qualitative analysis of stochastic model has attracted the attention of many mathematicians and biologists due to the fact the natural extension of a deterministic model is stochastic model [15]. In the aspect of life sciences, most phenomena are basically modeled as suitable stochastic processes, where relevant parameters are modeled as suitable stochastic processes. Furthermore, the ecological systems are often characterized by the fact that they experience a sudden …