Stability And Permanence Of Impulsive Biological Dynamical Systems In Patchy Environment

In many models of population dynamics and epidemic, due to spatial heterogeneity, each region because of different geographical environment and so on, which leads to different birth rate, mortality rate. As a result, more and more scholars have studied the patch effect. In addition, pulse phenomenon often exists in real life, for example, the regular fishing of fish, regular migration of birds and periodic inoculation of infectious diseases and so on. In this paper, we obtain investigation of stability and permanence of impulsive biological dynamical systems in patchy environment. The main content as follows:In the first part, we study dynamical behaviors of stage-structure predator-prey model with harvesting effort and impulsive diffusion. By the impulsive comparison theorem and the discrete dynamic system determined by the stroboscopic map, we obtain some sufficient conditions on the existence and global attractiveness of predator-eradicated periodic solution for the system.In addition, the permanence of the system is also derived, and estimate how many members the population has at large time in every patch, this is important to manage effectively the species.In order to verify the validity of the theoretical results, an example and its numerical simulations are given.In the second part of this paper, we study dynamics of SI epidemic model in a patchy environment with pulse vaccination and quarantine. On the basis of in previous publications,we consider the disease diffusion and extend two patch model to m- patch model, and consider that the situation with pulse vaccination and quarantine at different time. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ’infection-free’ periodic solution. By using impulsive type comparison results,Floquet theory, small amplitude perturbation techniques and the theory of matrix spectral radius, sufficient condition ensuring the globally asymptotical stability of the periodic solution is derived. In addition, the permanence condition of the system is obtained by the persistence theorem. An example and its numerical simulations are given to verify the effectiveness of the theoretical result.