The Studies On Population Dynamical Systems And Epidemic Models With Impulsive Effects

The impulsive differential equations have obtained much attention from many authors, and deeply developed during the past a few years. It is widely applied in various domains such as biological technology, medicine dynamics, physics, economy, population dynamics and epidemiology. It is well-known that many natural phenomena and human activities do exhibit impulsive effects in the fields of population dynamics and epidemiology. In Chapter 1, the paper gives some definitions and fundamental theories of impulsive differential equations and difference equations. In Chapter 2, 3 and 4, by using the theories of discrete dynamics, continuous dynamics and impulsive differential equations, and by incorporating with methods of nonlinear analysis, operator theory, mathematical simulation, we study and deal with the dynamical behaviors and impulsive control strategy in discrete population dynamical models and epidemic models.In Chapter 2, we study the effects of birth pulses on the dynamical complexity of stage-structured discrete models. Firstly, we propose and study the single-species discrete population model with stage structure and birth pulses. Using the stroboscopic map, we obtain an exact cycle of system and the threshold conditions for their stabilities. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single-species discrete model with birth pulses is complex, including period-doubling bifurcations, period-halving bifurcations, chaos and non-unique dynamics. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for period-doubling bifurcations (period-halving bifurcations) route to chaos. Secondly, we further investigate an exploited single-species discrete population model with stage structure for the dynamics in a fish population for which birth occurs in a single pulse once per time period. By studying the discrete dynamical system determined by the stroboscopic map, we investigate the stability of periodic solution of impulsive difference system, and obtain the threshold conditions for its stability. Numerical simulation shows the complex dynamical behaviors of the model, including period-doubling bifurcations, period-halving bifurcations, chaos, chaotic bands with periodic windows, non-unique dynamics, attractor crisis and basins of attraction. Moreover, we show that the timing of harvesting and harvest effort has a strong impact on the persistence of the species, the volume of mature fish stock and the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulse, the earlier culling the mature fish, thelarger harvest effort can be tolerated. Our theoretical results imply that carrying out the policy of time closures is beneficial to the sustainable development of fishery resource.Epidemic models with impulsive effects are considered in Chapter 3. Firstly, pulse vaccination of a kind of SIRS epidemic model with saturation infectious force and constant recruitment is analyzed. The dynamics of the epidemic model is globally investigated by using Floquet theory and comparison theorem of impulsive differential equation and analytic method. We obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. Secondly, an SI epidemic model with density-dependent birth pulses is proposed and studied. By studying the discrete dynamical system determined by the stroboscopic map, we obtain the local and global stability of periodic solution. Numerical simulation shows that there is a characteristic sequence of bifurcations, leading to chaotic dynamics, and that there are many complex dynamical behaviors, including period-doubling bifurcations, non-unique dynamics, attractor crisis and basins of attraction.When epidemic models are constructed to describe the transmission of infectious diseases, infection period, immune period and latent period of diseases are not always neglected. Compared with the epidemic models without time delays, the epidemic models with these time delays can describe the features of the diseases diffusion more well and truely. In Chapter 4, we propose four kinds of epidemic models with pulse vaccination and time delays. Due to the coexistence of time delays and impulsive effect, the dynamical behaviors become more complex and are difficult to study. In this chapter, we analyze and study the four epidemic models, respectively. It is proved that the disease-free periodic solution is globally attractive if the period of pulsing is less than r? (or T?), and the disease is uniformly persistent if the period of pulsing is larger than r* (or T*). The permanence of the models is investigated analytically. Our results indicate that a long period of pulsing or a long infection period of disease or a short latent period of disease is the sufficient condition for the permanence of the models.