Global dynamics of some evolution equations with temporal and sptatial structure

点击翻译按钮获取中文摘要

This proposed research program consists of two parts: (A) investigate the dynamics of some dynamical systems arising from biology in the forms of delay differential equations and reaction-diffusion equations; (B) re-examine the predator-prey interactions by incorporating the anti-predator defence responses of the prey into the model systems. For (A), the focus will be on the impact of the temporal delay (maturation delay or latency delay) and spatial diffusion, and among the topics are the equilibria or steady states and their stability, multiplicity and spatial patterns of steady states, basins of attraction, various bifurcations, travelling wave solutions and spreading speeds. The new methodology developed and new results obtained will enhance our knowledge on the effects of temporal delays and spatial diffusion on solution behaviours and enrich the theory and methodology of dynamical systems; the results will also help us better understand the evolution processes of populations, their interactions and their spatial spreads, as well as infection and spread mechanisms of infectious diseases. All these will help us predict and control future populations of some species and some infectious diseases. For (B), some exiting well-know predator-prey models will be modified to include various anti-predator defence responses, such as reducing productions, decreasing parental care and increasing vigilance and mobility etc. The resulting new models will be analysed and compared to the existing ones. Since such responses widely exist but have been largely overlooked in existing models, such re-examinations will help us better understand predator-prey interactions and take effective measures to prevent some species from going to extinction due to direct (predation) and indirect effects (fear) from their predators.