Study On The Temporal And Spatial Dynamics Of Predator-Prey Interactions Models

Predators and prey is the phenomena of ecological nature. Feeding process contains rich ecological meaning determines the predation theory will be a central proposition of modern ecology. More and more evidences show that mathematical model is an effective way in those studies about the dynamical relationship between predator and their prey. In Chapter 1, we make a review about the common predator-prey models in mathematical biology literatures. Moreover, we summarize some results and conclusions in those studies, and find out the core and focus problems in this respect.In Chapter 2,we deal with a predator-prey model with a weak Allee effect on the prey population. The classification and stability of all equilibriums of the system are given. Using the normal form theory of Hopf bifurcation, it is shown that Allee effect can result in the existence of a stable limit cycle. Our results suggested that the weak Allee effect can destabilize the coexistence equilibrium and bring rich and complicated dynamics to the previous simple model, such as supercritical Hopf bifurcations, implying that weak Allee effect can be one of the simple reasons for periodic behavior in the predator-prey communities.In Chapter 3, we analysis a spatial system of tri-trophic species chain connected by density-dependent migration(metacommunity). We give a stability criterion for spatial synchrony involving the Lyapunov number of the local population system, migration strategies and the network configuration. Using this criterion, numerical simulations suggest that, if only one species can migrate, the density-independent migration of intermediate species in the tri-trophic food chain is the strongest synchronizer; the density-independent migration of basal species cannot synchronize entire metacommunity; a moderate density-independent migration of top species can synchronize entire metacommunity. Our results also show that density-dependent migration has relatively weak synchronizing potential than density-independent migration strategies.