Study On The Dynamics Of The Delay Diffusion Population Systems

Delayed reaction–difusion ecosystems, is one of the hot topics inthe study of population dynamics system, which is concerned aboutby biological mathematicians and ecologists. The results can helpus better understand the long time spatiotemporal behavior of thepopulations, such as persistence, evolution, extinction and so on.In this thesis, the main results are as follows:In chapter2, we investigate a modifed delayed difusive of Leslie–Gower model under homogeneous Neumann boundary conditions. Wegive the stability analysis of the equilibria, and show the existence ofHopf bifurcation at the positive equilibrium under some conditions.Furthermore, we investigate the stability and direction of bifurcatingperiodic orbits by using normal form theorem and the center manifoldtheorem.In chapter3, we consider a delayed difusive epidemic model andanalytically investigate how time-delay afects the stability and pat-tern formation. Based on the stability analysis, we demonstrate that,for an appropriate parameter space with diferential difusivity of thesusceptible-infectious species, delayed feedback may generate instabil-ity via Hopf and Turing instability under some conditions, resultingin spatial patterns. Moreover, we perform a series of numerical sim-ulations with a fve-point approximation scheme and fnd that themodel exhibits complex pattern replication. Pure Turing instabilitygives birth to spots, spots stripes-mixture, stripes, stripes holes-mixture and holes patterns, pure Hopf instability to spiral wave pat-tern, and both Hopf and Turing instability to chaotic wave pattern. Our results well extend the fndings of spatiotemporal dynamics inthe reaction-difusion epidemic model, and may be useful for otherdelayed difusive models.And in the last chapter, we give some discussions and remarks.