Nonlocal Time-delay Reaction-diffusion Population Models In Advective Environment

The purpose of this thesis is to study the dynamic behavior of a delayed reaction-diffusion population model in advective environments.In Chapter 2,we first propose a time-delay reaction-diffusion advection model with general growth rates.By using the Lyapunov-Schmidt reduction method,we analyze the existence of the spatial nonhomogeneous positive steady-state solution and the Hopf bifurcation near the steady-state solution.The branch directions of Hopf bifurcation are analyzed by normal form theorem and central manifold theorem,and the stability of bifurcating periodic solutions is obtained.The above theoret-ical results are well verified in two population models with Logistic growth rate and weak Alle effect growth rate respectively.Considering that aquatic organ-isms have certain age structures(instar stages),such as generation period and regeneration period,and unidirectional flow is common in aquatic living envi-ronment,a Logistic reaction-diffusion convection model with age structure is p-resented on the basis of Chapter 2.The existence of the positive steady-state so-lution of the model and associated Hopf bifurcations are obtained by the implicit function theorem and the space decomposition theorem.Taking the time delay?as the bifurcating parameter,we find that under different parameter conditions,the stability of the positive steady state solution may not switch,or switch once or more times.We call this phenomenon”n order stability switches”,which is a much more complex spatiotemporal dynamic behavior.At the same time,we analyze the effect of advection term on Hopf bifurcations and its stability switch.The analysis results show that under certain conditions,the advection rate may promote or inhibit the generation of Hopf bifurcations in the system.In fact,in spatially heterogeneous environments,species migration trends are complex and varied under the influence of habitat environment.Therefore,in Chapter 4 of this thesis,we present a two-species time-delayed reaction-diffusion competition model with general convection terms▽·[b(x)u] and ▽·[b(x)v].The cooperation of general advection term also makes the elliptic operator in this model differ-ent from the above two models.It is not self-adjoint operator,and can not be transformed into self-adjoint operator by variable transformation.Therefore,we systematically analyzed the dynamic behavior of the model by introducing a se-ries of adjoint operators,and found Hopf bifurcation caused by time delay in the model.