Bifurcaitons In Population Models Of Single Species

Population models described by differential equations arise frequently from popula-tion statistics, ecology, epidemiology and other sciences. Studying their nolinear dynam-ics such as stability and bifurcations is very important in the fields of both differentialequations and mathematical biology. Bifurcation occurs in structurally unstable systems,in which the dynamics change topologically as parameters pass through some criticalvalues. The study of bifurcation problems in population models will help us to better un-derstand the effect of parameters (e.g., survival space and maturation period of species) onthe population dynamics (e.g., any sudden oscillations as parameters change), and hence,to control the species invasion and disease spread. At the same time, in the study of spe-cific population models, we may find some new interesting phenomena, which may canpromote the development of the theory for differential equations. This thesis focuses onthe bifurcations of the time-delayed reaction diffusion population models with homoge-neous Dirichlet boundary conditions and a first-order partial differential population modelwith nonlinear boundary conditions.Firstly, we study a diffusive and time-delayed blow?y model with homogeneousDirichlet boundary condition. Using the phase plane analysis for planar systems, we showthat the steady state bifurcation occurs under the zero Dirichlet boundary condition andthat there is a unique positive steady state under a specific positive Dirichlet boundarycondition. Moreover, we find that diffusion can stabilize the model under the positiveDirichlet boundary condition. Furthermore, we establish the existence and direction ofHopf bifurcations as well as the stability of the bifurcating periodic solutions.Secondly, we consider a class of time-delayed reaction-diffusion population modelwith zero Dirichlet boundary condition where the growth function is of logistic or weakAllee type and only depends on the delayed density. We give the definitions of forwardand backward Hopf bifurcations and show the existence of forward Hopf bifurcationsfrom the non-constant steady state, respectively. That is, the logistic type model hasforward steady state bifurcations and the weak Allee type model has backward steadystate bifurcations.Thirdly, we investigate a time-delayed reaction-diffusion population model with zero Dirichlet boundary condition where the growth function depends on both delayed andinstantaneous densities. Under the assumption that the delayed density dominates thegrowth function, we use the maturation period (i.e., time delay) as the parameter to showthe existence and direction of Hopf bifurcations from the non-constant steady state as wellas the stability of the bifurcating periodic solutions. Moreover, by constructing a pair ofupper and lower solutions we prove the uniform boundedness of periodic solutions, andhence obtain the global Hopf bifurcation result using a global Hopf bifurcation theorem.Finally, we explore an age-structured malaria population model. Using the Hopfbifurcation theorem for general age-structured models, we prove the existence of the pe-riodic solutions when the maximum of replication ratio of Plasmodium falciparum comesacross a critical value, which means that malaria parasites enter and are released fromthese red blood cells at approximately the same times. This part gives a method to esti-mate the maximum of replication ratio, and reveals some inherent link between the burstrate and the critical value of replication ratio such that the phenomenon of synchronizationoccurs: the bigger the burst rate, the smaller the critical value of replication ratio.