Stability And Bifurcations Of Some Reaction-diffusion Systems

Population and chemical models can be widely described by (delayed) reaction-difusion systems in the field of ecology, epidemiology, chemistry and other sciences. It’swell-known that to study bifurcation problems and to analyze the nonlinear dynamical be-haviors are one of the key topics in the field of diferential equations. Bifurcation, mainlyfocusing on structurally unstable systems, is the drastic change of some qualitative andtopological structures of the system as the parameters undergo some minor changes andcross through some critical values, which mainly contains static and dynamic bifurcation-s. The study on the efect of some parameters (e.g. difusion coefcients, reproductionperiod and delayed feedback) on the dynamics (e.g. stability and bifurcation when pa-rameters change) is of great significance in both theory and practice.In this thesis, we mainly investigate several kinds of chemical models and predator-prey systems with diferent functional responses influenced by difusion and delay byapplying the center manifold reduction, norm form methods and local Hopf bifurcationand global steady state bifurcation theorem for partial (functional) diferential equationsand some mathematical methods such as maximum principle of parabolic equation andcomparison principle. The main results are as follows:(1) Using the local Hopf bifurcation and global steady state bifurcation theoremfor the general reaction-difusion system, we obtain the stability of the unique positiveconstant equilibrium of the difusive couple Brusselator system and the existence of ho-mogeneous and inhomogeneous periodic solutions. Combining the prior estimates andthe existence or nonexistence of the solutions, we give the existence of the global steadystate bifurcation of the system. Finally, the interaction between Hopf bifurcations andsteady state bifurcation is investigated in detail.(2) The dynamics of a difusive Brusselator model with delayed feedback controlsubject to Dirichlet boundary condition are investigated. It shows that difusion leads toTuring instability. And the unique positive coexistence switches finite times from stabilityto instability to stability and becomes unstable eventually as the delay crosses through aseries of critical values. The properties of inhomogeneous periodic solutions induced byHopf bifurcations are determined by the center manifold and normal form theory. Manynumerical simulations are carried out to illustrate the analysis results. (3) Applying the distribution of the roots of the characteristic equations and tran-scendental equations and combining the maximum principle, comparison principle, weobtain the dynamics of several delayed predator-prey systems with difusion subject toNeumann boundary condition. Under a certain conditions, the global stability of theboundary equilibria of the Holling-III predator-prey systems and the dissipativeness anduniformly persistence of the ratio-dependent system without delay are investigated. Thenwe mainly investigate the efect of the difusion and delay on the system and obtain theexistence of spatially homogeneous and inhomogeneous periodic solutions as the delaycrosses through the critical values. Finally we discuss the properties of Hopf bifurcations.(4) Combining the eigenvalue analysis and the implicit function theorem, we ob-tain the existence and instability of the subcritical positive steady state of the cooperativedelayed difusion system subject to zero Dirichlet boundary condition. It is difcult toanalyze the characteristic equations because the space variable is involved in the coef-cients. We show the existence of forward Hopf bifurcation bifurcating from the backwardnon-constant positive steady state.