Stability And Bifurcation Analysis On Some Biological Dynamics Models

This thesis mainly focuses on the stability and bifurcation for some biological dynamics models.The research on stability and bifurcation is helpful to learn more about the formation of spatial-temporal pattern in real world.Applying Lyapunov method,monotonicity methods,global bifurcation theory and the theory of uniform persistence,we study the uniform persistence of system,the global attractivity of the steady state,the steady-state bifurcation and the Hopf bifurcation.Firstly,in the case of non-immune response and immune response,we respectively obtain the positivity and ultimate boundedness of the solutions in the in-host viral model with delays and general incidence functions.Furthermore,when the basic reproduction numbers satisfy certain conditions,we show the global attractivity of disease-free equilibrium and disease-infected equilibrium by virtue of LaSalle invariance principle.Secondly,for the diffusive in-host viral model with general incidence functions,subject to zero-flux boundary condition,the model is a degenerate reaction-diffusion equation,and the solution semiflow is not compact,however,the asymptotically compact of it can be proved by the Arzela-Ascoli theorem.Through the Kuratowski measure of noncompactness,we demonstrate that solution semiflow of the system is ?-contracting,and obtain the existence of global attractor;then by the comparison principle and consistent persistence theory,we prove the global attractivity of free-disease steady state and uniform persistence of system under different conditions;In homogeneous environments,by constructing Lyapunov functions,we certify the global attractivity of disease-free equilibrium and disease-infected equilibrium.Thirdly,we study the diffusive in-host viral model with delay,subject to Neumann boundary condition.Due to the effects of delay,the phase space of the delayed system is different from that of the system without delay.In inhomogeneous environments,according to the relation between the basic reproduction number and the principal eigenvalue,in view of monotonicity methods,persistence theory and the relative conclusions of the system without delay,we prove the global attractivity of free-disease steady state and uniform persistence of system under different conditions;In homogeneous environments,by the invariance principle,we prove the global attractivity of disease-free equilibrium and disease-infected equilibrium.Moreover,the simulations are applied to verifying our results for the system with heterogeneous and homogeneous feeding function.Finally,we consider phytoplankton-zooplankton model with toxic substances effect.We mainly analyze the dynamic properties of the system with or without diffusion.In the absence of diffusion,by applying Poincar?e-Bendixson Theorem,we obtain the bistable structure of system;in the presence of diffusion,under homogeneous Neumann boundary condition,we establish the prior estimate of positive steady state and obtain the existence and non-existence of non-constant positive steady state,meanwhile,we find that diffusion can lead to the formation of stationary pattern under certain conditions.Furthermore,we give the existence conditions of the corresponding steady-state bifurcation and Hopf bifurcation.