Qualitative Analysis Of Three Mathematical Models In Biology

Mathematical models can describe phenomena qualitatively and quantitatively and simplify complex realistic situations into mathematical problems,they have been widely applied in various fields,especially in theoretical epidemiology and population ecology.This dissertation investigates the improvement and promotion of some classical math-ematical models in epidemiology and ecology.We utilize the fundamental theorem of ordinary differential equations and parabolic equations,the contraction mapping theorem,the principal eigenvalues theory,and combined with numerical simulation,to analyze the dynamic properties of the multi-group epidemic model,the partially degenerate nonlocal diffusion model,and the locally-nonlocal coupling diffusion model.These obtained re-sults provide a theoretical basis for investigating,explaining,predicting,and controlling the changing regulations of objects in epidemics and ecology.Firstly,an SVIR epidemic model that contains multi-groups of host populations to reflect different risk levels of infection and transmissibility with limited immune resources are considered.The well-posedness and the existence of compact global attractor are first obtained,and then the basic reproduction number R₀is determined.Besides,we discuss global asymptotic stabilities of the disease-free and endemic equilibriums.When the disease is endemic,an optimal control problem is formulated to minimize infection scale and economic costs in a resource-constrained environment.Moreover,numerical examples and simulations are further presented to verify and support the validity of the theoretical results.We also compare the optimal vaccination coverage strategies with the constant one under limited immunization resources.As special cases of our model,we discuss the effects of seasonal factors on constant and optimal coverage strategies for some diseases highly affected by the season.Secondly,a class of partially degenerate nonlocal diffusion systems with free bound-aries are investigated.Such problems can describe the evolution of one species with non-local diffusion and the other without diffusion or with much slower diffusion.The ex-istence,uniqueness,and regularity of global solutions are first proven.The criteria of spreading and vanishing are also established for the Lotka-Volterra type competition and prey-predator growth terms.Moreover,we investigate long-time behaviors of the solution and propose estimates of spreading speeds when spreading occurs.Finally,a class of local and nonlocal diffusion systems with double free boundaries possessing different moving parameters concerned.We firstly obtain the existence,u-niqueness and regularity of global solution and then prove that its dynamics are governed by a spreading-vanishing dichotomy.Then the sharp criteria for spreading and vanishing are established.Of particular importance is that long-time behaviors of solution in this problem are quite rich under the Lotka-Volterra type competition,prey-predator and mu-tualist growth conditions.Moreover,we also provide rough estimates of spreading speeds when spreading happens.