Dynamic Studies Of Several Biomathematics Models

This dissertation focuses on the improvement and extension of several classicalmathematical models of the ecology and medicine. Using the qualitative theory ofdifferential equation, bifurcation theory and functional differential equation theory,we study their dynamical behaviors, such as the persistence of the system, thestability of the equilibrium as well as the existence and stability of the periodicsolutions. Moreover, the parameters'influence on the dynamical behaviors is alsoexplored here. It, therefore, sets up some theoretical foundations for explaining,predicting and controlling some phenomena in both ecology and medicine. To bespecific, our research is as follows:First, because of the complexity of the mode of transmission, it is difficultto accurately describe the actual contact rate, incidence rate and the number ofpopulation in disease latent period, based on the work of [112], we consider a classof SIRS epidemic model with more general nonlinear incidence and introduce timedelay to describe the disease latent period. By the qualitative analysis of wholeparameters, we give the expression of the basic reproductive number of the modelfor the spread of the disease. When the basic reproductive number is less than1, the disease free equilibrium is globally asymptotically stable. But when thebasic reproductive number is greater than 1, dynamical behaviors depend on thetime delay and one parameters of the nonlinear incidence. When these parametersare taken certain values, the unique endemic equilibrium in the phase space isglobally asymptotically stable. When they are taken some other parameters, theunique endemic equilibrium is unstable and the periodical oscillation phenomena ofpatients occur. And we prove the existence of Hopf bifurcation. Our work extend the work of [112].Second, based on the work of [57], we further study the uniform persistenceand the global asymptotic stability of the infected equilibrium for a class of delayedHIV-1 infection model. And we obtain the sufficient and necessary conditions forthe uniform persistence of the model and the sufficient conditions for the globalasymptotic stability of the infected equilibrium, which further improve the resultsof [57].Third, the developmental times of the different stages of schistosoma is veryimportant for schistosome transmission, we divide the definitive hosts populationand the intermediate snails population into two disjoint classes: susceptible andinfected, respectively. And we incorporate effects of the developmental times ofthe different stages of schistosoma into the classic Barbour model and consider amore general mathematical model, which is a four dimensional model containingfour time delays. We give the expression of the basic reproductive number of themodel. Using the invariant manifold of the model, we reduce it to a planar system.We obtain the sufficient conditions for the global asymptotic stability of equilibriumof the model without time delays. When the basic reproductive number is greaterthan 1, we find that dynamical behaviors depend on the time delay. When the timedelay crosses some critical values, the stability of the unique endemic equilibriumchanges from stable to unstable, and Hopf bifurcations occur.Four, we first discuss a predator-prey model with discrete and distributeddelays, give conditions for the asymptotic stability of the positive equilibriumand the Hopf bifurcation occurring. Next, based on the classic Lotka-Volterrapredator-prey model, we introduce age-structure into the predator species satisfiedthe McKendrick-Foerster equation, which is more tally with the actual situation.Change the model into a delayed system. We give sufficient conditions for thepositive invariance and boundedness of the solution and the global asymptotic sta-bility of the boundary equilibrium. We find that when the time delay gradually increases to some critical values, the positive equilibrium loses its stability andHopf bifurcations occur. Moreover, we give the explicit formulas to determine thestability and the direction of periodic solutions bifurcating from the Hopf bifurca-tions. Finally, we discuss a class of more realistic non-autonomous predator-preymodel with more general functional response and give the sufficient condition onthe existence of positive periodic solutions of the model.