Complex Dynamic Analysis In A Biological Model

In this paper,we analyze the complex dynamics of a schistosome model by applying the bifurcation theory in time-domain and frequency domain and numerical simulation,including the transcritical bifurcation,Hopf bifurcation,the existence and stability of periodic orbits.We find the change rule of important dynamic variables in schistosomiasis model with the change of branch parameters,which provides a theoretical basis for schistosomiasis prevention and control.This article is composed of the following four chapters.In the first chapter,we introduce the preliminary knowledge of bifurcation theory of nonlinear dynamical systems and the development history of the mathematical models of schistosoma.At the same time,we introduce the nonlinear dynamical system of local bifurcation theory and the main work in this paper.In the second chapter,we study the existence and stability of the equilibrium point of schistosome in the schistosome model by using the probability of snails infected with delta from latency to easy to infect period δas the bifurcation parameter.The Hopf bifurcation theory in the frequency domain is used to analyze the existence of the Hopf bifurcation,the fourth-order harmonic balance method is used to give the approximate expression,frequency and amplitude of the periodic orbit generated by the Hopf bifurcation.At the same time,the stability of the periodic orbit is judged.The corresponding branch diagram is given by using the Auto software numerical simulation.The phase diagram and time series diagram of the periodic orbit are given by Matlab numerical simulation.These numerically simulated images verify the results of the theoretical analysis.Finally,combined with the biological relevance,corresponding explanations are provided for the dynamic changes in the model.In the third chapter,we still use the schistosomiasis model in the second chapter as the research object,changing the branching parameter to the reproduction rate of snails v.We apply the branching theory in time domain and frequency domain to study the schistosomiasis model mentioned above and obtain more abundant results.Using the branching theory in the time domain,we obtain that the system will have two transcritical branches with the change of the reproduction rate v of the snails.Using the branching theory in the frequency domain,we obtain that the system will generate two Hopf branches,which can lead to the occurrence and disappearance of periodic oscillations.We performe a corresponding numerical simulation of this analysis,and give a biologically relevant explanation for the changes in the system at the same time.In the fourth chapter,we summarize the research in the full text,and we not only point out the deficiencies in the text,but also establish a future research direction.