Dynamics Of Several Classes Of Mathematical Models In Biology

Biologic mathematical models can describe biological phenomena qualitatively and quantitatively,thus the complex biological problem can be simplified as a mathematical problem by models.Using the theory of differential equations?dynamical systems and numerical simulations,we can analyze dynamic behaviors of the models to seek the op-timal strategies of prevention and control.This dissertation mainly dissects dynamics of several classes of mathematical models in biology.Details are follows:Chapter 2 aims at investigating asymptotical behavior of a 3-dimensional host-macroparasite system.Through dynamical system theory and matrix related knowledge,we obtain the basic production number R0 as a threshold which determines the outcome of parasites and the sufficient conditions for the Hopf bifurcation.The global dynamics of the system is obtained biologically.The results show that if the parameter r of negative binomial distribution of parasites within host is small enough,the endemic equilibrium is globally asymptotically stable which means that the level of aggregation of parasites within host may affect the persistence of disease.In chapter 3,we explore a model that describes the transmission of Wolbachia in mosquitoes population.Based on the theory of differential equations,the existence and classification of the equilibria are examined and then the local stability of the hyperbolic equilibrium are further obtained.Through sensitivity analysis of the basic reproduc-tion number for mosquito ROM and for outbreaks of Wolbachaia RMW,the result is provided theoretical support for anti-mosquito measures and the mechanism of Wolbachia transmis-sion.We assume that the mating between Wolbachia-infected males and Wolbachia-free females cannot produce offspring successfully which is the effect of CI.The results show host populations tend to produce infected female mosquitoes,restrain non-infectious fe-male mosquitoes leading to a reproductive advant,age of Wolbachia-infected mosquitoes which coincides with the spread of Wolbachia in mosquitoes.At the same time,the above parameters are the main factors influencing the spread of Wolbachia.Chapter 4 discusses the competition between Wolbachia-infected and Wolbachia-free mosquitoes,the qualitative behavior of the reduced system for the transmission of Wol-bachia in mosquitoes population is studied.Based on dealing with complex singular point of Poincare and Briot-Bouquet transform,the relatively complex topological structure near a complex critical point EOO is decomposed into simpler topological structures of several hyperbolic points.The results show that there can exist numerous kinds of topo-logical structures in the neighborhood of EOO including the parabolic orbits,the elliptic orbits,the hyperbolic orbits and any combinations of them.These complicated structures imply complex dynamics of the mosquito population with Wolbachia.Global qualitative analysis is further studied including the condition of the extinction of mosquitoes,suc-cessful invasion of the Wolbachia infected mosquitoes,coexistence of the two types of mosquitoes,and numerical simulations are finally conducted to confirm and extend the analytic results.In chapter 5,a mathematical model is formulated for the transmission dynamics of West Nile virus(WNv)between a stage-structured mosquito and bird population,the existence and classification of the equilibria are examined.Through sensitivity analysis of the basic reproduction number for mosquito RM and for outbreaks of WNv RO,preventive and control of anti-mosquito and WNv disease transmission are discussed.The results show that the best way to control the spread of WNv is to decrease the number of mosquitoes and control mosquitoes at the larval stage.Besides the decreasing of the recruitment rate of the uninfected birds is beneficial to the prevalence of the WNv.This observation suggests that it is a risk factor for the spread of WNv to control the birds during the period that the WNv prevails.By contraries,we should increase the recruitment rate of the uninfected birds.Chapter 6 discusses the dynamics of the transmission dynamics of West Nile virus(WNv)between a stage-structured mosquito and bird population.By using the theory of monotone dynamical systems,we carry out rigorous mathematical analysis on the global dynamics.The results show that if RM is not greater than 1,the mosquito will not survive,and the WNv will die out in this case due to the extinction of the vectors;if RM is greater than 1,the mosquito can survive,and the global dynamics of the system is completely determined by the number of the positive equilibria.The basic reproduction number alone is not enough to determine the dynamics of the model,we should pay more attention to the initial state of WNvChapter 7 deals with a class of bidirectional associative memory neural networks with time delays.Using the boundless of fundamental solution matrix?Lyapunov function and some analysis techniques,sufficient conditions are obtained for the existence and global exponential stability of the anti-periodic solution.An example and numerical simulations are also given to illustrate the theoretical results.