Study On Some Dynamic Models Of Strategy Evolution

Nowadays,human society is facing serious many threats of ecological envi-ronment and public health issues,such as environmental pollution,invasion of alien species,extinction and survival of population,and the emergence of new diseases or regional aggregation of epidemics,as well as re-emerging of infectious diseases.Using mathematical models to study these problems will not only deep-en human's understanding of the problems but also provide theoretical guidance for settling related problems.Based on some ecological environmental and pub-lic health realities,this thesis establishes some mathematical models,and the dynamic behaviors of the models are studied.Since maturity time plays an important role on survival and diversity of population,in order to study the optimal life history strategy determined by ma-turity time,the second part of this thesis establishes and analyzes an adaptive evolution model of life-history strategy about maturity time in seasonal environ-ment.Firstly,the dynamic behaviors of the two population competitive model for the same species with different life history strategies in seasonal environment is analyzed,and a complete classification of the competitive income in seasonal environment is obtained.It is found that the length of breeding season has an important influence on the extinction or survival of the species.Secondly,adap-tive dynamical system theory is used to obtain the evolutionary strategy of the maturity time of the invading population.For invading populations,the evolu-tionary stability strategy?ESS?about maturity time is locally existing,unique and a convergent stability strategy?CSS?.In addition,the results show that the ESS of the invasive species is different from its maximum fitness strategy.Final-ly,through numerical simulation of the influence of the length of breeding season on the maturity strategy,it is concluded that the evolutionarily stable maturity strategy always decreases with the increase of the breeding season length,but it always does not exceed its maximum fitness strategy.A new research confirms ZIKA virus can survive in an external water envi-ronment out of its host body for a period and infect immature mosquitos in the aquatic environment.Therefore,a vector-borne infectious disease mathematical model is established with two transmission strategies:vector transmission and virus transmission in the aquatic stage.Firstly,we define the basic reproduction number R₀,and if the virus released function of host and infection rate function of vector in aquatic stage both are sublinear,the globally dynamics of the system is decided by R₀.On the other hand,if one of the released function or infection rate function is not linear,the system may occur backward bifurcation.At last,using numerical simulations to discuss the dynamic behaviors of the system when the release rate function and the infection rate function at aquatic stage are not linear at the same time.When the system appears backward bifurcation,with increasing of R₀there would occur Hopf bifurcation at the neighborhood of which a local stable period solution exists and the system tends to the unique stable positive equilibrium with larger R₀.Besides,if there is no backward bifurcation,and R₀>1,there would be two Hopf bifurcations and between of them a period solution exists.Finally,a virus dynamics model is established according to the different in-fection strategies and death modes of HIV in vivo.The model considers both cell to cell infection and free viral infection strategies,and apoptosis and natural death caused by both viral and infected cell receptors.Firstly,we obtain the ba-sic reproduction number R₀.Then we obtain the infection-free equilibrium of the system is locally asymptotically stable if R₀<1.And if R₀>1,the system has a unique positive equilibrium,which is locally asymptotically stable.It is found that when the apoptosis rate of infected cells is large enough,the system may occur backward bifurcation.If the apoptosis rate of infected cells is moderate,the infection-free equilibrium is globally asymptotically stable at R₀<1.When we ignore the influence of apoptosis rate,the backward bifurcation would disappear,and dynamic behaviors of system are discussed by numerical simulation.