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Dynamical Analysis Of A Delayed Eco-epidemiological System With Holling Type ? Functional Response

Posted on:2017-05-04Degree:MasterType:Thesis
Country:ChinaCandidate:T R GongFull Text:PDF
GTID:2370330566952911Subject:Mathematics
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The states of current and historical moment have been considered in the mathematical models with delay differential equation(DDE).Hence,delay differential equations can describe the natural phenomena more accurately than ordinary differential equations in many areas.In the investigation about prey-predator models with delay,it is found that the functional response of predators to the prey is one of the important factors,which restricts the growth of the population.Therefore,many biological mathematicians have focused their attentions on establishing mathematical models of delayed eco-epidemiological systems.They are used to study the dynamic behaviors,such as the stability and bifurcation of equilibrium points.These results not only enrich the population dynamics,but also contribute to prevent and control the transmission of disease in populations.The dynamics of a delayed prey-predator system with Holling type II functional response are investigated in this dissertation.The boundedness of the solution is proved in the prey-predator system.Based on the existence of solutions for equilibriums,we obtain three boundary equilibriums and an internal equilibrium.By regarding the time delay as the bifurcation parameter and analyzing the linear system,the local stability of the internal equilibrium point is discussed,and Hopf bifurcation is demonstrated.By using the normal form theory and center manifold theorem,we derive some formulae for determining the stability and the direction of the period solutions bifurcating from Hopf bifurcations.Finally,some numerical simulations are carried out to verify the conclusions by using Matlab.The organization of this dissertation is as follows:1.The significance of this research has been analyzed form the ecological perspective of the species diversity and disease prevention.The backgrounds and the research progress of eco-epidemiological systems are summarized in both domestic and international.2.Some basic concepts and conclusions are given,which would be used in this dissertation.3.The mathematical model with time delay and Holling type II functional response has been given.We mainly discuss the stabilities of the equilibrium points,the direction of Hopf bifurcation and the stability of the periodic solution.Finally,some numerical simulations are carried out to verify the conclusions.
Keywords/Search Tags:time delay, stability, Hopf bifurcation, periodic solutions, Holling type ? functional response
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