Study And Application On Eco-epidemiological System

In this paper, we considered three eco-epidemiological systems and discussed their dynamical behaviors, by the methods of Hurwitz criterion、Mawhin’s continuation theorem、Lyapunov function etc, to study the stability of equilibria, the existence of positive periodic solutions, and the asymptotic behavior of solutions of stochastic system.The organization of this paper are as follows:In the ?rst chapter, introduced the background and signi?cance of eco-epidemiological system and given some preliminaries.In the second chapter, an epidemic model of Lotka-Volterra predator-prey system with epidemic in predator is discussed, both the susceptible and infected predator populations have the ability to predate, and converted to the susceptible population, with the bilinear incidence rate. By the method of the characteristic root, the conditions and threshold to the existence of various equilibriums are established. By the methods of Hurwitz criterion and Lyapunov-Lassel invariant set theorem, local stability and global stability of the system are proved.In the third chapter, we established an epidemic model of a Holling-Tanner predator-prey system with epidemic in predator, the infected predator populations lost the ability to breed and predate. The solution with initial data always remains positive in its region.The su?cient criteria are easily established for the existence of positive periodic solutions,by the methods of coincidence degree and Mawhin’s continuation theorem.In the forth chapter, we considered a stochastic SIR model incorporating media coverage, and showed that this model had a unique global positive solution. Furthermore, we investigated the asymptotic behavior of this stochastic solution. Finally, some numerical simulations were presented to illustrate our mathematical ?ndings.