Dynamical Analysis Of Predator-prey Model With A Transmissible Disease And Delay

Population biology plays an important role in theoretical ecology. It is a science that focuses on explaining and predicting change in the size of populations. The dynamics of the biological populations are captured by mathematical systems that are mainly expressed by differential equations. Mathematical models can depict accurately the factors affecting the population's rate of growth by theoretical arguments. Moreover, analysis of such model enables forecasting the profound economic implications of renewable resources management. In the natural world, the species can not exist alone, they will become subject to disease that are contracted by mutual contact with the environment. An organism which harms agriculture by means of feeding on crops or parasitizing livestock is known to us as pest. The term pest is used to refer to harmful animals. The word 'pest' we mean those livelihood which are involved to the cause of the damages to the human populations directly or indirectly. Hence, it is important to research that these species which predates those pests. The disease spreads among the predator species can weaken the capture capacity of predators. And sometimes it is seen that the pest populations are affected by some diseases. In this case those disease affected pests are more vulnerable to the predators because they are easy to catch than the healthy one. Hence, the research of a predator-prey model with a transmissible disease have profound implications. In addition,time delays play an important role in population dynamics. The predator-prey model with time delay can be more realistic in the ecosystem. Then based on this motivated above,we derive three predator-prey models, predator-prey model with a transmissible disease in the prey species, predator-prey model with a transmissible disease in the predator species,and Hopf bifurcation in a predator-prey system with ratio-dependent response and time delays. Then present some boundedness and permanence results.Its main contents can be summarized as follows:The first section is introduction, in which we present research background, purpose and significance of predator-prey system, given the model of predator-prey research present situation and the results, Finally the organization of this paper is presented.In Section 2, we mainly discussed the positively and boundedness of the model with a transmissible disease in the prey species, and then sufficient conditions for the model of local and global stability of the boundary equilibria, persistence and extinction are obtained. By applying Descartes' rule of signs, we get the sufficient conditions for existence of interior equilibria.In Section 3, we mainly discussed the positively and boundedness of the model with a transmissible disease in the predator species. By evaluating the eigenvalues of their associated Jacobian matrices of system at the boundary equilibria, we are able to obtain the sufficient conditions of the local stability of these boundary equilibria. Global stability of subsystem be analysed. Meanwhile, we discussed the global stability of the boundary equilibria by the establishment of appropriate Liapunov function. In addition,the sufficient conditions of the persistence and extinction of populations are provided.By applying Descartes' rule of signs, we get the sufficient conditions for existence of interior equilibria. The interior equilibrium is locally asymptotically stable by applying Routh-Hurwitz criterion.In Section 4, we discussed the Hopf bifurcation in a predator-prey system with Ratiodependent response and time delays. We show that existence, positive and boundedness of the solution, and then sufficient conditions of permanence for system. By evaluating the eigenvalues of their associated Jacobian matrices of system at the interior equilibria, and the establishment of appropriate Liapunov function, we are able to obtain the sufficient conditions of the local and global stability of the interior equilibria without delay. Finally,provides some Hopf bifurcation analysis on the system with delays.