Analysis Of A Class Of Ecotoxic Diseases With Two Populations

Mathematical Biology is a frontier discipline overlapping bioscience and mathematics. It studies quantities and spatial structure in the life science by using mathematics theory and computer technology, investigates the essential characteristics of ecological system, and elaborates the law of biological information by mathematical analysis to biological experimental data.As we know, the epidemic is a serious threat of the human health and others species survival. Prevention and treatment of infectious diseases concern with hundreds of millions of people’s well-being. Epidemic dynamics is a special subject for infectious diseases. Firstly, based on the mechanism about the spread of infectious diseases, the natural environment and the related social factors, the mathematical models can be established to reflect the characteristics of infectious diseases. Secondly, one can analyze the causes of epidemic occurring and main factors of its spreading, one can also reveal the prevalence of disease by studying the key parameters, which affect the spreading or vanishing of infectious diseases, and then find the best strategy of prevention and treatment. But usually most papers only assume that epidemic occurs and transmits for one population. For example, the SIR model and the SEIR model.Since 16th century, population dynamics has become an important branch of mathematical biology. It mainly analyzes the relationships between one population and another one, or the relationships between the population and given environment, finding out the principle involving the quantity and structure of population. There are many models to study the population dynamics in mathematical biology, such as the Lotka-Volterra predator-prey model, competition model and mutualistic model.Infectious disease can spread among populations, however, only a few recent papers have dealt with this problem. Considering epidemic dynamics and population dynamics, this paper mainly studies two species eco-epidemics model which the diseases only in predator.In order to study two species eco-epidemics model which the diseases in predator with the diffusion, this present paper focus on key parameters in the survival and the extinction of the predator populations. Firstly, an ordinary differential equation (ODE) model is established and the uniform bound is given. We also present the existence conditions of four nontrivial equilibria and prove their local stabilities. We prove that the global stability of coexistence equilibrium by establishing the Lyapunov functional. Numerical simulations show that the contact rate, the susceptible and incubational predator capture rate have an important effect on the long-time behavior of populations. Furthermore, a partial differential equation (PDE) model is established by introducing the diffusion term to describe the movement of populations. The uniform bound is firstly given. We also get the sufficient conditions for the coexistence equilibrium and prove its local stability by the method of subspace decomposition. We also show the global stability of the coexistence equilibrium by establishing the Lyapunov functional. Our theoretical results imply that the disease will spread and the susceptible predator will extinct if the contact rate is big; while if the contact rate is small, the disease will vanish and the infective predator extinct, the disease will become endemic if the contact rate is moderation. Our numerical simulations also prove the theoretical results.