Dynamics Of Two Systems

In this paper, single population in a polluted environment and a non-autonomous multispecies ecological competition predator-pray system with Holling III type functional response are studied respectively.1. Study on the Single Population in a Polluted EnvironmentRecently, a first attempt to consider a spatial structure has been carried out by Buonomo, where a reaction-diffusion model is proposed to describe the dynamics of a living population interacting with a toxicant present in the environment (external toxicant) through the amount of toxicant stored into the bodies of the living organisms(internal toxicant). However, as the authors pointed out, even if the resulting model presents many features which make stimulating its study, such a modelling approach is a rough approximation to the biological phenomena at hand. In 1999, Buonomo et.al. viewed the internal toxicant as drifted by the living population and then, by balance arguments, they derived a PDE system consisting into two reaction diffusion equations coupled with a first order convection equation, and the corresponding ODE system was obtained as well. This model is the most realistic by now but Buonomo pointed out that the analysis of this system is rather difficult, especially with respect to the possible use of the comparison techniques. Therefore, Buonomo only used approximation methods and simple anatytical results and numerical simulations are given. Obviously, the more clear work is deserved to do.we mainly used the comparison theory and qualitative analysis to investigate the model made by Buonomo et al. Both the necessary and sufficient criterion of survival and the sufficient condition of the extinction are obtained. We also carry numerical simulations for two concrete models, which strongly support our theoretical studies. Finally, we establish sufficient criteria for the existence of a positive periodic solution.2. Study on the Multispecies Ecological Competition Predator-Pray System with Holling III Type Functional ResponseRecently, the ecological predator-prey systems with Holling III type functional response have been studied by many authors, their works are concerned with two or three species model. However, in nature, multispecies system is common and universal. Simu-taneously, it is more complicated and realistic than the two or three system.The main purpose of this paper is to study the persistence of the general multispecies competition predator-pray system with Holling III type functional response. In this system, the competition among predator species and among prey species are simultaneously considered. We obtain sufficient conditions for uniform strong persistence. Then we carrynumerical simulations for two concrete models, which strongly support our theoretical studies.