Several Nonlinear Dynamical Systems On Biology In A Polluted Environment

This paper consists of three parts.Firstly, we consider a chemostat model with impulsive toxicant input and nutrient recycling.By Floquet theory and small amplitude perturbation method, we obtain the sufficient conditionfor the global asymptotic stability of boundary periodic solution. Furthermore, the sufficientcondition for the permanence of the system is obtained. The results show that poisonousenvironment will lead the microorganism species to be extinct.Secondly, we discuss a predator-prey system with antagonistic effect for prey. Bycharacteristic equation, we obtain the sufficient conditions for the local asymptomatic stability ofthe equilibria. By iterative sequence, the conditions for the global asymptomatic stability of theequilibria are obtained. Finally numerical simulations are carried out to illustrate our results.Thirdly, we study an SI epidemic model in a polluted environment. By the method ofaverage analysis and theory of impulsive differential equation, two thresholds between weakaverage persistence and local extinction for susceptible and infected population are obtained. Ourresults indicate that the amount of toxicant input in a pulse period will directly affect bothsusceptible and infected population. Numerical simulations are shown to prove that our resultsare correct.