Study On The Dynamic Behaviors Of Three Kind Of Population Dynamic Systems

This paper consists of the following three parts:Firstly, we propose and study a stage-structured model with toxicant and harvesting.By applying Bendixson-Dulac theroem, we show that the system admits no limit cyclein the first quadrant, thus the global stability of the interior equilibrium is followed. Theexistence of the bionomic equilibrium is studied. After that, we consider the system withtime-varying harvest effect, sufficient conditions are obtained for the global stability ofbionomic equilibrium by constructing a suitable Lyapunov function. The optimal policyis also investigated by using Pontryagin’s maximal principle. Some numeric simulationsare carried out to illustrate the feasibility of the main results.Next, a predator-prey system with refuge, toxicants and harvesting is proposed andstudied. By constructing a suitable Lyapunov function, the globally stability of theequilibriums is proved. By using the methods on mathematical analysis, we consider theimpact of refuge, toxicants and harvesting on the system, respectively. Numericalsimulations imply the feasibility of the main results.Finally, we propose a more reasonable three species model. By applying thecomparison theorem of differential equation, a set of sufficient conditions which ensurethe permanence of the system is obtained. We also investigate the extinction of thepredator species. Some numerical simulations are carried out to show the feasibility ofthe main results.