The Stability Research For Two Classes Of Population Systems With Toxins

With the development of economy and society,human over-exploitation and utilization of natural resources has caused a series of environmental pollution problems.The deterioration of the environment has led to an increase in toxins deposited in biological individuals,which seriously threatens the survival of the population and even destroys the ecological balance.Therefore,How to effectively control the influence of toxins on populations has become a topic that people urgently need to study,and using the mathematical stability theory to study the mathematical model,the survival status of the populations can be accurately analyzed,so as to achieve the purpose of saving the endangered species and controlling the growth of the harmful populations.This paper mainly studies the stability of two kinds of population systems with toxins.In the first chapter,we introduce the background and significance of the topic selection.The research status of bait-prey system,cooperative system and population system with toxin are described.In the second chapter,the optimal harvesting problem for a prey-predator system with prey refuge in the presence of toxicity is studied.By using the characteristic value of Jacobi matrix and constructing appropriate Lyapunov function,the local stability and global stability of equilibrium point of the system are proved.Moreover,the equilibrium solution of optimal harvesting is obtained by using the Pontryagin s maximal principle.In the third chapter,the stability of a mutually-beneficial symbiosis survival cooperative sys-tem with feedback control under the influence of toxins is analyzed,and the local and global stability of the positive and boundary balance points of the system are proved by Jacobi matrix and the appropriate Lya.punov function.The global stability,and an example verify that when the system has a balance point,the feedback control does not affect the stability of the positive equilibrium point,only the position of the positive equilibrium point is changed.