Persistent Existence And Stability For Several Classes Of Mutualism Models

Persistent existence and stability for several classes of mutualism models are studied in this paper. Such as, the stability of equilibrium,the persistence of ecology, the existence, stability and attractivity of periodic solutions,the existence for Hopf bifurcation, and so on.This is an important direction of mathematical ecology research,many ecologist and mathematician have paid much attention in this field. The main method is using qualitative theory of differential equation.In Chapter 1,We introduce the background of this article.In Chapter 2,We investigate a mutualism model with saturating terms and the effects of toxic substance, Based on the comparison theorem,the permanence of the system is considered, local or global stability of the positive equilibrium is studied by establishing appropriate method,nonexistence of the limit cycle is also discussed. An example of numerical simulation is given to explain the conclusions.In Chapter 3,A non-autonomous mutualism system with stage-structure is dis-cussed.The sufficient condition of the permanence for system is given. Furthermore, the existence and global asymptotical stability of periodic solution are proved by using Brouwer fixed point theorem and Liapunov function for periodic system. At last,an example is given to explain our conclusions.In chapter 4, A mutualism model with delay is investigated.Based on the com-parison, the permanence of the system is considered.The local stability of positive equilibrium and existence of Hopf bifurcation with delayΤas parameter is estabil-ished.Finally, to verify our theoretical results,some numerical simulations are also included.