Qualitative Analysis Of The Chemostat Model With Different Removing Rate

In this paper,an annular model which included competition relation and predator-prey relation is discussed.Under the conditions that different populations with different dilution rates and the parametersδ_i(i = 1,2,3)of the consume rates are defined by constants or a linear function,the quality of the system is analyzed with the qualitative theorem of ODE.When all the parametersδ_i(i = 1,2,3)of the consume rate are defined by constants, the existence and the local stability of the semi-trivial equilibrium are considered.By constructing a Lyapunov function the global stability of the the semi-trivial equilibrium is proved;On the assumption that m₃ = D₂ and a₁m₂//a₂m₁>D₂/D₁,the existence of the positive equilibrium and the local stability of the positive equilibrium are considered. In the end,the uniform persistence of the system is analyzed.Under the conditions that one of the parametersδ_i(i = 1,2,3)of the consume rate is defined by a linear function.The existence and the local stability of the semi-trivial equilibrium are considered.The global stability of the the semi-trivial equilibrium is proved;the existence of the Hopf bifurcation is considered and the related equilibrium is proved to be one order's fine focus,further the stability of the periodic equilibrium is proved.Considering the model of the parametersδ_i(i = 1,2,3)of the consume rate is defined by a linear function.The existence and the local stability of the semi-trivial equilibrium are considered.The global stability of the the semi-trivial equilibrium is proved.Numerical simulation of the corresponding equlibria's existence and stability is presented by Matlab.