Dynamical Analysis About Two Types Predator-prey Model With Time Delay

Predator-Prey Model is an important research aspect of mathematical ecology,and there are a lot of factor impact species fluctuation,and time-delay is an important kind of factors.As we all know,Delay can have a sizeable impact for the ecosystem.Theoretical ecologists have generally found that in the interaction between population,delay is inevitable, and a longer time-delay system would undermine the stability of equilibrium position,even a number of unstable Phenomena,such as unstable equilibrium position and the cycle volatility can be interpreted as in the model generated by the introduction of time-delay consequences.In this paper,two kinds of typical predator-prey models with time-delay are studied,through analyzing,we found that the stability of time-delay model has a very important role.The key points are obtained as follows:1.The SIS model with time delay is estabished,the boundness of solution and the sufficient condition of locally asymptotically stable of the equilibria are studied,and the point obtained in the equilibrium is no Hopf branch.Furthermore,the global stability of the equilibria is obtained,when Border equilibria or Positive equilibria have the global stability of the situation,with the relative rates for different conditions.2.For a similar the SIS model with time delay,through analysis of the equilibrium point,we find that Time delayÏ„changes would give rise to the stability of the equilibrium point Changed under certain conditions.Whenτ∈[0,Ï„₀),Positive equilibrium point is asymptotically stable,whenÏ„>Ï„₀ is instability,whenÏ„=Ï„₀,the Hopf branch system.3.We set up the Holling-Leslie Model with delay,and the boundness of solution is discussed.For the time delayÏ„,The changes in the stability of equilibrium point result intop changes in Hopf branch.By constructing a suitable Lyapunov function,when under certain conditions,the positive equilibrium is globally stable.The above conclusions show that,delay on the predator-prey model has an important role in the the stability of the equilibrium poin.For ecological protection,you can adjust the time delay to ensure that predators prey species continue to achieve a permanent deposit.