Analysis For The Dynamic Behaviors Of Predator-prey Model With Harvesting

Predator-prey system plays an important role in ecology, the dynamic property of this model is studied by many scholars. By analyzing the predator-prey model we know that it can guide people to protect the environment, meet the largest demand of people and maintain the rapid development of the population without population extinction and ecological balance destroy. Thus it has an important significance to protect the ecological environment and sustain the development.Because predator needs some time to digest after capturing prey and the human’s trapping is interrupted, introducing delay to the system can better reflect the actual situation. This paper studies a predator-prey model with two ratio-dependent response functions. The first part of this paper is to analyze the predator prey model without time delay. Positivity and boundedness of the solution are discussed in this paper under the initial conditions and we also obtain the equilibrium point of this model. Then we study the stability of each equilibrium point of this model, analyze the sufficient condition of Hopf bifurcation and positive equilibrium point when τ=0and τ≠0. The related calculation expression which decides the Hopf bifurcation direction and stability of periodic solutions on the positive equilibrium is given, by using the center manifolds theorem from Hassard and normal forms theory. The second part takes external capture delay into consideration and we analyzes the positivity and boundedness of the system solution. Stability of the equilibrium and the condition of local Hopf bifurcation in delay system are also analyzed in this part. Based on the above theorem, the Hopf bifurcation direction, stability of periodic solutions and other related expressions are obtained. Analyze the impact of predator-prey system when external delay is introduced. Finally, numerical simulation of a set of parameters can support our theoretical analysis based on Matlab software.