The Dynamics Of A Predator-Pray Model With Double-Delays

With the increasing lack of resources, people's environmental awareness is growing. The extinction of animals and plants and the continuous survival of the ecosystem have become the focus of social attention. How many enemies should we put into the ecological system to enable wildlife to continue to survive? How should the fishermen control the fishing rate in order not to prevent fishes from breeding? For these reasons, the research on Predator-prey system is of high practical value.In ecological system, the existence of delay influences the change rule of population to different degrees. For example, the growth from cubs into adulthood needs a duration of time, and the prey also needs some time to transform the food into self-growth. The predator-prey's pregnancy and breast-feeding are also important factors. Predator-prey model with delay can make a reasonable and more accurate prediction In the real world population trends.This paper takes predator-prey systems with double delay as its research object. Firstly, the background and the present situation of the predator-prey model are researched. Secondly, we respectively chose time delayτ₁ ,τ₂as parameter, and analyze the stability of the predator-prey model with double delay. The specific practices are as follows: first, making a delay parameterτ₁ = 0and chosingτ₂=τas parameter, according to the distribution of the linear root which closes to the system balance to find out the stability of the system balance and the conditions of existence for Hopf-bifurcation; second, when the delay parameterτ₂=τwithin the stability region, we makeτ₁ as parameter, by the same way we get the stability of the system balance and the conditions of existence for Hopf- branch; third, using the theory and normal form method in manifold center to give Hopf-bifurcation's branch direction and the specific calculation formula of the stability of periodic solution. At last, a numerical simulation is provided to further support our conclusion.