Influence Of Prey Refuge On The Stability And Species Evolution Under The Delayed Leslie-Gower Type Predation-prey System

The paper describes a delayed predator-prey system with modified Leslie-Grower scheme,in which the time delays are regarded as bifurcation parameters.The functional response is considered to be of Holling-type II and incorporates a constant proportional prey refuge.The system is analyzed from the point of view of persistence and stability for this particular functional response.The asymptotic stability of coexist equilibrium is examined by using constructed Lyapunov function.The delayed system is analyzed with focusing on the gestation delay of predator .The occurrence of Hopf bifurcation in the neighborhood of the positive interior equilibrium is investigated.Then,numerical simulations are carried out to verify the theoretical results of the paper.On the process of prey species evolution,the evolutionary singular strategy of prey is worked out and is proved to be an evolutionary stable strategy.The direction of body size trait evolution is influenced by the maximal of population growth rate and maximal catching ability of predator species.The analysis result shows if the prey species can have much more bigger growth rate at the minor body size,the stable strategy will choose the smaller trait even though it will easily be catched.Which is contrary to catching ability,that makes the prey favour bigger trait value.Refuge has great significance to prey because investing on avoiding predator is economic than struggle to escape from them.