Research On A Multiple-delayed Predator-prey System With Habitat Complexity And Harvesting Effects

Based on the predator-prey system with Holling type functional response7,this paper incorporates the effect of habitat complexity,time delay and harvesting effort and proposes a multiple-delayed predator-prey system with habitat complexity and harvesting effort.In this paper,the dynamic behavior of the system is studied by using the relevant knowledge of dynamical system and some numerical simulations are done to prove the results of theoretical analysis.Firstly,a kind of multiple-delayed predator-prey system with habitat complexity and linear harvesting effort is studied,including the positivity and boundedness of the solu-tions,the stability of equilibrium and the properties of the bifurcating periodic solutions.The positivity and boundedness of the solutions that satisfy initial conditions are verified when we didn't consider the effect of time delay.In the following,it is observed that the positive equilibrium will be changed from unstable state to globally asymptotically stable state when the complexity of the habitat reached a certain threshold.Based on this,we also considered the effect of time delay in three cases.The effect of time delay on dynamic behavior is studied and the conditions which guarantee the existence of Hopf bifurcation are obtained.And the explicit formulae for determining the properties of the bifurcating periodic solutions are derived by using the central manifold theorem and the normal form theory,including the direction of Hopf bifurcation,the stability and period of the bifurcating periodic solutions.In order to verity the results of theoretical analysis,some numerical simulations are done.Secondly,a kind of predator-prey system with habitat complexity and nonlinear harvesting effort is studied,including the stability of positive equilibrium and optimal taxation policy.The stability of positive equilibrium is analyzed by using the relevant knowledge of dynamical system and the optimal tax is obtained by Pontryagin's maximum principle.Finally,some numerical simulations are done by Matlab and the effect of taxation policy on the system is analyzed.