Stability,bifurcation And Simulation Of Two Predator-prey Systems

The bifurcation theory of differential equations not only plays an important role in the field of applied mathematics,but also has many applications in other disciplines and even life.Therefore,the study of bifurcation problems have both theoretical and practical significances.The bifurcation problems exist not only in continuous systems but also in discrete systems.Moreover,discrete systems have more dynamical behaviors than the continuous systems.In this thesis,on the basis of^[1]and^[2],the branch problems of codimension 1 and codimension 2 of two discrete systems are further studied.In known literature^[1],we think that the stability given by the author has obvious limitations,and find its shortcomings by giving two counterexamples.We propose a new stability theorem,and find that flip bifurcation of codimension 1 and 1:2 strong resonance bifurcation of codimension 2 occur in the system at the positive equilibrium point.We not only give the detailed formula derivation,but also use Matlab to carry out the numerical simulation to prove the conclusion,and give the phase diagram,bifurcation diagram,and Lyapunov exponential diagram respectively.In known literature^[2],the author proved that the system can produce fold,flip,Neimark-Sacker and B-T bifurcation,and gave corresponding parameter conditions.However,in^[2],the author assumed the parameter D=1 and adopted Euler method to take step size h=1,which violated the accuracy requirement.Therefore,in Chapter 3,we first simplify the continuous system without assuming the parameter D=1;then the semi-discretization method is adopted without assuming h=1 for the corresponding system;after that,not only the stability condition,but also the condition of the supercritical bifurcation and 1:1 strong resonance bifurcation are given;Finally,numerical simulation is given to verify the theoretical results.In Chapter 4,we consider the modified Leslie Gower predator-prey model,giving the parameter conditions of stability,and find that the system has Flip bifurcation and Neimark-Sacker bifurcation,and combine numerical simulation to prove our existing conclusions.In the system studied in Chapter 2,specific conditions for fold-flip bifurcation have not been given,and there is no good control method for the bifurcation to chaos in the system studied.These problems are the main work to be considered in the future.