Dynamics Analysis Of Two Types Of Predator-Prey Models With Crowley-Martin Functional Response

This paper discusses the dynamic behavior of two types of predator-prey models with Crowley-Martin functional response functions.The predator-prey model has always been an important research field in biomathematics,and the study of complex predator-prey models with Crowley-Martin functional response can help to understand the development of biological populations to a large extent.The first chapter of this paper mainly introduces the research background and current situation of the classic Lotka-Volterra predator-prey model,and sorts out different types of functional responses.In addition,it also introduces the special status of the Hopf bifurcation,delay term,and stage structure phenomenon in differential models and their development history.In chapter 2,we study a predator-prey model with the Crowley-Martin functional response function in which the densities of the predator population and the prey population differ greatly,and the stability of the system equilibrium point is analyzed by the characteristic root method.Then,by the Poincare-Andronov-Hopf theorem,we give the conditions for the Hopf bifurcation.We calculate the normal form limited on the central manifold and give the criterion determining the properties of Hopf bifurcation.Also,we investigate the existence of the Flip bifurcation.Finally,the above theoretical results are verified by numerical simulation,and the examples given are subcritical Hopf bifurcation and periodic solutions are stable.In chapter 3,we study a three-dimensional coccinellid-aphid model with stage structure in predator including maturation and gestation delays,in which coccinellids are predatory ladybugs,and its interaction with aphids is characterized by Crowley-Martin functional response.In this part,we analyze the equilibrium points of the non-delay system,and the local and global asymptotic stability of internal equilibrium points are investigated.For the delayed system,we not only analysis the stability of equilibria,but also study the existence of Hopf bifurcation.The two delays of maturation delay and gestation delay are selected as bifurcation parameters,and the properties of Hopf bifurcation are studied by using the normal form and the central manifold theorem.Finally,the numerical results are used to verify the theoretical results.In the fourth chapter,some biological conclusions are drawn based on the theoretical results above,and future work has prospected.