Stability Analysis Of Two Fractional Predator-prey Models With Two Delays And Incommensurate Orders

Fractional calculus is the generalization of integer-order calculus to arbitrary order.Due to the effects of memory and hereditary properties of fractional calculus,it is more accurate to describe the complex dynamic behavior of system rather than integer-order calculus.With the improvement of fractional calculus,it has a wide range of applications in physics,chemistry,electricity,economics,biology,economics and other fields,and has progressively become one of the research hotspots.Since both biological evolution and physical processes require a certain amount of time to complete,the development of system is not only dependent on the current state,but also related to the state in the previous period.Thus,time delay widely exists in nature.Populations in nature are not independent of each other,but are linked in some way.The interaction between prey and predator is one of them and plays an important role in maintaining the ecological balance of nature.The fractional predator-prey model with time delay is often used to explain the dynamic behavior of predator population,such as persistence,stability,existence of periodic solutions,and so on.The stability of system is the focus of many researchers.Therefore,it is meaningful to study the stability of fractional predator-prey models with time delays.This thesis study the stability and the existence of Hopf bifurcation of two kinds fractional predator-prey models with two time delays and incommensurate orders.The main work is as follows:In the first part,we consider a fractional two-delays predator-prey model with incommensurate orders and Holling-Ⅲ functional response.We discuss the stability of the system in the following four cases by taking delay as bifurcation parameter:(ⅰ)Whenτ1=τ2=0,the local stability of the positive equilibrium of system is analyzed by RouthHurwitz criterion.(ⅱ)When τ1>0,τ2=0,the critical value τ10 of Hopf bifurcation at the postive equilibrium of system is calculated by taking τ1 as bifurcation parameter.(ⅲ)When τ1=0,τ2>0,similar to(ⅱ),taking τ2 as bifurcation parameter,we can calculate the critical value τ20 of Hopf bifurcation at the postive equilibrium of system.(ⅳ)Whenτ1,τ2>0 and τ1≠τ2,using the method of stability switching curves,the feasible region of the two-delays system is calculated,then the stability switching curves is given.The directions of crossing of the eigenvalue on the stability switching curves are determined,hence,the stable region and the conditions of the existence of Hopf bifurcation of system are obtained.Finally,the accuracy of the given theoretical results is verified by numerical simulations.In the second part,we consider a fractional predator-prey model with two discrete and one distributed time delays and incommensurate orders.By introducing a new variable into the system,the system becomes a two-delays system with integer order and fractional order.By Routh-Hurwitz criterion,the method of stability switching curves and local Hopf bifurcation theory of fractional delay differential equations,we discuss the stability and the existence of Hopf bifurcation of the positive equilibrium of system in τ1=τ2=0 and τ1,τ2>0 and τ1≠τ2,respectively.Finally,some numerical simulations are carried out to illustrate the consistency with our results.