Invasion Dynamics Analysis Of A Polyphagous Predator-prey Model In An Open Convection Environment

In this paper,the classical comparison principle,global bifurcation theory,the uniform persistence theory and numerical simulations are used to explore the dynamical behavior of a generalist predator-prey system in open advective environments.It turns out that compared with the specialist predator-prey model,the dynamics of this system are more complex.The detail contents list as follows:In Chapter 1,the known research results of predator-prey system under the environments with unidirectional flow are introduced.The existing results on predatorprey system are relatively rare comparing with the competitive system.We introduce the research progress of two main predation models(specialists as well as generalists),and further state the significance of our research problem.In Chapter 2,as preliminaries,we list some basic results which will be useful in the chapters follows,such as the related eigenvalue theory and the important results of the positive solution to the single species model.Some estimations on the positive solutions are also established.In Chapter 3,we give a comprehensive analysis of the dynamical behaviors in q-a plane,where q denotes the advection rate and a is the predation rate,by means of the global bifurcation theory and the uniform persistence theory.Firstly,we analyze the stability of the semi-trivial solutions;secondly,taking the predation rate a as the bifurcating parameter,we study the existence and uniqueness of positive steady states,which bifurcate from the semi-trivial solutions;finally,the global dynamical behaviors are investigated.It turns out that there exist two critical advection rates q1*,q2*and two critical predation rates a0(q),a*(q),which divide the dynamical behaviors into three or four scenarios:(i)survival of the prey only,or(ii)the predator only,(iii)coexistence,and(iv)extinction of two species.It is noteworthy that predators take-over(i.e.(ii))can not occur in the specialist predator-prey system.The effects of the diffusion rates d1,d2 on critical curves q=q1*,q=q2*,a=a0(q)and a=a*(q)are firstly shown in Chapter 4.The shape evolution of the critical curve a=a*(q),which separates persistence of generalist predators only from coexistence,is observed with the changing of diffusion rates.Sccondly,we present some numerical simulation results which complement the analytic results.Numerical results suggest that as the specialist predator-prey system,the random dispersals of both prey and generalist predators are dramatically in favour of the invasion of the generalist predators.Meanwhile,numerical results also illustrate significant differences between the specialist predator-prey system and the generalist predator-prey system,including the shape evolution of the critical predation rates,the shrinkage of the range of prey persistence alone and the existence of optimal advection rate for predators to take over prey.These interesting phenomena have important biological implications on the invasion of generalist predators in open advective environments.