Dynamics Of Predator-prey System With Strong Allee Effect

The understanding of patterns and mechanisms of spatial dispersal of interactingspecies is an issue of significant current interest in conservation biology and ecology, andbiochemical reactions. A typical type of interaction is the one between a pair of predatorand prey, or more generally, a pair of consumer and resource. Mathematical models of theinteraction can be written as reaction-diffusion equations, ordinary differential equations,and functional differential equations, which are important in the research of nonlineardifferential equations. The methods of our studies include the classical dynamical systemtheory, analytical semigroup and topological methods, etc.For the predator-prey system with strong Allee effect in prey, the dynamical prop-erties of some special systems have only been obtained by numerical simulations. Thispaper aims to the complete analytical analysis. Unlike the Logistic growth, the strongAllee effect induces bistability, for which currently known methods cannot be appliedto, this paper greatly improves some methods and technical constructions. Details are asfollows:1. A completely global bistability analysis has been obtained for the system of ordi-nary differential equations, which is independent of spatial variables. By taking a compo-nent of positive coexistence equilibrium as parameter, two global bifurcation values aregiven: heteroclinic loop bifurcation point and Hopf bifurcation point. When the parameterchanges, the global stable zero equilibrium, the unique heteroclinic loop, the unique limitcycle as an alternative stable state, the coexistence equilibrium point as alternative stablestate are obtained successively. In addition, an improved Dulac criteria is given whenproving the nonexistence of periodic solution. Also a predator-prey systems with Alleeeffect could have even richer dynamic structure like multiple limit cycles. Our rigorousanalysis can be applied to most existing predator-prey models with strong Allee effect.Our analytical analysis are mostly new and complete, and the results do not depend onthe specific algebraic forms or parametrization of the nonlinear functions in the models.The analysis also provide new method and approach for the dynamical behavior of otherplanar systems.2. By constructing super-sub solutions for predator-prey system with strong Allee effect, we obtain the basic dynamical behavior, including the existence of global solutionand asymptotic behavior. Also the a priori estimate bounds of solutions and the spatial ho-mogeneous and nonhomogeneous periodic solutions are given. Especially, a large enoughinitial predator population will always lead to the extinction, i.e. the convergence to thesteady state (0, 0), which implies that (0, 0) is always a locally stable steady state withbasin of attraction including all large ??0 for a given ??0. Thus the system is bistable (ormulti-stable) if there is another locally stable steady state solution or periodic orbit. More-over, if the amount of initial prey population is less than the threshold value, (0, 0) willalways is globally asymptotically stable, which is also the characteristic of strong Alleeeffect. All the results show that the strong Allee effect essentially enlarges the complexityof spatial-tempo behavior of predator-prey reaction-diffusion systems.3. The elliptic system independent of time is analyzed in detail. A priori estimatesand nonexistence of positive steady state solutions are obtained. The stability and bifur-cation of semi-trivial steady state solution are given thorough. It is difficult to obtain thepositive lower bound of positive steady state solutions since the bistability deduced bystrong Allee effect and the semitrivial steady state solutions, the classical Leray-Schauderdegree cannot be applied to obtain the nonconstant positive steady state solutions. There-fore we make use of the global bifurcation theory generalized by Shi and Wang[1] to getthe nonconstant positive steady state solutions, i.e. spatial pattern formation.4. The stability and bifurcation analysis of the functional and partial functional dif-ferential equations with two discrete delays are considered. The delays will increase theinstability of predator-prey system with strong Allee effect. Based on this, the in?uenceof diffusion is considered. Since the appearance of Laplace operator, the characteristicequation of linearization equals to a sequence of countably many transcendental equa-tions. While each transcendental equation may generate countable Hopf bifurcation criti-cal value ?? , which makes the ordering of the critical value sequence difficult. The paperclarifies the sequence of critical value ?? corresponding to eigenvalues of the Laplace oper-ator. Moreover the spatial homogeneous and nonhomogeneous Hopf bifurcation periodicsolutions and their properties are analyzed.