The Study on Predator-prey Models with Allee Effect and Nonlinear Density Dependent
|School||Xi'an University of Engineering|
|Keywords||nonlinear density response Allee effect refuge effect limit cycle uniqueness persistent existence|
Mathematics bionomics is the most complete and systematic branch of the mathematicalbiology. with the rapid development of ecological mathematics, its researchs of the dynamicproperties concerning the predator-prey systems have become the focus of attention, fothermore, biologists and mathematicians arrived at a compromise agreement on an importantissue, in this paper, we consider the differences between the nature species and humanintervention, and other comprehensive factors, for more objective reflect predators and preyspecies interactions, we construct two kinds of predator-prey system model and discuss itfully.The first kind is a predator-prey system model with nonlinear density dependent， thissection separately studies:(1) the autonomy predator-prey system model of prey species havenonlinear density dependent and HollingⅢ functional response, equilibrium and limit cycleare discussed by using qualitative stability theory of differential equations, and when positiveequilibrium is unstable，the sufficient conditions of the existence and uniqueness of limitcycle got is obtained;(2) nonautonomous predator-prey system model with nonlinear densitydependent for predators and prey, the sufficient condition persistent existence is obtained, andthe sufficient conditions of the existence and uniqueness of the periodic solutions is obtainedby using the method of constructing Lyapunov function;(3) nonautonomous predator-preysystem model with stage structure for predators and prey, the sufficient condition of theexistence of periodic solution is obtained by the coincidence degree theory.The second type is a predator-prey system model with Allee effect and refuge effect, thissection studies:(1) a autonomy predator-prey system model with Allee effect and refugeeffect, the stability of equilibrium point and existence and uniqueness of limit cycle isdiscussed by differential equation qualitative stability theory;(2) a autonomy predator-preysystem model with Allee effect and HollingII function response, the stability of equilibriumpoint is discussed and the sufficient conditions of the existence of limit cycles is obtained.(3)a autonomy predator-prey system model with Allee effect and HollingⅢ function response,the stability of equilibrium point is discussed and the sufficient conditions of the existence oflimit cycles is obtained.This paper studies two kinds of predator-prey systems which consist of six models, which enrich the predator-prey system model theory. Refuge effect has a positive impactthrough the data simulation confirm. Theoretical foundation is provided for protecting rareand endangered species to maintain ecological balance as well as safeguarding biodiversity. Inreality, the research of the two kinds of predator-prey system models can provided effectiveways of guiding the protection for rare vegetation and fauna, especially for endangeredspecies. Because of the time limit, in the later study, for the system model with nonlineardensity dependent, we can make the density dependent generalization; and for thepredator-prey system model with Allee effect and refuge effect, we can further study theuniqueness of limit cycles and the existence of the boundary cycle.