The Study on Predatorprey Models with Allee Effect and Nonlinear Density Dependent 

Author  LuoLiGui 
Tutor  ZhengWeiWei 
School  Xi'an University of Engineering 
Course  Applied Mathematics 
Keywords  nonlinear density response Allee effect refuge effect limit cycle uniqueness persistent existence 
CLC  O175 
Type  Master's thesis 
Year  2012 
Downloads  24 
Quotes  0 
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 predatorprey 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 predatorprey system model and discuss itfully.The first kind is a predatorprey system model with nonlinear density dependent， thissection separately studies:(1) the autonomy predatorprey 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 predatorprey 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 predatorpreysystem 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 predatorprey system model with Allee effect and refuge effect, thissection studies:(1) a autonomy predatorprey 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 predatorpreysystem 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 predatorprey 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 predatorprey systems which consist of six models, which enrich the predatorprey 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 predatorprey 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 thepredatorprey system model with Allee effect and refuge effect, we can further study theuniqueness of limit cycles and the existence of the boundary cycle.