The Qualitative Analysis Of A Family Of Predator-prey System With Holling Type-IV Functional Response

In population dynamics,a predator-prey system with Rolling type-IV function response is the following system,which is used to model the phenomenon: " group defense". ([3],[4], [6], [7], [8]) ,here,x(t) å’Œy(i) are functions of time representing population densities of prey and predator,respectively. r, K, n, D, a and 6 are all parameters which have biological meanings,.^ is the carrying capacity of the prey,D is the death rate of the predator,r is the maximum growth rate of the prey,/z is the maximum predation rate,a is the so-called half-saturation constant,and x/a+bx+bx2 is Holling Type-IV functional response function.So r, K, a, p. and D are all positive parameters. From the point of view of biology, we suppose x(t) > 0, y(t) > 0. The parameter b is such that the denominator of above system does not vanish for non-negative x,in fact,we only need suppose system(l)has very strong biological background and challenging complex dy-namics,so many biologists and mathematicians are interested in it.In late decade,a lot of articles are concerned about it or more general system, and some interesting dynamical phenomena are found: " paradox of enrichment " , for example,and the existence of homoclinic orbit,etc.(see [3], [4], [6], [7], [8]). We study the global qualitative analysis depending on all parameters, and get the partition of parameter space. We show the global dynamics of system (1) for different parameters.This article consists of two sections,in the first section,we study the global dynamics of system(l) when b > .Depending on the global qualitative analysis of parameters, we show that the system exhibits " paradox enrichment " for some parameters,a global stable attractor (ie,positive equilibrium)for some parameters and a unique stable limit cycle for some other parameters.The bifurcation analysis of system (1) indicates that it exhibits numerous kinds of bifurcation phenomena including the saddle-node bifurcation.Hopf bifurcation,and the homoclinic bifurcation. When and we show that system (1) has the cusp bifurcation of codimension 2(ie,Bogdanov-Takens bifurcation); when ,we show that this system has the cusp bifurcation with codimension at least S.Then we choose two parameters as the bifurcation parameters and show that system (1) exhibits the Bogdanov-Takens bifurcation,which proves the existence of homoclinic.In the second section,we study the multiplicity of weak focus in system (1) when b = 0,then system (1) becomes:In[7],the authors show that system (2) has rich global dynamics depending on all parameters,including a weak focus with multiplicity at least 2,but the authors didn't calculate the multiplicity.In the second section of this article,we calculate the order of the weak focus with formative series method and Mathematica 4.0,and show that the order of the weak focus is two for some parameters.We also get the stability ,then by Hopf bifurcation theorem,we show that system (2) has at least two limit cycles for some parameters.