Two Categories With Holling Type-iv Prey - Predator System Qualitative Analysis

The Holling type-IV functional response function reflects a phenomenon which the concentration of the biological nutrient base not only can't promote the growth of creatures, but also can be holded-up. The proposition of the Holling type-IV functional responsing function is very significant for the research on prey-predator system of biological population. There are two kinds of the present Holling type-Ⅳfunctional responsing function:(1) the simplifiedⅣtype:φ(x)=(mx)/(a+(x~2)); (2) the generalⅣtype:φ(x)=(mx)/(a+(bx)+(x~2)).This paper is devoted to discuss the qualitative theory and the dynamics properties of the two Holling-IV type predator-prey systems.First of all, we introduce the background and the current situation of the Holling-Ⅳtype functional responsing function. For the more, the qualitative analysis on simplified Holling type-Ⅳfunctional response system under sparse effect is considered. By using the qualitative theory of differential equation, the system with Holling type-Ⅳfunctional response under sparse effect is investigated. The positive boundedness of solutions and the global stability of the system are discussed, the sufficient conditions for the existence and uniqueness of the limit cycle are obtained, and the prey density dependent function of the system in Wang Ji-hua's paper is extended to the four order. This part of the result obtained about such a system of the existing has been enriched and popularized. At last, the predator-prey with delay Holling type-Ⅳfunctional response is studied, and the predator's numerical response has Leslies form. The time delay as the parameter is chosen, the sufficient condition of positive equilibrium point is obtained and the existence of the Hopf bifurcation is also found by ordinary differential and stability method. Then this research methods are applied to the predator's numerical response have general form of Holling type-IV. Thus discovery system in the positive equilibrium point also can produce the near Hopf branch phenomenon. And from two aspects of form and meaning are compared Leslies type and Holling type-IV. This branch of the system is the promotion of the studied existing system.