期刊：
Journal of Mathematical Biology,2001年43(3):268-290 ISSN：0303-6812
通讯作者：
Xiao, DM
作者机构：
[Xiao, DM] Cent China Normal Univ, Dept Math, Wuhan 430079, Hebei, Peoples R China.;Dalhousie Univ, Dept Math & Stat, Halifax, NS B3H 3J5, Canada.
通讯机构：
[Xiao, DM] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Hebei, Peoples R China.
关键词：
Attractor;Critical point of higher order;Limit cycle;Predator-prey system;Ratio-dependent response
摘要：
Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey interactions where predation involves serious searching processes. However, such models have set up a challenging issue regarding their dynamics near the origin since these models are not well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey system at the origin in the interior of the first quadrant is studied. It is shown that the origin is indeed a critical point of higher order. There can exist numerous kinds of topological structures in a neighborhood of the origin including the parabolic orbits, the elliptic orbits, the hyperbolic orbits, and any combination of them. These structures have important implications for the global behavior of the model. Global qualitative analysis of the model depending on all parameters is carried out, and conditions of existence and non-existence of limit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.
作者机构：
[Zhang, XA] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;Acad Sinica, Inst Math, Beijing 100080, Peoples R China.;Bar Ilan Univ, Fac Life Sci, IL-52900 Ramat Gan, Israel.
通讯机构：
[Zhang, XA] C;Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.
关键词：
Permanence;Extinction;Threshold of the harvesting;Maximum sustainable yield
摘要：
In this paper, we establish a mathematical model of two species with stage structure and the relation of predator-prey, to obtain the necessary and sufficient condition for the permanence of two species and the extinction of one species or two species. We also obtain the optimal harvesting policy and the threshold of the harvesting for sustainable development.
