文档介绍：Qualitative Analysis of a Delayed and Stage-Stuctured Predator-Prey System
Lingshu Wang1 Guanghui Feng
School of Mathematics and Statistics, Department of Mathematics,
Hebei University of Economics & Business Shijiazhuang Mechanical Engineering College,
Shijiazhuang, Shijiazhuang,
wanglingshu@ fengguanghui@
Abstract—A delayed and stage-structured predator-prey system density of the predator at time t ; τ≥ 0 representing a time
with Holling type-II functional response is discussed. By using representing a time delay due to the gestation of the predator;
the normal form theory and center manifold theorem, the linear the parameters aa, , a , bb, , mrr, , , r are positive
stability of the system is investigated and Hopf bifurcations are 12 1 12
established. Formula determining the direction of bifurcations constant; x /(1+ mx ) is the Holling type-II response function.
and the stability of bifurcating periodic solutions are given.
Numerical simulations are carried out to illustrate the theoretical II. STABILITY AND HOPF BIFURCATION
results. In this section, we discuss the stability of the positive
Keywords- time delay; stage-structure; predator-prey system;
stability; Hopf bifurcation. equilibrium and the existence of Hopf bifurcations for (1) with
time delay τ as a parameter. Assume
−> −−+> +
I. INTRODUCTION ()H12