Abstract
A delayed predator-prey system with stage structure is investigated. The existence and stability of equilibria are obtained. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the normal form and the center manifold theory. Finally, a numerical example supporting the theoretical analysis is given.
Highlights
The age factors are important for the dynamics and evolution of many mammals
Xt x t r − ax t − by[2] t, 1 mx t y 1 t kbx t y2 t 1 mx t v1 y1 t, 1.1 y 2 t Dy1 t − v2y2 t, Fixed Point Theory and Applications where x t denotes the density of prey at time t, y1 t denotes the density of immature predator at time t, y2 t denotes the density of mature predator at time t, b is the search rate, m is the search rate multiplied by the handling time, and r is the intrinsic growth rate
It is assumed that the reproduction rate of the mature predator depends on the quality of prey considered, the efficiency of conversion of prey into newborn immature predators being denoted by k
Summary
The age factors are important for the dynamics and evolution of many mammals. The rates of survival, growth, and reproduction almost always depend heavily on age or developmental stage, and it has been noticed that the life history of many species is composed of at least two stages, immature and mature, with significantly different morphological and behavioral characteristics. In 7 , they employ the theory of competitive systems and Muldowney’s necessary and sufficient condition for the orbital stability of a periodic orbit and obtain the global stability of the positive equilibrium for the general system. Time delay due to gestation is a common example, because generally the consumption of prey by a predator throughout its past history governs the present birth rate of the predator. We introduce the delay τ due to gestation of mature predator into system 1.2 and consider the following system: xt n x t − f x t y2 t , y 1 t kf x t − τ y2 t − τ − D v1 y1 t , 1.4 y 2 t Dy1 t − v2y2 t , where all coefficients are positive constants and the detailed ecological meanings are the same as in system 1.2.
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