Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Abstract

In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

“…Secondly, compared with continuous-time models, the advantage they offer is that they are generally more direct, more convenient and more accurate to formulate. Thirdly, recent works have shown that for discrete-time models the dynamics can produce a much richer set of patterns than those observed in continuous-time models (Busłowicz, 2010;Busłowicz and Ruszewski, 2012;Duda, 2012;Feedman, 1980;Holling, 1965;Huang and Xiao, 2004;Hsu, 1978;Raja et al, 2011;Xu et al, 2011;Zhang et al, 2011). At last, we can get more interesting dynamical behaviors and more accurate numerical simulations results from the discrete-time models; moreover, numerical simulations of continuous-time models are obtained by discretizing the models.…”

Contribution of time delays to p53 oscillation in DNA damage response

“…It is widely suggested that negative feedback loop can induce biochemical oscillation [9–12], and the functional mechanism of such feedback loop in the p53 network has been studied by many researchers [6, 13–15]. It was reported that p53 dynamics depend on the p53‐murine double minute 2 (Mdm2) and ataxia telangiectasia mutated (ATM)‐wild‐type p53‐induced phosphatase 1 (ATM‐p53‐Wip1) negative feedback loops [16].…”

Dynamics of a predator-prey system with stage structure and two delays

“…Predator-prey systems are very important in population dynamics and have been investigated by many authors [2,4,14,15,18,23,24]. It is well known that there are many species whose individual members have a life history that takes them through immature stage and mature stage.…”