Hopf bifurcation analysis in a predator–prey model with two time delays and stage structure for the prey

Abstract

In this paper, a stage-structured predator-prey model with Holling type III functional response and two time delays is investigated. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are studied. Based on the normal form method and center manifold theorem, the explicit formulas are derived to determine the direction of Hopf bifurcation and the stability of bifurcating period solutions. Finally, the effectiveness of theoretical analysis is verified via numerical simulations. This study may be helpful in understanding the behavior of ecological environment.

Hopf bifurcation analysis in a predator–prey model with time delay and food subsidies

“…We further considered the global continuation of local Hopf bifurcations and obtained the global existence of periodic solutions. Usually, the conversion delay induces Hopf bifurcation and destabilizes the equilibrium, such as reported by [7]. Our theoretical findings suggest that incorporating food subsidy into a prey-predator model leads to a large stable interval of the coexistence equilibrium.…”

“…In the year 2012, Nevai et al established a model about prey, predator, and subsidies [5] and discussed the dynamical behavior of such systems. Recently, time delay effect has been proved to significantly affect the dynamics of predator-prey models [6,7]. Thus, in this paper, we consider the effect of time delay τ on the model given in [5], which leads to the delay differential equation model ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dx(t) dt = rx(t) (1 -x(t) K ) -θx (t) x(t)+s(t)+e y(t), ds(t) dt = iγ s(t) -ψs (t) x(t)+s(t)+e y(t), dy(t) dt = [ εθx(t-τ )+ηψs (t-τ ) x(t-τ )+s(t-τ )+e ]y(t)δy(t),…”

Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

“…By the similar argument as Corollary2.7 of [26], we knoww that the eigenvalue with positive real parts of (12) for large n are determined by the roots for equation (20). Then we study the distribution of the roots of the limiting equation (20). Suppose that λ = ϱ ± iσ are the roots of ( 20 ), and…”

“…Based on the research ideas in [2], Chen and Shi [5], Su et al [30] studied the dynamic behavior near the non-constant steady-state solution of the time delay diffusion logistic equation. And other investigation of bifurcation analysis with time delay, such as Xu et al [36], Zuo and Wei [38], Yang [37], Peng and Zhang [20], Wei and Wei [34].…”

Bifurcation analysis and control of a delayed stage-structured predator–prey model with ratio-dependent Holling type III functional response

“…Llibre and Vidal (2016) studied a predator-prey model with stage structure for the prey, and they proved that the limit cycles when the unique equilibrium point at the positive octant is unstable. Peng and Zhang (2018) also considered the influence of stage structure and derived the relevant nature of Hopf bifurcation.…”