Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion

Abstract

In this paper, the dynamics of a class of modified Leslie-Gower model with diffusion is considered. The stability of positive equilibrium and the existence of Turing-Hopf bifurcation are shown by analyzing the distribution of eigenvalues. The normal form on the centre manifold near the Turing-Hopf singularity is derived by using the method of Song et al. Finally, some numerical simulations are carried out to illustrate the analytical results. For spruce budworm model, the dynamics in the neighbourhood of the bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations. Then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found, which can be used to explain the phenomenon of periodic outbreaks of spruce budworm.

Double Hopf Bifurcation in Delayed reaction–diffusion Systems

“…Song et al [49] investigated the Turing-Hopf bifurcation from the point of view of analysis of bifurcation and normal forms, in which they extended the method given in [19,58], and deduced the normal form with parameters of Turing-Hopf bifurcation. Xu and Wei [56] applied this method to study the Turing-Hopf bifurcation of a predator-prey model. Turing-Hopf bifurcation can be seen as a special case of Hopf-zero bifurcation of dynamic system from the point of view of classification of bifurcation.…”

“…where I 3 is the 3 × 3 identity matrix and M n = − n 2 l 2 D, n ∈ N 0 . That is, each characteristic value λ is a root of an equation (λ − αe −dτ e −λτ + 2βy * + d 3 n 2 l 2 ) · ∆ n (λ, τ ) = 0, (56) where ∆ n (λ, τ ) = λ 2 + (d + d 1 n 2 l 2 + d 2 n 2 l 2 )λ + d 2 n 2 l 2 (d + d 1 n 2 l 2 ) + e −λω (µI * λ + µI * d 2 n 2 l 2 + µI * d),…”

Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel

“…Recently, to show an accurate dynamic classification at this singularity, Song et al [54] applied the normal form theory proposed by Faria [18] to a general reaction-diffusion equation, and obtained a series of explicit formulas for calculating the normal forms associated with the Turing-Hopf bifurcation. This spatiotemporal dynamics induced by the Turing-Hopf bifurcation were observed in several reactiondiffusion models [4,53,59], see also [1,50] for the reaction-diffusion system with delay. Another typical codimension-two bifurcation is the double Hopf bifurcation.…”

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

“…The innovation of this work lies in the complete analysis of Hopf bifurcation, Turing instability, Turing-Hopf bifurcation, and double Hopf bifurcation. In previous work, the authors often studied one type of codimension-two bifurcations, such as Turing-Hopf bifurcations or double Hopf bifurcations [2,42,36]. In our work, the time delay in prey growth and the diffusion of population could induce the emergence of both two types of codimension-two bifurcations.…”