Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

Abstract

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

Dynamic Exploration and Control of Bifurcation in a Fractional-Order Lengyel-Epstein Model Owing Time Delay

“…Considering that there are different feedback delays in the development of the concentrations of both chemical reactants, Li and Zhang [23] built the following delayed Lengyel-Epstein model:…”

Existence of positive periodic solutions for a predator-prey model

“…By using the iterative technique and further precise analysis, sufficient conditions on the global attractivity of a positive equilibrium for a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge were obtained [4]. On the other hand, many researchers have considered delayed prey and predator models [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. For example, Nindjin et al have discussed the following delayed predator-prey model [5]:…”

“…It is too hard to discuss equation (1.8). To my best knowledge, even if for a two delays bifurcation equation, it was considered three cases: (i) τ 1 = 0, (ii) τ 1 = τ 2 , (iii) τ 1 ∈ (0, τ 0 ) was fixed, those cases indicate that there is one delay in the bifurcation equation (see [7][8][9][10][11][12][13]). By means of the extended Chafee's criterion, the present paper investigates the existence of periodic solutions for the model (1.7).…”

Patern Formation and Diffusion-Driven Instability of a Modified Lengyel-Epstein Model

“…In [18], conditions for Hopf bifurcation and local stability of equilibrium point of Lengyel-Epstein model involving time delay in the self-decomposition of the activator were examined and some numerical simulations for certain examples was performed by Zhang and He. In [9], Lengyel-Epstein model with two different time delays was studied, moreover the existence and direction of Hopf bifurcation and stability of periodic solutions were determined by Li and Zhang. In [2], the stability analysis of equilibrium point and direction of Hopf bifurcation were obtained for inőnite dimensional system with two different discrete delay terms, moreover numerical simulations to support the theoretical conclusions were given by Bilazeroğlu and et al…”