Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction–diffusion model

Abstract

We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of the center-manifold in which the solution asymptotically evolves. Numerical simulations illustrate the phenomenon.

The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

“…From Theorem 2.1 in [11] and Theorem 3.1 in [10] we obtain the following results on system (2) without delayed feedback (τ = 0).…”

“…In addition, the authors also derived the conditions for the occurrence of Turing instability. In [10], Guo et al supplemented and improved the results in [11] and further established the Turing instability region determined by diffusion coefficients. In addition, the authors discussed the effect of diffusion coefficients on the existence of Hopf bifurcation.…”

“…The derivation of the model and more details can be found in [6,18]. In [11], Guo et al proved the existence of Hopf bifurcation and steady state bifurcation of system (1) by taking a as a parameter. In addition, the authors also derived the conditions for the occurrence of Turing instability.…”

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

“…In recent years, many researchers studied reactiondiffusion systems in point of bifurcation [13][14][15][16][17][18][19][20][21][22][23][24][25]. The aim of this paper is to study the stability and Hopf bifurcation of the system (1.6).…”

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

“…The diffusion-driven instability of the equilibrium leads to a spatially inhomogeneous distribution of species concentration, which is the so-called Turing instability. Although Turing instability was first investigated in a morphogenesis, it has quickly spread to ecological systems [3,4,6,[13][14][15][16][17][18][19][20][21][22][23][24], chemical reaction system [25][26][27][28][29][30] and other reaction-diffusion system [31][32][33][34][35][36][37][38]. From [39], we know that the phenomenon of spatial pattern formation in (1) with diffusion can not occur under all possible diffusion rates.…”