Pattern Formation in a General Degn-Harrison Reaction Model

Abstract

Abstract. In this paper, we study the pattern formation to a general Degn-Harrison reaction model. We show Turing instability happens by analyzing the stability of the unique positive equilibrium with respect to the PDE model and the corresponding ODE model, which indicate the existence of the non-constant steady state solutions. We also show the existence periodic solutions of the PDE model and the ODE model by using Hopf bifurcation theory. Numerical simulations are presented to verify and illustrate the theoretical results.

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

“…The Degn-Harrison system describes the reaction scheme between the oxygen u and the nutrient v, taking into account that the excess of oxygen inhibits the respiration according to a nonlinear rate of the type u/(1 + u 2 ), see for details [7]. In [16] the author, having in mind that other phenomena can be described by similar reaction schemes, generalizes the inhibitory law using an arbitrary function ϕ(u). For the resulting generalized Degn-Harrison reaction-diffusion system, the existence of periodic solutions has been established in [16], using the Hopf bifurcation theory.…”

“…In [16] the author, having in mind that other phenomena can be described by similar reaction schemes, generalizes the inhibitory law using an arbitrary function ϕ(u). For the resulting generalized Degn-Harrison reaction-diffusion system, the existence of periodic solutions has been established in [16], using the Hopf bifurcation theory. OMISSIS…”

Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems

“…The Degn-Harrison system (1) has been studied extensively in the literature, but most of the researches focus on the dynamics of this model including the local and global asymptotic stability of the steady-state solutions [25], [26], Turing instability [27], [28] and Hopf bifurcation [29], [30]. However, as far as we know, this is the first work deal with control synchronization of the model (1).…”