Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

Yan Zhang,Zhenhua Bao

doi:10.1155/2014/574921

Abstract

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.

Highlights

In 1952, Turing published his pioneering paper describing the chemical process between signaling molecules that spread away from their source to form a concentration gradient within a series of cells [1]
By using the center manifold theory, normal form methods, and the abstract results in [12], we are able to prove that the spatially homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point are unstable
By using the standard Hopf bifurcation theorem, we are able to prove the existence of periodic solutions of the system

Summary

Introduction

In 1952, Turing published his pioneering paper describing the chemical process between signaling molecules that spread away from their source to form a concentration gradient within a series of cells [1]. Since the time when the model was proposed by Turing, many researchers have studied the model extensively (see [11] and the references therein for great details) They mostly focus on the pattern formations of Turing type. In [12], the authors derived a simplified Hopf bifurcation theorem for the general semilinear reaction-diffusion equations on the one-dimensional domain and used the abstract theorem to prove the existence of oscillatory patterns emerging from Hopf bifurcations. By using the center manifold theory, normal form methods, and the abstract results in [12], we are able to prove that the spatially homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point are unstable.

Existence of Hopf Bifurcations

Conclusions