Oscillatory Turing Pattern Formation from the Interactions between Hopf and Turing Bifurcations

Abstract

Coupled positive and negative feedback loops are not only a fundamental building block of more complex networks but also an object of engineered gene regulatory networks in synthetic biology. Here, we analyze a two-component reaction-diffusion system of coupled positive and negative feedback loops. By Hopf and Turing bifurcation analyses both identifying system parameter regions where spatial patterns emerge, we find that the interactions between the two kinds of bifurcation modes can lead to oscillatory Turing patterns, such as holes and stripes as well as their coexisting modes, whereas the pure Turing instability only can lead to paralleling stripes or hexagons. Our analysis indicates that the formation of complex patterns depends on the bifurcation types as well as the interactions among bifurcation modes.