Bifurcation and Turing instability for genetic regulatory networks with diffusion

Abstract

In this paper, a diffusive genetic regulatory network under Neumann boundary conditions is considered. First, the criteria for the local stability and diffusion-driven instability of the positive stationary solution without and with diffusion are investigated, respectively. Moreover, Turing regions and pattern formation are obtained in the plane of diffusion coefficients. Second, the existence and multiplicity of spatially homogeneous/nonhomogeneous non-constant steady-states are studied by using the Lyapunov–Schmidt reduction. Finally, some numerical simulations are carried out to illustrate the theoretical results.