Unilateral sources and sinks of an activator in reaction-diffusion systems exhibiting diffusion-driven instability

TL;DR

This paper investigates how unilateral source and sink terms in reaction-diffusion systems affect the conditions for pattern formation via Turing instability, showing that these terms restrict the parameter space for bifurcations.

Contribution

It introduces unilateral source and sink terms into reaction-diffusion models and analyzes their impact on the bifurcation domain for spatial patterns, extending classical Turing analysis.

Findings

Unilateral terms reduce the domain of diffusion parameters allowing bifurcations.

Inhibitory unilateral terms can induce bifurcations where none occur without them.

Boundary conditions influence the bifurcation domain in the presence of unilateral terms.

Abstract

A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type $s_{-}(x)u^{-}$, $s_{+}(x)u^{+}$ describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet-Neumann) boundary conditions as well as that of pure Neumann conditions is described.