An influence of unilateral sources and sinks in reaction-diffusion systems exhibiting Turing's instability on bifurcation and pattern formation

Abstract

We consider a general reaction-diffusion system exhibiting Turing's diffusion-driven instability. In the first part of the paper, we supplement the activator equation by unilateral integral sources and sinks of the type $\left(\int_{K} \frac{u(x)}{\left| K \right|} \; dK \right)^{-}$ and $\left(\int_{K} \frac{u(x)}{\left| K \right|} \; dK \right)^{+}$. These terms measure an average of the concentration over the set $K$ and are active only when this average decreases bellow or increases above the value of the reference spatially homogeneous steady state, which is shifted to the origin. We show that the set of diffusion parameters in which spatially heterogeneous stationary solutions can bifurcate from the reference state is smaller than in the classical case without any unilateral integral terms. This problem is studied for the case of mixed, pure Neumann and periodic boundary conditions. In the second part of the paper, we investigate the effect of both unilateral terms of the type $u^{-},u^{+}$ and unilateral integral terms on the pattern formation using numerical experiments on the system with well-known Schnakenberg kinetics.