Bifurcation in skew-symmetric reaction-diffusion systems with unilateral terms

The paper deals with skew-symmetric reaction-diffusion systems satisfying assumptions guaranteeing Turing's instability and supplemented by unilateral terms of type v(-) and v(+). Existence of critical and bifurcation points is proved for diffusion rates, for which it is excluded without any unilateral term. These results are achieved by rewriting the skew-symmetric system as an abstract equation with positively homogeneous potential operator. General theorems about a variational characterization of the largest eigenvalue for positively homogeneous operators in a Hilbert space and bifurcation in equations with potentials are proved and subsequently applied to the reaction-diffusion systems, yielding the desired conclusions. (C) 2021 Elsevier Inc. All rights reserved.