Necessary conditions for Turing instability in the reaction–diffusion systems associated with replicator dynamics

Spatial pattern formation via Turing instability in the reaction–diffusion system associated with the replicator dynamics is concerned with the long-term effects of perturbations, whereas the notion of reactivity describes the transient behaviour of perturbations to an asymptotically stable equilibrium point. This article establishes the connection between these two concepts—Turing instability and reactivity—in the context of the reaction–diffusion system associated with game replicator dynamics. In particular, we show that for Turing instability to occur in the reaction–diffusion system, the smallest diffusion coefficient of the system must be strictly less than the ratio of positive reactivity of the stable equilibrium point and square of wavenumber. This connection is also explored in terms of elements of the symmetric part of the associated stability matrix.