General autocatalysis criterion for diffusion-driven instability in go-or-grow models

Formulate a general autocatalysis condition that guarantees diffusion-driven instability for the go-or-grow family of ODE–PDE systems with linear diffusion in the migrating population u and local kinetics for the proliferating population v, specifying how the partial derivative of the v-kinetic term with respect to v generalizes across transition-rate functions α(u,v), β(u,v), and growth g(u+v)v.

Background

The paper reviews how diffusion-driven instabilities arise in ODE–PDE coupled systems and presents an autocatalysis condition in specific settings, leading to high-frequency instabilities and pattern formation.

The authors highlight the unresolved task of formulating a general autocatalysis criterion applicable across the go-or-grow model class, which would clarify instability mechanisms under broad switching and growth functions.