Backward bifurcations in spatial replicator models:when invasion criteria fail to predict coexistence

What the math gives to ML

The paper identifies a nonstandard failure of invasion-based stability reasoning in spatial replicator systems: the principal eigenvalue of the linearized invasion operator can be negative while a stable finite-amplitude coexistence state already exists. The mechanism is a backward bifurcation, represented locally by an amplitude equation with a saddle-node fold and hysteresis. A transferable neural-network construction is a spatially or feature-index coupled competitive latent field whose nonlinear interaction is deliberately placed in this backward-bifurcation regime, creating robust finite-amplitude memory or multimodal representations that cannot be eliminated by infinitesimal perturbations.