Metacommunity persistence on spatially heterogeneous landscapes

What the math gives to ML

The paper provides a persistence mechanism for two competing populations in spatially heterogeneous environments: stable coexistence is guaranteed when each species can invade the other's monospecific equilibrium, provided both species persist in isolation. The continuous model uses nonlocal colonization operators, while finite habitat averaging produces a lower-persistence approximation, showing that coarse-graining can destroy survival margins. A transferable neural-network construction is to treat competing experts or subnetworks as populations sharing a normalized capacity resource, and enforce mutual-invasibility margins computed from linearized growth rates. This yields a falsifiable stability test for expert collapse and a way to decide when habitat-like heterogeneity should be preserved rather than averaged away.