A minimal model for a differentiation phase transition in coupled plastic systems

Coupled systems whose components are plastic often respond to a load they cannot accommodatein an undifferentiated configuration by assuming distinct, complementary roles—a division of laborseen across slime molds, insect colonies, and vertebrate collectives. We ask whether three structuralconditions suffice to produce such differentiation and, if so, what kind of transition it is. We introducea minimal agent-based model in which agents hold allocations on a probability simplex and evolve byreplicator dynamics under three terms: an attractive coupling to the population mean (coherence),a repulsive term of tunable strength (back-pressure), and a weak demand-matching term. A mean-field stability analysis yields a closed-form critical point βc = K, set by the coherence coupling,and identifies the transition as a supercritical pitchfork. Three independent numerical routes—the stability spectrum, the amplitude growth rate, and finite-size scaling of the order-parametersusceptibility—locate the same critical point and reproduce the mean-field growth-rate slope 1/D.The critical point scales linearly with the coupling, and an ablation separates the roles of thethree conditions: back-pressure drives the transition, coupling is necessary for it to occur, andheterogeneity orients the broken state as a symmetry-breaking field. We also note that simplexsaturation can make the continuous transition appear discontinuous, so its order must be read fromthe linear regime.