A Matrix Formalism for Competing Affiliations

We develop a compact matrix formalism for competition between two social affiliations for adherents in a well-mixed population. The share dynamics is represented by a one-dimensional replicator equation driven by a two-by-two interaction matrix, whose qualitative regimeâdominance of either affiliation, stable coexistence, or bistabilityâis fully characterized by two scalar parameters derived from the matrix entries. The central contribution is an explicit regime-switch analysis of a rank-one matrix alteration step. We show that this alteration acts on the regime parameters solely through a single control quantity determined by the desired minus current fitness difference and the current population state. Threshold conditions identify which fitness targets induce a qualitative change of the replicator flow, and a recovery condition quantifies the minimal alteration required to drive an otherwise losing affiliation to full dominance. A persistent hard core of loyal adherents plays a dual role. First, it prevents complete extinction of the disadvantaged affiliation. Second, in the reduced fluid-population model introduced here, it lowers the intervention required for recovery under matrix alteration. The interplay between hard-core size and alteration strength is made explicit: A larger hard core reduces the intervention required for recovery, and vice versa. The framework is illustrated in three stages of increasing complexity and connects to sociophysical models of opinion dynamics, language competition, and religious affiliation.