Chaos and coexisting attractors in replicator-mutator maps

Abstract

Abstract Mutation is an unavoidable and indispensable phenomenon in both biological and social systems undergoing evolution through replication-selection processes. Here we show that mutation in a generation-wise nonoverlapping population with two-player-two-strategy symmetric game gives rise to coexisting stable population states, one of which can even be chaotic; the chaotic state prevents the cooperators in the population from going extinct. Specifically, we use replicator maps with additive and multiplicative mutations, and rigorously find all possible two dimensional payoff matrices for which physically allowed solutions can be achieved in the equations. Subsequently, we discover the various possibilities of bistable outcomes—e.g., coexistences of fixed point and periodic orbit, periodic orbit and chaos, and chaos and fixed point—in the resulting replicator-mutator maps.