Open AccessChaos and coexisting attractors in replicator-mutator maps
Author(s) -
Archan Mukhopadhyay,
Suman Chakraborty,
Sagar Chakraborty
Publication year - 2021
Publication title -
journal of physics. complexity
Language(s) - English
Resource type - Journals
ISSN - 2632-072X
DOI - 10.1088/2632-072x/abf232
Subject(s) - replicator equation , attractor , chaotic , population , fixed point , stochastic game , statistical physics , orbit (dynamics) , multiplicative function , bistability , mutation , mathematics , mathematical economics , computer science , biology , physics , mathematical analysis , genetics , artificial intelligence , quantum mechanics , demography , sociology , engineering , gene , aerospace engineering
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.