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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Special Issue: AUS-ICMS 2020   > Article

On replicator equations with nonlinear payoff functions defined by the Ricker models

Sur les équations de réplicateur avec fonctions à gain non-linéaire définies par les modèles de Rick


Mansoor Saburov
American University of the Middle East
Kuwait



Published on 28 July 2021   DOI : 10.21494/ISTE.OP.2021.0740

Abstract

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An evolutionary game is usually identified by a smooth (possibly nonlinear) payoff function. In this paper, we propose a model of evolutionary game in which the nonlinear payoff functions are defined by the Ricker models. Namely, in the proposed model, the biological fitness of the pure strategy will increase according to the Ricker model. Motivated by some models of economics of transportation and communication networks, in order to observe the evolutionary bifurcation diagram, we also control the nonlinear payoff functions in two different regimes: positive and negative. One of the interesting feature of the model is that if we switch the controlling parameter from positive to negative regime then the set of local evolutionarily stable strategies (ESSs) changes from one set to another one. We also study the dynamics and stability analysis of the discrete-time replicator equation governed by the proposed nonlinear payoff function. In the long-run time, the following scenario can be observed: (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever.

An evolutionary game is usually identified by a smooth (possibly nonlinear) payoff function. In this paper, we propose a model of evolutionary game in which the nonlinear payoff functions are defined by the Ricker models. Namely, in the proposed model, the biological fitness of the pure strategy will increase according to the Ricker model. Motivated by some models of economics of transportation and communication networks, in order to observe the evolutionary bifurcation diagram, we also control the nonlinear payoff functions in two different regimes : positive and negative. One of the interesting feature of the model is that if we switch the controlling parameter from positive to negative regime then the set of local evolutionarily stable strategies (ESSs) changes from one set to another one. We also study the dynamics and stability analysis of the discrete-time replicator equation governed by the proposed nonlinear payoff function. In the long-run time, the following scenario can be observed : (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever ; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever.

Replicator Equation Evolutionarily stable strategy Nash equilibrium Ricker model

Replicator Equation Evolutionarily stable strategy Nash equilibrium Ricker model