A Structure-Preserving Rational Integrator for the Replicator Dynamics on the Probability Simplex

Abstract

In this work, we introduce a quadratically convergent and dynamically consistent integrator specifically designed for the replicator dynamics. The proposed scheme combines a two-stage rational approximation with a normalization step to ensure confinement to the probability simplex and unconditional preservation of non-negativity, invariant sets and equilibria. A rigorous convergence analysis is provided to establish the scheme's second-order accuracy, and an embedded auxiliary method is devised for adaptive time-stepping based on local error estimation. Furthermore, a discrete analogue of the quotient rule, which governs the evolution of component ratios, is shown to hold. Numerical experiments validate the theoretical results, illustrating the method's ability to reproduce complex dynamics and to outperform well-established solvers in particularly challenging scenarios.

References in corpus (12)

• Coupled Replicator Equations for the Dynamics of Learning in Multiagent Systems
• An introduction to Lie group integrators -- basics, new developments and applications
• Geometric integration on spheres and some interesting applications
• A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance
• Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes
• A non-standard numerical scheme for an age-of-infection epidemic model
• Replicator dynamics with diffusion on multiplex networks
• Modified Patankar Linear Multistep methods for production-destruction systems
• A rare mutation model in a spatial heterogeneous environment: the Hawk and Dove game
• Rare Mutations in Evolutionary Dynamics
• Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics
• A non-local rare mutations model for quasispecies and Prisoner's dilemma: numerical assessment of qualitative behaviour