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Stability and convergence analysis of a variable order replicator-mutator process in a moving medium.

Emile Franc Doungmo Goufo1

  • 1Department of Mathematical Sciences, University of South Africa, Florida, 0003 South Africa.

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|May 19, 2016
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Summary

This study introduces a generalized variable order derivative to model replicator-mutator dynamics in moving media, finding that learning accuracy thresholds increase with derivative order, while limit cycle amplitudes are affected by transport processes.

Keywords:
Limit circleReplicator–mutator equationStability and convergence of a numerical schemeTransportVariable order derivative

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Area of Science:

  • Mathematical Biology
  • Dynamical Systems
  • Evolutionary Game Theory

Background:

  • Replicator-mutator dynamics are fundamental to understanding evolutionary processes.
  • Existing models often assume constant order derivatives, limiting their applicability.
  • Variable order derivatives offer a more generalized framework for complex systems.

Purpose of the Study:

  • To investigate replicator-mutator dynamics in a moving medium using a variable order derivative approach.
  • To explore the biological relevance of this generalized model in contexts like social language learning.
  • To analyze the stability of fixed points and the impact of derivative order on system dynamics.

Main Methods:

  • Application of the variable order derivative concept to replicator-mutator dynamics.
  • Numerical solution using the Crank-Nicholson scheme for stability and convergence analysis.
  • Simulation of a three-strategy population model with varying transport parameters.

Main Results:

  • The learning accuracy threshold is a monotonically increasing function of the derivative order.
  • Limit cycle amplitudes increase with derivative order (γ) and position (r), but system stability is maintained.
  • Transport processes significantly influence bifurcation dynamics, causing limit cycles to appear and disappear.

Conclusions:

  • The variable order derivative model provides a more nuanced understanding of replicator-mutator dynamics in moving environments.
  • The findings highlight the crucial role of transport phenomena in shaping evolutionary trajectories and system stability.
  • This generalized approach offers valuable insights into biological systems like social learning and population dynamics.