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

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.