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

A more generalized approach, the concept of variable order derivative, is used to study the well-known replicator-mutator dynamics taking place in a moving medium. The biological relevance of the variable order context is explored via the language learning in social groups and stability of fixed points for the generalized model is recalled and discussed. Related graphs are plotted for different values of the derivative order γ. It happens that the threshold condition for learning accuracy symbolized by a function of payoff is a monotonically increasing function irrespective of the value of the time derivative order. Also, the limit cycles and their amplitudes are shown to vary with the value of the derivative order γ. These amplitudes become bigger as γ grows but the stability of the system is not affected. The generalized model, namely the variable order replicator-mutator dynamics in a moving medium is numerically solved via Crank-Nicholson scheme whose stability and convergence results are provided in details. An application to a variable order replicator-mutator dynamics of a population with three strategies is presented and numerical simulations are performed for some fixed values of the position variable r and the grid points. They display limit cycles appearing and disappearing in function of the values of the position r. The amplitudes of limit cycles are also proved to proportionally depend on r and the stability of the system remains unaffected. This shows the impressive effect of the transport process on the bifurcation dynamics of the model.