Numerical simulation of replicator systems

Background. Replicator systems often arise when evolution is concerned. Mathematical models of population dynamics, game theory, economics and biological and molecular evolution lead to systems of partial differential equations. Due to the absence of analytical solutions for a vast majority of such problems, approximate solutions obtained via numerical simulation are required. Hence, construction of efficient algorithms for solving spatial and time-dependent replicator systems is crucial for understanding the dynamics and properties of evolution. Results. We gave an overview of the existing approaches to numerical simulation of replicator systems arising in various fields. We described a mathematical model of population dynamics with explicit space in game theory setting with an asymmetric conflict and a model of biological evolution in presence of a mutator-gene. Both models led to nonlinear systems of partial differential equations that we reduced to the same general form. Then we described the numerical method based on the finite volume framework to solve the system, and provided some numerical examples that demonstrate the method's validity. Conclusions. We conclude that the constructed numerical method is suitable for simulation of replicator systems of general form.