A reaction-diffusion-ODE model in anthropology
A reaction-diffusion-ODE model for the Neolithic spread of farmers in Europe has been recently proposed by Elias, Kabir and Mimura. This model involves hunter-gatherers and farmers, which are divided into two subpopulations, namely sedentary and migrating farming populations. The conversion between the farming subpopulations depends on the total density of the farmers and it is superimposed on the classical Lotka-Volterra competition model; it is therefore described by a three-component reaction-diffusion-ODE system. In this talk, we study its singular limit as the conversion rate tends to infinity and prove that solutions of the three-component system converge to solutions of a two-component system with linear diffusion in one of the equations and nonlinear degenerate diffusion in the other. This is joint work with Jan Elias, Masayasu Mimura and Yoshihisa Morita.
We then study the limiting two component system. From an ecological viewpoint, the nonlinear diffusion takes into account the population density pressure of the farmers on their dispersal. The interaction between farmers and hunter-gatherers is of the Lotka-Volterra prey-predator type. We present an alternative method for proving the existence and uniqueness of the global-in-time solution and study its asymptotic behavior as time tends to infinity. This is joint work with Jan Elias and Masayasu Mimura.