The Fokker-Planck equation and the master equation in the theory of migration

In the theory of migration two mathematical models are regarded as very important. One is described by a quasilinear partial differential equation of parabolic type called the Fokker-Planck equation. The other is described by a nonlinear integro-partial differential equation called the master equation. Both the models are employed frequently at the same time in the theory of migration. Hence we need to investigate whether the descriptions given by the models are close to each other or not. The purpose of the present paper is to mathematically prove that if the effort required in moving is large, then the models are close to each other in the following sense: the mixed problem with the periodic boundary condition for the master equation has a unique solution that is very close to a solution of that for the Fokker-Planck equation, where the effort is a sociodynamic quantity that represents a cost incurred in moving. By making use of the result of the paper, we can apply both the models to movement of human population at the same time.