A geometrical similarity between migration of human population and diffusion of biological particles

The purpose of this paper is to prove that migration of human population and diffusion of biological particles (e.g., cells, bacteria, chemicals, animals and so on) have a close similarity such that if we geometrically reduce the geographical movement of human population, then the reduced movement is very close to diffusion of biological particles. We construct mathematical models that describe these two kinds of phenomenon, i.e., we derive a nonlinear integro-partial differential equation whose solution represents the density of human population, and we derive a quasilinear partial differential equation of parabolic type whose solution represents the density of biological particles. We refer to the former equation and the latter equation as the master equation and the Fokker–Planck equation, respectively. We prove the close similarity by demonstrating that if we transform a solution of the master equation in terms of geometrical reduction of the space variable of the solution, then the solution thus transformed is close to a solution of the Fokker–Planck equation.