A geometrical similarity between migration of human population and diffusion of biological particles
In this paper, the authors consider the following nonlinear integro-partial differential equation, known as the master equation, and the quasilinear partial differential equation of parabolic type, known as the Fokker-Planck equation, respectively: \[ \begin{aligned} \displaystyle \partial f(t,X){/}{\partial t} &= -\omega (t,X)f(t,X)+\int_{Y\in\Omega(\varepsilon)} W(t;X|Y)f(t,Y)\,dY, \cr \displaystyle \partial g(t,x){/}{\partial t} &= a(g(t,x))\Delta g(t,x)+ b(g(t,x))\nabla g(t,x).\nabla g(t,x), \end{aligned} \] where \(\Omega\) is a bounded domain in \(\displaystyle\mathbb{R}^2 \) in which biological particles diffuse; the bounded domain \(\displaystyle\Omega(\varepsilon):=\{X\in\mathbb{R}^2 ; X=x/\varepsilon , x\in\Omega\},\) \(\displaystyle\varepsilon\in (0,1]\) is where human population migrate; \(a\) and \(b\) are given functions; \(\displaystyle W(t;X|Y)\) represents the transition probability of human from a point \(\displaystyle Y\in\Omega(\varepsilon)\) to a point \(\displaystyle X\in\Omega(\varepsilon)\) at time \(t\geq 0\); and \(\displaystyle\omega(t,X):= \int_{Y\in\Omega(\varepsilon)} W(t;Y|X)\,dY.\) The authors, under certain assumptions, prove that the master equation, which describes the migration of the human population, and the Fokker-Planck equation, which describes the diffusion of the biological particles, are similar when they geometrically reduce the geographical movement of the human population.
scientific article; zbMATH DE number 5116313