A mathematical modeling approach to the formation of urban and rural areas: Convergence of global solutions of the mixed problem for the master equation in sociodynamics

被引：8

作者：

Tabata, Minoru ^{[1
]}

Eshima, Nobuoki ^{[2
]}

Takagi, Ichiro ^{[3
]}

机构：

[1] Osaka Prefecture Univ, Grad Sch Engn, Dept Math Sci, Osaka 5998531, Japan

[2] Oita Univ, Fac Med, Dept Stat, Oita 8795593, Japan

[3] Tokai Univ, Sch Business Studies, Dept Business Management, Kumamoto 8628652, Japan

来源：

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS | 2011年 / 12卷 / 06期

关键词：

Urbanization; Human migration; Master equation; Nonlinear integro-partial differential equation; Convergence; Global solutions; PARTIAL DIFFERENTIAL-EQUATION; CAUCHY-PROBLEM; HUMAN-POPULATION; MIGRATION;

D O I：

10.1016/j.nonrwa.2011.05.025

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Urban and rural areas are formed by human migration from thinly populated areas to densely populated areas. It is known in sociodynamics that human migration is described by a nonlinear integro-partial differential equation whose unknown function denotes the population density. This equation is called the master equation. The master equation has its origin in statistical physics, and is regarded as one of the most fundamental equations in natural sciences, as its name suggests. We describe the formation of urban and rural areas by making use of global solutions of the mixed problem for this equation. In this paper we prove sufficient conditions for the mixed problem to have a unique global solution that converges to a two-tier step function as the time variable tends to infinity. This step function is a stationary solution of the master equation, and the higher (lower, respectively) step represents a stationary urban (rural, respectively) area. This result mathematically describes the formation of urban and rural areas in the real world. (C) 2011 Elsevier Ltd. All rights reserved.

引用

页码：3261 / 3293

页数：33