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 |
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Authors: | Minoru Tabata Nobuoki Eshima Ichiro Takagi |
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Institution: | 1. Department of Mathematical Sciences, Graduate School of Engineering, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan;2. Department of Statistics, Faculty of Medicine, Oita University, Oita 879-5593, Japan;3. Department of Business Management, School of Business Studies, Tokai University, Kumamoto 862-8652, Japan |
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Abstract: | 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. |
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