Turing Instability in an Economic-Demographic Dynamical System Can Lead to Pattern Formation on Geographical Scale

Abstract

Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the emergence of the densely populated areas at their present locations is widely believed to be linked to more favourable environmental and climatic conditions. In this paper, we challenge this point of view. We first identify a few areas at different parts of the world where the environmental conditions (quantified by the temperature, precipitation and elevation) are approximately uniform over thousands of miles. We then examine the population distribution across those areas to show that, in spite of the homogeneity of the environment, it exhibits a clear nearly-periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there exists an inherent mechanism that can lead to pattern formation even in a uniform environment. We consider a mathematical model of the coupled demographic-economic dynamics and show that its spatially uniform, locally stable steady state can give rise to a periodic spatial pattern due to the Turing instability. Using computer simulations, we show that, interestingly, the emergence of the Turing patterns eventually leads to the system collapse.