Turing instability in an economic–demographic dynamical system may lead to pattern formation on a 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 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) show a relatively small variation in space on the scale of thousands of kilometres. We then examine the population distribution across those areas to show that, in spite of the approximate homogeneity of the environment, it exhibits a significant variation revealing a nearly periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there may exist an inherent mechanism that may 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, the spatial scale of the emerging pattern being consistent with observations. Using numerical simulations, we show that, interestingly, the emergence of the Turing patterns may eventually lead to the system collapse.

Self-organization of modular activity in immature cortical networks

“…Self-organization based on local facilitation and lateral inhibition was first mathematically described by Turing 30 . While not the only mechanism for spontaneous pattern formation, it appears to be ubiquitous in nature 33,51,52 in both animate and inanimate systems across vastly different scales (eg: [53][54][55][56] ). Our results are consistent with such a mechanism operating at neural activity timescales to shape the structure of spontaneous activity via the selective amplification of patterns near a system specific characteristic wavelength.…”

Turing patterns by supramolecular self-assembly of a single salphen building block

“…These principles have been observed not only in biological morphogenetic processes but are also widely present in many more dynamic systems than thought earlier ( Kuznetsov and Polezhaev, 2020 ; Leyshon et al., 2021 ; Scholes et al., 2019 ), including demographic, sociolinguistic, psychologic, economic, ecologic, or epidemiologic phenomena ( Batabyal, 2021 ; Chakraborty et al., 2021 ; Chen, 2019 ; Iskarous, 2019 ; Lacalli, 2020 ; Mimar et al., 2021 ; Pal and Poria, 2021 ; Putra et al., 2019 ; Vandermeer and Perfecto, 2020 ; Zincenko et al., 2021 ). Nevertheless, in chemistry, these RD principles have been typically used to explain how reactions concerted at different scales can result into oscillatory and out-of-equilibrium systems, such as the Belousov-Zhabotinsky (BZ) reaction ( Zaikin and Zhabotinsky, 1970 ) ( e .…”

Modelling the emergence of cities and urban patterning using coupled integro-differential equations

“…Lastly, a number of recent papers on a model of reactiondiffusion equations of population and wealth distribution have been released [19][20][21]. In their statistical analysis of the population landscape [19], the authors show spatial correlation across Canada, Australia and Mongolia that cannot be explained by environmental factors alone, highlighting the need for explanatory modelling. In all three papers, they model population wealth growth.…”