A Generalized Population Dynamics Model of a City and an Algorithm for Engineering Regime Shifts

Measures of wealth and production have been found to scale superlinearly with the population of a city. Therefore, it makes economic sense for humans to congregate together in dense settlements. A recent model of population dynamics showed that population growth can become superexponential due to the superlinear scaling of production with population in a city. Here, we generalize this population dynamics model and demonstrate the existence of multiple stable equilibrium points, showing how population growth can be stymied by a poor economic environment. This occurs when the goods and services produced by the city become less profitable due to a lack of diversification in the city’s economy. Then, relying on critical slowing down signals related to the stability of an equilibrium point, we present an algorithm for engineering regime shifts such that a city at a stable equilibrium point may continue to grow again. The generality of the model and the algorithm used here implies that the model and algorithm need not be restricted to urban systems; they are easily applicable to other types of systems where the assumptions used are valid. Cities are large and dense spatial agglomerations of humans and their socioeconomic activities. The growth of cities results in distinct spatial patterns of settlement and human activity that have been the subject of extensive research over the past decades. There is now a growing consensus that the processes that give rise to urban spatial patterns are localized, resulting in urban growth driven 1 from the bottom up [1, 2]. There is also a considerable amount of research effort focused on the morphology of urban growth where form is more emphasized rather than function [3–6]. In urban economics, a few well-known results exist concerning the optimal town size [7, 8]. However, these studies are usually more concerned on the spatial rather than the temporal aspect of urban growth. To obtain new insights into the evolution of a city and why some cities thrive where other cities fail, consideration must also be given to the temporal aspect of urban growth and the factors that drive this growth. Examples of work that has been done in this area are the modeling of retail and residential spaces of a city using difference equations by Beaumont, Clarke and Wilson [9], and the modeling of population migration within a city by Weidlich and Haag [10]. In this paper, we build upon previous work by Bettencourt et al. and consider a simple population dynamics model driven by the population migration that may occur to take advantage of newly emerged economic opportunities [11]. Cities represent places of economic opportunity for the population migration of humans. Individuals and corporations come together for the exchange of goods and services in close proximity [12–16]. Indeed, empirical data from cities indicate that measures of wealth and production scale superlinearly with the population of a city [11]. The scaling appears in the form of power laws Y = Y0N [11] where Y is a property of the city, Y0 is a constant, N is the population of the city and β is the scaling exponent. Superlinear scaling occurs when β > 1 and sublinear scaling occurs when β < 1. There has been some controversy surrounding this result because it has been observed that the scaling exponent varies quite sensitively to the definition of a city’s boundaries over which properties of a city like wealth and production are aggregated [17]. However, a consistent scaling exponent can be observed across multiple cities if the definition of a city’s boundaries is able to capture urban functionality [18,19]. Furthermore, the empirical exponents can be theoretically predicted by considering the social interactions of its residents on a spatial network [20, 21]. For properties related to production and growth, β is theoretically estimated at β ≈ 1.17. While properties related to production and growth might scale superlinearly with the population, undesirable properties like crime can also scale superlinearly with city size [11, 22–25]. These are obvious trade-offs that economic migrants must make when choosing to settle in a city. Therefore, a city will not grow if the disadvantages that come with agglomeration outweighs the advantages that come with it. Clearly, this has not been