Rate effects on the growth of centres

Fry, Hannah; Smith, F. T.

doi:10.1017/s0956792516000231

Cited by 1 publication

(1 citation statement)

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“…The transition from small corner shop to large supermarkets [ 17 ] is explained by modelling the advantage of larger floor area compared to the travel costs to such sites. Fry & Smith [ 18 ] recently extended this approach to develop a time-dependent model; they use entropy to define the profit of a configuration and hence drive growth of each retail location. Simplifications of their model allow asymptotic analysis on customer preference towards larger floor areas, showing a bifurcation from a homogeneous state where there are no differences between centre sizes, to a ‘winner takes all’ dynamic, whereby the centre with the original maximum size is the site that dominates the market.…”

Section: Mathematical Models For Urban Population Densitymentioning

confidence: 99%

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

Whiteley

Avitabile

Siebers

et al. 2022

J. R. Soc. Interface.

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Human residential population distributions show patterns of higher density clustering around local services such as shops and places of employment, displaying characteristic length scales; Fourier transforms and spatial autocorrelation show the length scale between UK cities is around 45 km. We use integro-differential equations to model the spatio-temporal dynamics of population and service density under the assumption that they benefit from spatial proximity, captured via spatial weight kernels. The system tends towards a well-mixed homogeneous state or a spatial pattern. Linear stability analysis around the homogeneous steady state predicts a modelled length-scale consistent with that observed in the data. Moreover, we show that spatial instability occurs only for perturbations with a sufficiently long wavelength and only where there is a sufficiently strong dependence of service potential on population density. Within urban centres, competition for space may cause services and population to be out of phase with one another, occupying separate parcels of land. By introducing competition, along with a preference for population to be located near, but not too near, to high service density areas, secondary out-of-phase patterns occur within the model, at a higher density and with a shorter length scale than in phase patterning. Thus, we show that a small set of core behavioural ingredients can generate aggregations of populations and services, and pattern formation within cities, with length scales consistent with real-world data. The analysis and results are valid across a wide range of parameter values and functional forms in the model.

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Section: Mathematical Models For Urban Population Densitymentioning

confidence: 99%