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Self-Organisation in Spatial Systems-From Fractal Chaos to Regular Patterns and Vice Versa.

Banaszak M, Dziecielski M, Nijkamp P, Ratajczak W - PLoS ONE (2015)

Bottom Line: This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations.Such developments can range from chaotic to fully ordered.Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland.

ABSTRACT
This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

No MeSH data available.


Related in: MedlinePlus

Force field for a metropolitan system with M = 3 and with an equal population for each city (see also [21]).
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pone.0136248.g002: Force field for a metropolitan system with M = 3 and with an equal population for each city (see also [21]).

Mentions: Since the pioneering works of [20, 21] and [14], it has been widely recognised that the attraction forces leading to an agglomeration can be derived from a gravitational-like potential, which can be expressed as follows:U(r)=-kmr,(4)where r is the distance from the metropolitan centre, m is its population (also referred to as mass), and k is an empirical constant which depends on a variety of system-specific details, such as, for example, objects being attracted (people or goods, or maybe both). The singularity at r = 0 can be easily eliminated by the regularization, which introduces an additional lengthscale, h:U(r)=-kmr2+h2,(5)which not only eliminates infinity at r = 0, but also takes into account the basic observation that cities are not mathematical points: they have a size. The attractive metropolitan force, which is a vector, can be calculated by the standard formula known from classical mechanics:F→(r→)≡-∇→U(r)=-kmr2+h2r→r2+h2,(6)where is a 2-dimensional vector, since the towns in our spatial system are placed on a 2D plane. As in physics, the potentials are additive scalars, and therefore can be used for systems with M metropolitan centres, yielding the following expression:U=-k∑i=1Mmiri2+h2,(7)where mi and ri are the population and the distance from the ith city, respectively. Similarly, the attractive force can be written as:F→=-k∑i=1Mmiri2+h2r→iri2+h2.(8)As an example, we show in Fig 2 the force field calculated from Eq (8), for a metropolitan system with M = 3 and an equal population for each city.


Self-Organisation in Spatial Systems-From Fractal Chaos to Regular Patterns and Vice Versa.

Banaszak M, Dziecielski M, Nijkamp P, Ratajczak W - PLoS ONE (2015)

Force field for a metropolitan system with M = 3 and with an equal population for each city (see also [21]).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4581867&req=5

pone.0136248.g002: Force field for a metropolitan system with M = 3 and with an equal population for each city (see also [21]).
Mentions: Since the pioneering works of [20, 21] and [14], it has been widely recognised that the attraction forces leading to an agglomeration can be derived from a gravitational-like potential, which can be expressed as follows:U(r)=-kmr,(4)where r is the distance from the metropolitan centre, m is its population (also referred to as mass), and k is an empirical constant which depends on a variety of system-specific details, such as, for example, objects being attracted (people or goods, or maybe both). The singularity at r = 0 can be easily eliminated by the regularization, which introduces an additional lengthscale, h:U(r)=-kmr2+h2,(5)which not only eliminates infinity at r = 0, but also takes into account the basic observation that cities are not mathematical points: they have a size. The attractive metropolitan force, which is a vector, can be calculated by the standard formula known from classical mechanics:F→(r→)≡-∇→U(r)=-kmr2+h2r→r2+h2,(6)where is a 2-dimensional vector, since the towns in our spatial system are placed on a 2D plane. As in physics, the potentials are additive scalars, and therefore can be used for systems with M metropolitan centres, yielding the following expression:U=-k∑i=1Mmiri2+h2,(7)where mi and ri are the population and the distance from the ith city, respectively. Similarly, the attractive force can be written as:F→=-k∑i=1Mmiri2+h2r→iri2+h2.(8)As an example, we show in Fig 2 the force field calculated from Eq (8), for a metropolitan system with M = 3 and an equal population for each city.

Bottom Line: This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations.Such developments can range from chaotic to fully ordered.Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

View Article: PubMed Central - PubMed

Affiliation: Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland.

ABSTRACT
This study offers a new perspective on the evolutionary patterns of cities or urban agglomerations. Such developments can range from chaotic to fully ordered. We demonstrate that in a dynamic space of interactive human behaviour cities produce a wealth of gravitational attractors whose size and shape depend on the resistance of space emerging inter alia from transport friction costs. This finding offers original insights into the complex evolution of spatial systems and appears to be consistent with the principles of central place theory known from the spatial sciences and geography. Our approach is dynamic in nature and forms a generalisation of hierarchical principles in geographic space.

No MeSH data available.


Related in: MedlinePlus