Coalescing colony model: Mean-field, scaling, and geometry

Abstract

We analyze the coalescing model where a 'primary' colony grows and randomly emits secondary colonies that spread and eventually coalesce with it. This model describes population proliferation in theoretical ecology, tumor growth, and is also of great interest for modeling urban sprawl. Assuming the primary colony to be always circular of radius r(t) and the emission rate proportional to r(t)^{θ}, where θ>0, we derive the mean-field equations governing the dynamics of the primary colony, calculate the scaling exponents versus θ, and compare our results with numerical simulations. We then critically test the validity of the circular approximation for the colony shape and show that it is sound for a constant emission rate (θ=0). However, when the emission rate is proportional to the perimeter, the circular approximation breaks down and the roughness of the primary colony cannot be discarded, thus modifying the scaling exponents.

Modeling cities

“…We consider two variants of this process [37] -in a first version (model M 0 ) we assume that the primary colony remains circular after the coalescence with a secondary colony. In contrast, we can consider a modified version of the process (model M 1 ) where after the coalescence with a secondary colony, the shape of the primary colony does not remain circular.…”

“…5. If we denote by r(t) the radius of the primary colony at time t and by x(t) the radius of the colony absorbed at time t, we can show [37] that the equations governing the evolution of these quantities are [35,37]  …”

“…This simple Shigesada-Kawasaki coalescing model is based on the circular approximation and if we drop this assumption, analytical calculations seem out of reach. We can however investigate this problem numerically [37] and assume that the area A and the perimeter L obey to a power law scaling of the form…”

“…For cities, we can imagine that at least the perimeter or the surface are the relevant variables for triggering new towns in its surrounding and that θ ≥ 1 is therefore probably the most relevant for urban systems. We consider two variants of this process [37] -in a first version (model M 0 ) we assume that the primary colony remains circular after the coalescence with a secondary colony. In contrast, we can consider a modified version of the process (model M 1 ) where after the coalescence with a secondary colony, the shape of the primary colony does not remain circular.…”

“…In M0, the primary colony (in red) remains circular and the area of the secondary colony is evenly distributed on the rim; in M1, the shapes are simply 'concatenated'. Figure taken from [37].…”