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

Area of Science:

• Theoretical Ecology
• Mathematical Biology
• Urban Dynamics

Background:

• Colony growth models are crucial for understanding population proliferation, tumor development, and urban sprawl.
• The coalescing model describes a primary colony emitting secondary colonies that merge back into it.
• Previous models often assume simplified colony shapes, like perfect circles.

Purpose of the Study:

• To analyze a coalescing colony model with a primary colony emitting secondary colonies.
• To investigate the impact of emission rate proportionality (θ) on colony dynamics and scaling exponents.
• To critically assess the validity of the circular approximation for colony shape under varying emission rates.

Main Methods:

• Derivation of mean-field equations for primary colony dynamics.
• Calculation of scaling exponents as a function of the emission rate parameter θ.
• Comparison of theoretical results with numerical simulations.
• Testing the circular approximation's validity for different emission rate scenarios.

Main Results:

• The circular approximation for colony shape is valid when the emission rate is constant (θ=0).
• When the emission rate is proportional to the perimeter, the circular approximation breaks down.
• Colony roughness becomes significant for non-constant emission rates, altering scaling exponents.

Conclusions:

• The assumption of a circular primary colony is a simplification that holds only under specific emission rate conditions.
• For emission rates dependent on colony size or perimeter, incorporating colony roughness is essential for accurate modeling.
• This research refines coalescing colony models, improving their applicability to ecological, biological, and urban systems.