Part II: Urban Growth PDEs
Continuum models of urban expansion from logistic saturation through reaction-diffusion waves to rare transition instantons. How does a city boundary advance, and what governs the speed of sprawl?
Part Overview
Continuum models of urban expansion from logistic saturation through reaction-diffusion waves to rare transition instantons. The Fisher-KPP equation gives traveling-wave fronts with minimum speed \(c_{\min} = 2\sqrt{Dr}\), while instanton calculus captures the exponentially rare fluctuations that trigger phase transitions in urban systems.
Key Topics
- • Logistic ODE analytical solution
- • Stability analysis
- • Fisher-KPP traveling wave
- • Minimum wave speed \(c_{\min} = 2\sqrt{Dr}\)
- • Instanton formalism
- • Fortran upwind solver
3 chapters | From ODEs to field theory | The mathematics of urban sprawl
Chapters
Chapter 1: Logistic ODE
The simplest model of bounded growth. Analytical solution, carrying capacity, inflection point, and stability analysis of the logistic equation as a foundation for spatial extensions.
Chapter 2: Fisher-KPP Waves
Adding diffusion to the logistic equation yields the Fisher-KPP PDE. Traveling wave solutions, the minimum wave speed \(c_{\min} = 2\sqrt{Dr}\), and numerical implementation with Fortran upwind solvers.
Chapter 3: Instanton Formalism
Rare but decisive transitions in urban systems. The instanton (saddle-point path) of the stochastic action gives the most probable trajectory for noise-driven phase transitions between urban states.