Invasiondiffusion

Invasion-diffusion is a modeling framework used to describe the spatiotemporal spread of biological populations, diseases, or other entities that disperse through space while undergoing local growth or interactions. The approach combines local growth dynamics with spatial diffusion to produce traveling fronts or spreading waves.

Mathematically, invasion-diffusion is often described by a reaction-diffusion equation of the form ∂u/∂t = D ∇^2 u + f(u), where u(x,t) represents a density or concentration, D is a diffusion coefficient, and f(u) captures local dynamics such as growth or decay. A common example is the logistic growth f(u) = r u (1 − u/K), giving the Fisher-KPP model. In homogeneous one-dimensional media, this setup yields traveling wave solutions with a minimal speed c = 2√(rD). Extensions and qualitative behavior depend on boundary conditions, initial data, and the precise form of f.

Applications are widespread. In ecology, invasion-diffusion models describe range expansion and spread of invasive species across landscapes. In epidemiology, they help explain how infectious diseases propagate through space. In oncology, reaction-diffusion formulations model tumor cell invasion into surrounding tissue, guiding understanding of growth patterns and treatment effects. These models also support studies of pattern formation and interface dynamics.

Extensions include adding advection to represent directed movement, nonlocal or long-range dispersal terms, and spatial heterogeneity to reflect heterogeneous environments. Limitations include the assumptions of homogeneous, isotropic diffusion and simple local dynamics, which may not capture complex landscapes or multi-species interactions. Numerical methods such as finite differences or finite elements are commonly used to simulate invasion-diffusion systems.