Diffsus

Diffsus, short for diffusion-susceptibility framework, is a term used in applied mathematics and theoretical modeling to describe systems in which diffusion processes interact with local nonlinear dynamics to produce spatial and temporal patterns. It emphasizes how disturbances propagate through space, and how diffusion can either dampen or amplify local activity.

Mathematically, a diffsus model couples diffusion terms with reaction kinetics within partial differential equations. A typical formulation is ∂u/∂t = D∇^2u + F(u, x, t), where u(x,t) is the state vector, D is a diffusion coefficient tensor, and F collects local reaction terms. Analysis often focuses on linear stability around homogeneous steady states. For small perturbations, the growth rate depends on the spatial frequency k, leading to conditions for diffusion-driven instability; such conditions can yield pattern formation, including Turing-type patterns, when diffusion destabilizes a previously stable state.

History and usage: the term diffsus is informal and not a standardized label in major textbooks, but it appears in lecture notes and some research discussions to distinguish the role of diffusion in shaping dynamics from purely local kinetics. It is used in various domains where spatial coupling is important.

Applications: diffsus concepts appear in ecology (spatial spread and patterning of populations), epidemiology (spread of infections with movement), chemical engineering (catalytic reactors and reaction-diffusion systems), and materials science (diffusion-mediated phase separation). Related ideas include reaction-diffusion systems, Turing patterns, and general spatial dynamical systems.