Keller-Segel system

Short description: Mathematical model of chemotaxis

The Keller–Segel system is a class of mathematical models describing the collective movement of cells or organisms in response to chemical signals, a process known as chemotaxis. It was first introduced in the 1970s by Evelyn Keller and Lee Segel to explain the aggregation behavior of Dictyostelium discoideum, a slime mold that migrates and forms clusters in response to chemoattractants.

Mathematical formulation

The most common form of the Keller–Segel model is a system of coupled nonlinear partial differential equations (PDEs). In its simplest parabolic–parabolic version, it is written as:

$\begin{array}{rl}\frac{\partial u}{\partial t}& =D_u\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v),\\ \frac{\partial v}{\partial t}& =D_v\mathrm{\Delta }v-\alpha v+\beta u,\end{array}$

where

• $u(x,t)$ represents the density of cells,
• $v(x,t)$ is the concentration of the chemoattractant,
• $D_u,D_v>0$ are diffusion coefficients,
• $\chi >0$ denotes chemotactic sensitivity,
• $\alpha ,\beta$ are parameters for signal decay and production.

Applications

The Keller–Segel system has been widely used in mathematical biology to study:

• bacterial chemotaxis,
• slime mold aggregation,
• tumor angiogenesis,
• and ecological population dynamics.

It has also served as a prototype model in applied mathematics, illustrating how nonlinear PDEs can capture pattern formation, blow-up phenomena, and self-organization.

Mathematical properties

A major research focus concerns the global existence and blow-up of solutions. In two dimensions, the system exhibits a critical mass phenomenon: below a certain threshold of initial cell density, solutions remain globally bounded, while above it, solutions may blow up in finite time, modeling cell aggregation into singular clusters.

See also

• Chemotaxis
• Reaction–diffusion system
• Nonlinear partial differential equation