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The Keller-Segel model is a mathematical model of chemotaxis, proposed in 1970 by Evelyn Fox Keller and Lee A. Segel to describe chemotaxis-driven aggregation of populations of unicellular organisms.

In its original version the Keller-Segel model is written in terms of a system of two coupled partial differential equations, one describing the evolution in time of the density of a population of amoebae and the second describing the evolution in time of a chemical signal that mediates chemotaxis, called acrasin. More generally, the Keller-Segel model can be used to describe the aggregation of a population of unicellular organisms (such as bacteria) by the means of chemical stimuli, called chemoattractants in case of positive chemotaxis or chemorepellents in case of negative chemotaxis. The Keller-Segel model is well-known in mathematical biology and has been considered vastly as a model to describe a variety of applications, for instance in ecology and economics, other than a number of different phenomena in biology, such as neurodegenerative deseases and cancer growth.

The Keller-Segel model is also known in mathematics as the Patlak-Keller-Segel model. In fact, the model proposed by Keller and Segel was anticipated in a different form and for a different application by the work of Patlak in 1953.

Original formulation
In their first work, Keller and Segel describe from a mathematical point of view the aggregation of a population of amoebae which is able to self-organize thanks to the production of two chemical signals: acrasin and acrasinase. Both of these signals are emitted by the amoebae. Moreover, the system accounts for the presence of a fourth substance, a complex that results from the chemical reaction of acrasin and acrasinase. The original formulation of the Keller-Segel model takes the form of a system of four partial differential equations for the evolution of amoebae, acrasin, acrasinase and the complex. These four variables are represented in terms of their concentrations, respectively $$ a(t,x), \rho(t,x), \eta(t,x), c(t,x) $$ at time $$ t$$ and at position $$ x $$ in space. The system has the general form:

$$\begin{align} \begin{cases} \partial_t a &= -\nabla \cdot \left( D_1 \nabla a \right) + \nabla \cdot (D_2 \nabla a)\\ \partial_t \rho &= D_\rho \nabla^2 \rho - k_1 \rho \eta + k_{-1} c + a f(\rho) \end{cases} \end{align}$$