Numerical Simulation of Chemotactic Bacteria Aggregation via Mixed Finite Elements

We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria. Mathematics Subject Classification. 35Q, 65M, 92B, 92C. 1. Problem formulation The equations for the collective motion of the bacteria can be derived (with no free parameters) from the underlying biochemistry. The basic equations for the bacterial density ρ and the attractant concentration c are (see [1–3,13]) ∂ρ ∂t = Db ∇ρ−∇ · (kρ∇c) + aρ, (1) ∂c ∂t = Dc∇c+ αρ, (2) where Db is the bacterial diffusion constant, k is the chemotactic coefficient (or chemotactic sensitivity), a is the rate of bacterial division, α is the rate of attractant production, Dc is the chemical diffusion constant. The terms in equation (1) include the diffusion of bacteria, chemotactic drift and division of bacteria. Equation (2) expresses the diffusion and production of attractant. In this model, the production of attractants is taken proportional to the bacterial density, as a first approach. But other models, certainly more realistic exists, see for example the model given by equation (7).