Recent Developments in the Numerical Methods for the Chemotaxis Models

Here, ρ(x, y, t) is the cell density, c(x, y, t) is the chemoattractant concentration, χ is a chemotactic sensitivity constant, Ω is a bounded domain in R, ∂Ω is its boundary, and n is a unit normal vector. Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. There exists an extensive literature about chemotaxis models and their mathematical analysis a first place to start is [23], as well as [15, 16], and for a deeper background [2, 8, 21, 1, 7, 3, 4, 24]. The first descriptions of the mechanism owe to Keller and Segel, [17, 18, 19] and Patlak [22]. In this description, the organism or migrating enzyme chooses a direction upwards of a chemical signal which leads to aggregation. Although there is an extensive literature on this subject, only a few numerical methods have been proposed for these models. Chemotaxis models are usually highly nonlinear due to the density dependent cross diffusion term (attracting force) that models chemotactic behavior, and hence, any realistic chemotaxis model is too difficult to solve analytically. Therefore, development of accurate and efficient numerical methods is crucial for the modeling and analysis of chemotaxis systems. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. This blow-up represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems, see, e.g., [1, 2, 3, 4, 8, 24]. In either case, capturing such solutions numerically is a very challenging problem.