A Localized Meshless Method for a Two-Dimensional Chemotaxis Model

Chemotaxis is the directed movement of cells or organisms in response to chemical gradients, a process critical to biological functions such as immune responses and bacterial foraging. It is typically modeled using the Keller–Segel system, comprising a set of nonlinear advection–diffusion–reaction partial differential equations (PDEs). Numerically solving these models is challenging. Traditional finite difference methods are largely restricted to simple rectangular domains, while finite element methods, although capable of handling complex geometries, depend on computationally expensive mesh generation.

This work proposes a localized meshless method designed to overcome these limitations. The technique completely avoids mesh generation, making it equally effective for irregular domains with both uniform and non-uniform node distributions. Our problem-dependent approach uses the Laplacian as the main differential operator to derive radial basis function (RBF)-based particular solutions. These solutions are then used to approximate the spatial operators in the time-dependent PDE, thereby reducing the problem to a system of ordinary differential equations (ODEs). This resulting ODE system can be efficiently integrated in time using standard, high-performance solvers.

The proposed meshless framework is validated through numerical simulations of the Keller–Segel chemotaxis model. The results confirm the method's competitive performance in terms of numerical accuracy and computational efficiency, demonstrating its strong potential as a flexible alternative for solving complex biological PDEs.