We numerically investigate a nonlinear system of parabolic partial differential equations modeling the negative chemotaxis interactions between a biological species and a lethal chemical substance that is externally supplied. The work extends the knowledge regarding the solutions to an ODE system to which the solutions of the original PDE model converge, as well as the regime of this convergence beyond the existing analytical results. In particular, for a periodic supply of the substance, a threshold value for the periodicity of the solutions to the ODE system is determined through systematic numerical experiments. Under the obtained conditions – weaker than the current analytical characterization – the convergence and eventual periodicity of the solutions to the PDE model is verified by meshless numerical simulations using the Generalized Finite Difference (GFD) method. Lastly, an optimal control problem is considered, and an approximate solution is constructed. A Forward-Backward Sweep algorithm combined with the GFD resolution provides the approximate optimal states. • Solutions to a parabolic chemotaxis system are numerically investigated. • Under a periodic source term, a periodicity threshold is determined by experiments. • The Generalized Finite Difference Method is used for the numerical resolution. • Convergence properties are numerically obtained beyond analytical characterizations. • An approximate solution to an optimal control problem is built.
Herrero-Hervás et al. (2026) studied this question.