On a negative chemotaxis system with lethal interaction

In this work, we propose and analyze a mathematical model describing the interaction between a biological species and a lethal chemical substance, incorporating diffusion, negative chemotaxis, logistic growth, and toxicity effects. Motivated by the interaction between E. coli bacteria and hydrogen peroxide, the model accounts for a substance that simultaneously induces cell death and is self-produced by the population. The dynamics are described by a system of nonlinear parabolic partial differential equations with an external supply of the substance. For constant supply rates, we study the local stability of spatially homogeneous steady states, showing that the balance between the logistic growth rate and the supply determines the linearized behavior of the system. When the supply is asymptotically time-periodic, we establish threshold conditions for the existence of periodic solutions through the analysis of an associated ODE system. A numerical scheme based on the Generalized Finite Difference method is developed, and its convergence to the continuous solution is established. Numerical simulations are presented to validate the analytical results and to illustrate additional dynamical phenomena, including pattern formation.