On a Negative Chemotaxis System with Lethal Interaction

Abstract

.In this work, we propose and analyze a mathematical model describing the interaction between a biological species and a chemical substance through motility, negative chemotaxis, and lethality. The model is based on the interaction of \textit{E. coli} bacteria and hydrogen peroxide ($H_2O_2$), representing the dual role of $H_2O_2$: acting as a chemorepellent that causes bacterial cell death while also being produced by the bacteria themselves. The process is modeled using a system of parabolic partial differential equations, which in dimensionless form is given by$$\begin{cases} \displaystyle u_t = D \Delta u + \chi \nabla \cdot ( u \nabla v) + ru(1-u) - uv , & \quad x \in \Omega, ~ t > 0, \\\\ \displaystyle v_t = \Delta v + a u - v + f(x,t), & \quad x \in \Omega, ~ t > 0, \end{cases}$$where $u$ represents the population density of the species, $v$ is the concentration of the substance, $f$ is a source term representing an external supply of the substance and $D, ~ \chi,~ r >0$ and $a \geq 0$ are parameters, together with non-negative initial values and Neumann homogeneous boundary conditions. We first analyze the linear stability of spatially homogeneous steady states for a constant $f$. For $r < f$ , the only biologically relevant state, $(0, f)$ is locally asymptotically stable. Conversely, when $r > f$, a secondary equilibrium state $(u_*, v_*)$ emerges, which is locally asymptotically stable, while $(0, f)$ becomes unstable. The existence of periodic solutions is also investigated when $f$ represents an asymptotically time-periodic supply, deriving a threshold value for $r$ that guarantees its existence. Next, we develop a numerical scheme using the Generalized Finite Difference Method, providing a brief introduction to the method, along with proof of its convergence to the solutions of the system. The work is completed with a numerical study of the analytically studied cases and as well as some final remarks on pattern formation.