Reaction enhancement by flux-limited chemotaxis

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. Quite often it acts to improve efficiency of biological reactions; one example is the immune system signalling, where infected tissues release chemokines attracting monocytes to fight invading bacteria. Another example is reproduction, where eggs release pheromones that attract sperm. In this paper, we analyze a system of two reacting densities, one of which is chemotactic on another. Since the speed of any biological agents is limited, we employ flux limited chemotaxis model. Our main result is the rigorous derivation of the scaling laws showing how presence of chemotaxis affects the typical reaction time scale. This work builds on the results of \cite{kiselev2022chemotaxis}, which employed a classical Keller-Segel chemotaxis term (not flux limited) - leading to the effect of possible over concentration and restricting the results to radial data. The model presented here is more reasonable biologically and covers broader parameter regimes.