  Many phenomena in biology involve both reactions and chemotaxis. These
processes can clearly influence each other, and chemotaxis can play an
important role in sustaining and speeding up the reaction. In continuation of
our earlier work, we consider a model with a single density function involving
diffusion, advection, chemotaxis, and absorbing reaction. The model is
motivated, in particular, by the studies of coral broadcast spawning, where
experimental observations of the efficiency of fertilization rates
significantly exceed the data obtained from numerical models that do not take
chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes)
into account. We consider the case of the weakly coupled quadratic reaction
term, which is the most natural from the biological point of view and was left
open. The result is that similarly to higher power coupling, the chemotaxis
plays a crucial role in ensuring efficiency of reaction. However,
mathematically, the picture is quite different in the quadratic reaction case
and is more subtle. The reaction is now complete even in the absence of
chemotaxis, but the timescales are very different. Without chemotaxis, the
reaction is very slow, especially for the weak reaction coupling coefficient.
With chemotaxis, the timescale and efficiency of reaction are independent of
the coupling parameter.
