Reaction enhancement by chemotaxis

An interesting problem arising in many contexts of mathematical biology is the study of the relevance of chemotaxis in reaction-diffusion processes. We approach this problem through a mathematical model representing the fertilization process of corals. As a result we obtain a system of partial differential equations describing the cell dynamics being affected by three basic phenomena: diffusion, chemotaxis and a surrounding flux. First we prove that our model has in general global classical solutions. Next, we compare the asymptotic behavior with and without chemotaxis. We show the relevance of chemotaxis after making a systematic adaptation of the well-known moments technique on bounded domains (usually used for proving blow-up in Keller-Segel systems) to analyze the behavior of the cell dynamics when the chemotactic signal increases. In the context of the proposed model, our analysis shows that for suitable initial data, the remaining fraction of unfertilized eggs at any given time τ>0 becomes arbitrarily small if the chemotaxis signal is sufficiently large and the sperm and eggs are concentrated enough at the initial time.