Chemotaxis and Reactions in Anomalous Diffusion Dynamics

Abstract. Chemotaxis and reactions are fundamental processes in biology, often intricately intertwined. Chemotaxis, in particular, can be crucial in maintaining and accelerating a reaction. In this work, we extend the investigation initiated by Kiselev and Ryzhik [ Comm. Partial Differential Equations, 37 (2012), pp. 298–318] by examining the impact of chemotactic attraction on processes such as reproduction (broadcast spawning) in the context of anomalous diffusion of gamete densities. We analyze two contrasting scenarios: a chemotaxis-free case and a chemotactic case, the latter in two spatial dimensions. For that, we consider a partial differential equation, with a single density function, that includes advection (assuming a divergence-free velocity field to model ambient flow), chemotaxis, absorbing reaction, and diffusion, incorporating the fractional Laplacian [Formula: see text]. The inclusion of the fractional Laplacian is motivated by experimental evidence supporting the efficacy of anomalous diffusion models, particularly in scenarios with sparse targets. The fractional Laplacian accommodates the nonlocal nature of superdiffusion processes, providing a more accurate representation than traditional diffusion models. Our proposed model represents a step forward in refining mathematical descriptions of cellular behaviors influenced by chemotactic cues.