A Reaction-Diffusion-Chemotaxis Model for Human Population Dynamics over Fractal Terrains

Abstract

Advection of entities induced by gradients in attractant concentration fields is observed via diffusiophoresis in colloids and via chemotaxis in microorganisms. Mathematically, both diffusiophoresis and chemotaxis follow similar mathematical descriptions and display a variety of interesting behaviors that are not observed through other transport mechanisms. However, the application of such a mathematical framework has largely been restricted to soft matter research. In this article, we argue that this framework is more general and can be expanded to study human population dynamics. We assert that human populations also migrate chemotactically, but by sensing concentrations gradients in attractants such as resource availability, social connections, and safety indices. Therefore, we extend the Fisher-KPP reaction-diffusion model, foundational to human population dynamics, to incorporate chemotactic advection. Furthermore, we introduce a fractal terrain to better mimic the human dispersal phenomena. Simulations demonstrate that, by including chemotaxis of a population toward attractants which are dispersed heterogenously over fractal terrains, population hotspots can appear from from initially uniformly dispersed states whereas Fisher-KPP without chemotaxis predicts a persistent tendency toward population uniformity. Varying the chemotactic migration yields fine control over inter- or intra-population segregation and thus the population growth rates may be substantially altered by considering the population-attractant coupling. This framework may be useful for characterizing historical population separations, and furthermore is particularly pertinent for predicting emergence of new population hotspots as climate change is expected to cause large-scale human displacement, which may be dictated by chemotactic movement of humans due to evolving concentration gradients in safety indices.