Fisher-KPP model with chemotaxis over fractal terrains

The advection of entities induced by gradients in attractant concentration fields is observed through diffusiophoresis in colloids and chemotaxis in microorganisms. Mathematically, both diffusiophoresis and chemotaxis follow similar 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 entities influenced by forcings of a single scale. In this article, we argue that this framework is more general and can be expanded to study entities migrating along terrains that induce forcings of many scales. We extend the Fisher-KPP reaction-diffusion model, foundational to population dynamics for both small (cellular) and large (ecological) scales, to incorporate chemotactic advection. Furthermore, we introduce a fractal terrain to better mimic the population dispersal phenomena. Simulations demonstrate that, by including chemotaxis of a population toward attractants that are dispersed heterogeneously over fractal terrains, population hotspots can appear 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 intrapopulation segregation, and thus the population growth rates may be substantially altered by considering the population-attractant coupling. This framework may be useful for characterizing population hotspots over heterogeneous landscapes.