(A) study on optimal resource distribution maximizing total population

Abstract

This paper explores research in population dynamics, a field of mathematical biology. In particular, we research the effects of resource distribution on the total population of species living in a bounded habitat using logistic reaction-diffusion equations. One of the main challenges in this area of research is determining the optimal resource distribution that maximizes the population. This paper introduces two optimization questions and presents their respective answers. The first question concerns the ratio of total population to total resources. We show that this ratio can be infinitely large in high dimensions under certain environmental conditions. Next, we address the issue of species that spread at a slow rate. The concentration of resources is generally advantageous for a species’ survival, but for species with slower spreading speeds, fragmented resources may be more beneficial for maximizing populations. Using the bounded variation norm, we demonstrate that optimal resource distribution becomes fragmented as the diffusion rate decreases in a bounded habitat with a smooth boundary. Additionally, we introduce several related issues to which our results may be applied.