Population Interactions in Ecology: A Rule-Based Approach to Modeling Ecosystems in a Mass-Conserving Framework

Abstract

Mathematical biology/ecology teaching for undergraduates has generally relied on the Lotka--Volterra competition and predator-prey models to introduce students to population dynamics. Students are provided with an understanding of the application of dynamical system theory in simulating and understanding the behavior of the natural world, and they are provided with opportunities to practice phase plane analysis techniques such as determining the stability of equilibrium points and bifurcation analysis. This paper outlines a course in ecological modeling suitable for all students in the life sciences. The course is based on realistic ecological principles, such as using nutrient concentration to measure populations together with explicit resource availability to constrain population growth, and it considers simple Lotka--Volterra systems within this theoretical framework. An advantage of this approach is that the widely experimentally observed models of mixotrophy and mutualism can be naturally and simply introduced and analyzed. Continuous variation of models across a trophic level is now possible. Competitors can smoothly change to mutualist/mixotroph populations, which can further smoothly change to become predators, synthesizing in simple terms the relationships among trophic interactions within the Lotka--Volterra framework. Standard texts on mathematical ecology do not include mixotrophy, which is central to understanding trophic interactions.