# Population Interactions in Ecology

The SIAM Review Education section has many objectives. One of them is to bring contemporary applications to mathematics courses that would otherwise be outside of the mainstream curriculum. Another is to present topics in applied mathematics in new ways that provide a pedagogical advantage. In this issue, we feature an article that does both. In “Population Interactions in Ecology: A Rule-Based Approach to Modeling Ecosystems in a Mass-Conserving Framework,” authors Cropp and Norbury take a comprehensive and modern look at population modeling.

In many differential equations and dynamics textbooks, population dynamics begins and ends with the famous Lotka—Volterra model with perhaps some mild variations as exercises. Presented as an application of the analysis of systems of differential equations, this approach leaves the student and instructor with the sense that the deep questions of ecology can be answered with attractors and limit cycles. Unfortunately, those in the life sciences community do not accept the model as relevant. Indeed, if this is the only exposure mathematics students have to population models, it may leave students with an overly simplistic perspective on ecology. On the other side of the disciplinary divide, courses for nonmathematics majors tend to present population ecology in a “just in time” fashion where solutions to the Verhulst equation are presented as formulae without much explanation. These students will be left without the necessary modeling and analytical skills to quantitatively study ecological systems. This Education section feature takes on the problem directly through what they call the “conservative normal” (CN) framework. While the approach is a theoretical basis for understanding ecological systems, it has considerable merit as a way for students to construct and understand models of populations in a food web.

The CN framework is a system of rules that maps ecological principles to mathematical structures and formulae. For instance, the first rule is that one should measure a population by the limiting nutrient it contains (e.g., do not model the number of plankton cells, model the mass of nitrogen in the living population). It is both a modeling approach and a unified way for students to think about many interacting species. Other rules include mechanisms for change in populations, resource constraints, and other essential processes that govern ecological systems. The authors then apply CN to a variety of small but interesting problems. These include ecological systems of autotrophs where multiple species compete for a single resource, and systems of mixotrophs such as phytoplankton that can switch between growing inefficiently via photosynthesis and grazing on other phytoplankton. Given that phytoplankton consume a significant fraction of the carbon dioxide produced from burning fossil fuels, one could argue that it may be more relevant for nonmathematicians to learn about these mixotrophic systems before dwelling on cycles in the Canadian Lynx and the Snowshoe Hare populations. The CN framework puts whole categories of ecological systems on equal footing for students to explore.

In summary, this article is a notable contribution that spans a disciplinary divide between the mathematics and life sciences communities. As such, the CN framework may be a valuable resource in classes for mathematicians as well as classes for nonmathematicians. On one hand, it presents classic applications of differential equations in an exciting new way, and on the other, it links biological principles to mathematical structures in a way that will appeal to nonmathematicians. While the examples in this article are aimed at the undergraduate curriculum, the CN framework is useful in many broad contexts, including systems with spatial dependence that would likely end up in a graduate curriculum or more specialized course. Most importantly, for the relatively low cost of introducing the framework, this approach gives students a broader perspective on ecological systems and the modeling process in general than what is typically provided in differential equations and dynamical systems texts.

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*SIAM Review.*, 57(3), 437-465.