Modeling a Three-Species Predator-Prey Ecosystem

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Robert Hesse, Mathematics


Differential equations are useful for mathematically modeling an event in which values are changing simultaneously in time. An example of such an event is a naturally existing ecosystem. In a model of an ecosystem, a differential equation is used to model the growth and death rates of each species, creating a system with one equation per species. These models are used by institutions such as the Department of Natural Resources when determining hunting and fishing limits for a given year.

Modeling an ecosystem with differential equations is particularly helpful when finding equilibria and examining the long-term behaviors of the populations. Not only do equilibria tell a lot about populations in an ecosystem over a period of time, but the differential equations can also be used to find the eigenvalues of these equilibria, which describe the behavior of the populations around these specific values. Differential equations make these long-term behaviors easy to determine because equilibria and eigenvalues for an ecosystem modeled by differential equations can be found using algebra and linear algebra.

The goal of this research was to compare different models at similar points and determine whether or not all the models have similar behavior around the same equilibria, even when the models became more complex. This research first compared the basic two-species predator-prey Lotka-Volterra model and the three-species version of the same model. Next, an unexpected behavior of the Lotka-Volterra model was examined. Two variations of the three-species Lotka-Volterra model were then compared. Finally, the Arditi-Ginzburg model, which is a different way of modeling a predator-prey ecosystem was compared to the previous models. All of the models examined in this research displayed similar spiral sink behavior, even as the models became more complex. This indicates that it may not be worth the extra cost of computation by adding more information from the environment, as the models display similar behavior around their equilibria.

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