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  • Thumbnail for A three-species model with predator-prey, competition, and mutualistic interactions
    A three-species model with predator-prey, competition, and mutualistic interactions by Liu, Minqi

    This research is inspired by Brown, Bruder and Kummel’s research project on the predator-prey interaction of aphids and ladybugs on yucca plants. An important feature of this study system is that it contains ants as a third species. Therefore, this ecological system is composed of a predator-prey relationship between the ladybugs and aphids, a competitive relationship between the ladybugs and ants, and a mutualistic relationship between the aphids and ants. Most existing mathematical models study one type of interaction or they focus on three species and study a tri-trophic food chain. We develop and analyze a new mathematical model that includes the predator-prey interaction as well as the competitive and mutualistic aspects of the system. The predator-prey interaction is described by a Rosenzweig-MacArthur model, which assumes logistic growth of the predator. To build a mathematical model for the competitive and mutualistic relationships, we use a modified Lotka-Volterra model and include terms representing competition and mutualism. Since the three-species model is substantially harder to analyze, we first study the three submodels, i.e. the predator-prey, competition, and mutualism model. Then we use the submodel results to explore the three-species model and the significance of its parameter values. With the help of Mathematica and MATLAB, we construct phase planes and time series plots, find the equilibria of the systems, and determine the stability of each equilibrium.