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.
Invasive species have the potential to drastically shift the community composition of habitats through increased competitive interactions. In economically important ecosystems, this can cause populations to decline and economies to collapse. The European Green Crab (Carcinus maenas) is an omnivorous and prolific crab which has invaded much of both North American coasts, causing damage to some major bivalve fisheries. My study sought to identify the population trends of the green crab along a river estuary system in the Gulf of Maine. In addition, my study investigated the potential for interspecific competition between rock crabs (Cancer irroratus) and green crabs. Subtidal traps were placed biweekly along the shoreline of the Damariscotta River for the 2013, 2014, 2015, and 2018 summer seasons. Carapace width, total weight, sex, ovigerous status, and number of intact legs per individual were quantified for each individual of both species (N=1,208). A caged in-lab experiment was used to observe potential competition for food between rock crabs and green crabs of the same size. My study was unable to find a significant difference between 2015 and 2018 green crab catch rates, suggesting there was no population growth present. In addition, rock crabs were identified as the dominant species in my lab trials. This study suggests that competitive dominance in the native crab species could have the potential to shift the habitat of the invasive green crab to a higher position in the water column, which could limit any further population growth. As green crabs significantly contributed to the collapse of many bivalve fisheries (e.g. the soft-shelled clam), this result could help to inform more effective fisheries and aquaculture management.
In 2000, three seasons after the institution of a revenue sharing plan, Major League Baseball commissioned the Blue Ribbon Panel to assess competitive balance within the league. Their report found that small market teams are at a considerable disadvantage due to the larger revenue bases of teams located in more heavily populated areas. However, these results have often been challenged. This thesis builds upon existing models in an attempt to determine the extent to which market size and local revenue independently affect competitive balance. Additionally, it seeks to analyze any effects of baseball's revenue sharing plan on competitive balance. While the findings support the claim that revenue sharing enhances competitive balance, they fail to establish market size as a positive determinant of local revenue.