Mathematical Modeling of Fish Populations in Lake Ontario using Differential Equations

Abstract

The purpose of this research is to use mathematical models to study the connection between the rainbow trout fish population and the lamprey population in Lake Ontario. These species have a parasite/host relationship. The lamprey, a destructive and invasive species, give the rainbow trout scars and wounds that hinder their life spans. I chose to use models that are traditionally used for predator/prey relationships. It is an acceptable method because by definition predation includes parasitism [8]. Besides, mathematical models will only take the most dominant features into account. The predator/prey model quantifies what happens when the predators eat their prey. In this case the lamprey are not eating the trout, but they are still causing them harm. Because the harm tends to negatively impact the trout’s reproduction rate, the model seems well-suited for this situation. After studying available data I adopted a system of two differential equations that incorporate parameters measuring four factors. These factors include how aggressively the trout are being depleted by the lamprey, how much the lamprey benefit from the trout, and the natural growth of the trout (absent lamprey) and decline of the lamprey (absent trout). By using these equations to quantify the fish population levels as time passes, we find clues to which types of dynamics can be expected. As an experimental methodology, we modify each parameter to see the effect upon the population dynamics. We find mathematical evidence that the observed increases and decreases in lamprey population in [8] are due not just to human intervention, but arise naturally from the parasite-host relationship. The effects of various methods to help reduce the impact of the hurtful species in the environment will be also discussed.