Modeling Lotic Organism Populations with Partial Differential Equations

We present in this paper a mathematical model for a population of caddisfly larvae in the Upper Mississippi River, which can either live in the current of the water or fix themselves to large wood debris submerged throughout the river. The model consists of a system of partial differential equations which captures these coupled dynamics. After introducing the model, we give a qualitative analysis of the dynamics, which includes a steady state solution followed by a numerical solution to the system using a finite difference scheme implemented in R. Finally, we extend the model to a competitive system with the goal of capturing the dynamics of the interaction between native caddisfly larvae and invasive zebra mussels. Our results demonstrate that although analyzing the exact behavior of lotic organism populations remains a difficult task, utilizing mathematical models such as the one presented in this paper can lead to further knowledge regarding which characteristics of lotic organisms have greatest influence over population growth.