Traditional fine-scale mathematical models, such as ordinary differential equations (ODEs), have given valuable insights into the long-term behaviors of interacting species. However, they can be difficult to analyze. The higher the number of species and nonlinear interactions, the more challenging it is to find all the qualitative solutions to, say, a system of ODEs modeling a community. More species and interactions often introduce more unknown parameters into the model, where two choices of parameter values may lead to entirely different qualitative dynamics. Thus, increasing interactions and species often produces a community that is infeasible to rigorously and numerically explore.

To bypass this mathematical hurdle, I will take a step back from fine-scale dynamics and introduce a new mathematical framework: Widespread Ecological Networks and their Dynamical Signatures (WENDyS). I will show how WENDyS takes a system of species and their relative strengths of interactions and translates them into piecewise constant population growth models. In particular, at the cost of trading fine-scale for coarse-scale dynamics, we rigorously produce a finite library of all long-term outcomes that can occur between a community of species.

Joint work with Drs. Konstantin Mischaikow (Dept. of Mathematics; Rutgers), Marcio Gameiro (Dept. of Mathematics; Rutgers), and Juan Bonachela (Dept. of Ecology, Evolution, and Natural Resources; Rutgers).