Teaching dynamical systems modeling, analysis and simulation using a Lotka-Volterra Predation Narrative and a Numerus WebKit App

Wayne Marcus Getz , Richard Salter

The Lotka-Volterra prey-predator (LVPP) equations are among the few ecological models that have become a standard part of the undergraduate mathematics curriculum. Rather than looking elsewhere for fresh examples, we use this familiar story as a single narrative thread along which the toolkit of dynamical systems analysis and simulation is introduced, one modeling assumption at a time. Five linked teaching modules progress from the classical model (equilibria, isoclines, linearization, a conserved quantity and the failure of Euler's method on a neutral center), through logistic prey growth (structural stability, Lyapunov functions), a prey Allee effect (bistability, separatrices, basins of attraction) and Holling functional responses (Hopf bifurcations, limit cycles, the paradox of enrichment), to a culminating model with predator interference and ratio dependence, whose non-differentiable origin is attracting but not Lyapunov stable. Multispecies extensions then lead to Gause-Lotka-Volterra competition, May-Leonard heteroclinic cycles, chaos in the Hastings-Powell food chain, May's stability-complexity criterion, and metaphysiological consumer-resource ``trophic stacks'' in which food-chain length depends non-monotonically on predator self-interference and in which alternative stable states and emergent Allee effects arise. Nine worked examples pair analytical results with simulations in a Numerus WebKit app, a browser-based simulator whose presets require no programming and whose AI window lets students modify the models or build their own. We discuss how the narrative serves students in the biological, social and applied sciences, and argue that comparing alternative narratives of predation teaches students that models are arguments to be tested, refined and sometimes replaced.