The first mathematical model for elk–wolf interaction in Yellowstone National Park using the E-SINDy algorithm

The ecological dynamics between elk and wolves in Northern Yellowstone have been a focal point of long-term research, particularly following the reintroduction of wolves to the region. Although numerous studies have explored this prey–predator interaction from ecological and behavioral perspectives, there remains a lack of comprehensive analysis using mathematical modeling approaches capable of uncovering underlying dynamical patterns. In this study, we investigate the prey–predator dynamics of the elk–wolf system in Northern Yellowstone National Park, using a data-driven modeling approach. We used yearly population data for elk and wolves from 1995 to 2022 to construct a mathematical model using a sparse regression modeling framework. To the best of our knowledge, no previous work has applied this framework to capture elk–wolf interactions over this time period. Our modeling pipeline integrates gaussian process regression for data smoothing, sparse identification of nonlinear dynamics for model discovery, and model selection techniques to identify the most suitable mathematical representation. Stability and bifurcation analyzes are then performed to understand the system’s qualitative behavior. A saddle–node bifurcation identifies the parameter range in which both species can coexist, while values outside this range lead to the extinction of one or both species. Hopf and saddle–node bifurcations define regions of stable co-existence, periodic oscillations, and extinction. Bifurcations such as Bogdanov–Takens and cusp are examined by varying two parameters simultaneously. Ecologically, these bifurcations reflect the interplay between wolf pressure and elk defence strategies. They suggested that small changes in parameter can trigger sudden shifts between co-existence, oscillations, or extinction.