The First Mathematical Model for Elk Wolf Interaction in Yellowstone National Park Using the E-SINDy Algorithm

TL;DR

This work addresses elk–wolf dynamics in Yellowstone through a data-driven framework that combines Gaussian process regression for smoothing with ensemble SINDy (E-SINDy) to discover minimal nonlinear models from multi-decadal data. After model selection via AIC/BIC, the authors perform nonlinear stability and bifurcation analyses, revealing a cubic, interpretable system that supports a stable coexistence equilibrium and delineates parameter regions leading to extinction or oscillations. Key findings include a data-driven model that closely tracks observations and predicts future dynamics, an ecologically meaningful parameterization that yields a critical elk threshold $e_c = a_2/a_3$, and rich bifurcation structures such as saddle-node, Hopf, Bogdanov–Takens, and cusp points under parameter variation. The study provides a transparent, mechanistic framework for understanding long-term prey–predator interactions in a protected ecosystem and offers a template for applying similar data-driven approaches to other ecological communities.

Abstract

In this study, we investigate the prey predator dynamics of the elk wolf system in northern Yellowstone National Park, USA, 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. The resulting model is analyzed for its nonlinear dynamics with ecologically meaningful parameters. Stability and bifurcation analyzes are then performed to understand the systems qualitative behavior. A saddle node bifurcation identifies parameter ranges where both species can coexist, while regions outside this range may lead to the extinction of one or both populations. Hopf and saddle node bifurcations together delineate zones of stable co existence, periodic oscillations, and extinction scenarios. Furthermore, co dimension two bifurcations, including Bogdanov Takens and cusp bifurcations, are explored by varying two parameters simultaneously. Ecologically, these bifurcations reflect the complex interplay between wolf pressure and elk defence mechanisms, such as grouping or herd behavior. They suggested that small changes in ecological parameters can lead to sudden shifts in population outcomes ranging from stable co existence to extinction or oscillatory cycles.

Figures (17)

• Figure 1: Study area map
• Figure 2: Schematic diagram of the modeling and analysis framework
• Figure 3: Time series plot of the Northern Yellowstone National Park yearly elk and wolf population dataset. Elk and wolf photos are from Yellowstone Park NPS_YellowstonePhotography.
• Figure 4: Elk and Wolf populations after Z-score normalization of the original dataset
• Figure 5: Regularisation of (a) elk, and (b) wolf data. The solid lines are the mean values, and the shaded areas are the $95\%$ confidence interval of the GPR. The dots are the original time series data.