Tec de Monterrey, 2020

The project addressed a fascinating challenge: predicting the long-term balance of Yellowstone’s ecosystem by modeling the populations of caribou (prey), wolves (apex predators), and coyotes (mesopredators). The ultimate goal was to generate a tool that could inform park management decisions, such as hunting restrictions, reintroduction programs, or habitat protection, to preserve ecosystem stability.

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At the core of the project was a system of differential equations representing the growth and decline of each species:

• Caribou (K): affected by natural birth rates and predation from wolves and coyotes.

• Wolves (L): population growth tied to caribou availability, with a natural mortality rate.

• Coyotes (C): similarly dependent on caribou presence, but also in competition with wolves for territory and prey.

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Using historical population data from Yellowstone’s public database, the team performed multiple linear regression to estimate the coefficients (α, β, γ, δ, ε, η, λ) for the differential equations. These coefficients captured how strongly each species’ population influenced the others.

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The team implemented the model using Python with the Runge–Kutta method to numerically solve the differential equations. The code was published in Google Collab, allowing interactive modification of time steps, population sizes, and coefficients. This flexibility meant park managers could simulate various scenarios, such as gradual reintroduction of coyotes, and observe their long-term impact on the ecosystem.

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Graphical outputs showed:

• Cyclic behavior of caribou and wolves; when wolf populations rose, caribou declined sharply, followed by wolf starvation, which allowed prey populations to rebound.

• Coyote decline as wolves became more numerous, due to direct predation and competition.

• Equilibrium points where all three species coexisted, though with fluctuating numbers over time.

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Predictions were made for 50, 100, 150, and 200 years into the future:

• Caribou showed long-term recovery, reaching nearly 40,000 individuals by year 200.

• Coyotes dropped to near extinction (0.004×10³) unless actively reintroduced.

• Wolves stabilized at a moderate population (~5,700) after cycles of growth and decline.

The model recommended a gradual reintroduction of coyotes to avoid dramatic ecosystem shifts, as well as targeted hunting only of species predicted to grow exponentially and threaten balance.

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The team noted that the original caribou rate of change provided in the dataset was not consistent with observed population behavior. When recalculated manually, the model predicted complete caribou extinction in ~15 years if current trends continued unchecked; a catastrophic scenario that would also eliminate wolves and coyotes due to starvation. This highlighted the importance of verifying input data before relying on model outputs for policy decisions.

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Beyond academic analysis, the project delivered a working decision-support tool for park managers, allowing them to simulate interventions and plan conservation strategies proactively. By combining mathematics, programming, and ecology, the team created a resource with real potential to guide wildlife management and preserve the delicate balance of Yellowstone’s ecosystem.

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Please find attached below the relevant documents to this project. (Note: most, if not all documents, are in Spanish)