Grand Valley State Univ REU Summer 2022   --   Project Description -- Faculty Mentor: Norma Ortiz-Robinson

## Data driven modeling and optimal control of the wolf and moose populations in Isle Royale, Michigan.

Isle Royale is an island located in Lake Superior just north of Michigan’s Upper Penninsula. Although the populations of moose and wolves in the island have cycled through the years, recent declines in wolf population has resulted in the National Park Service’s plan to reintroduce between 20-30 wolves[5]. In this project we seek to fit and analyze a data-driven differential equations model of the Predator-Prey relationship between moose and wolves of Isle Royale. In addition, we will use tools from optimal control theory and the GEKKO Optimization suite for Python [3] to obtain and analyze optimal strategies that can achieve a sustainable wolf population in the island.

A predator-prey relationship between species can be expressed by the Lotka-Volterra system of differential equations of the form

${dx\over dt} =ax-bxy$ ${dy\over dt} = -cy+dxy$

in which the variable x is a function of time and represents the population levels of the prey and y similarly represents the predator. We thus seek to use the data available [5] to find the best possible fit for the parameters a,b,c, and d. Once the model is obtained, equilibrium points will be analyzed and classified. Finally, we introduce a control function u(t) to the model to represent the rate at which the population of wolves is increased and formulate an optimal control problem. Numerical optimal solutions will be obtained and analyzed in the context of current reintroduction practices. This study resembles, in part, the methods utilized in [2] while [1] and [4] will serve as references.

### Required Preparation:

Interested students should have successfully completed Undergraduate level courses in Differential Equations and basic programming (or show programming experience).

## Application and more information may be found on the GVSU REU website.

### References

[1]   Fister, K. R., & Lenhart, S., Optimal Harvesting in an Age-Structured Predator–Prey Model. Applied Mathematics and Optimization, 54(1), 1–15(2006).

[2]   C. Foster-Bey, N Ortiz-Robinson, Control strategies to contain SARS-CoV-2 in a data driven SIR model for the state of Michigan, USA, Letters in Biomathematics, (Accepted, August 2021)

[3]   Beal, L. D., D.C. Hill, R.A. Martin, and J.D. Hedengren, “Gekko Optimization Suit”, Processes 6, no 8: 106.

[4]   Hadhazy, A., The Wolf and the Moose: Natural Enemies that Need Each Other. Scientific American (2008, July).

[5]   Isle Royale-Wolf and Moose Populations. (2021, March 18). National Park Service.