**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.