Presentation Type
Poster
Presentation Type
Submission
Keywords
bifurcations, differential equations, mobile app, educational software
Department
Mathematics
Major
Computer Science and Mathematics
Abstract
Slopes is an interactive environment for exploring numerical methods and graphical solutions to ordinary differential equations. The app launched with five activities for exploration: slopefields, phase planes, oscillations, solutions to systems, and numerical methods for approximation. Bifurcations is a new sixth activity that we designed to investigate changes in the long term behavior of solutions to autonomous differential equations. This activity displays a slopefield and implements the ability to add solutions, but also introduces two new views that show how varying a single parameter impacts the values and stability of equilibrium solutions. We demonstrate the value of the new bifurcations activity by exploring a population model of invasive crayfish. Conservation groups have attempted to trap and manually remove crayfish because they have decimated local amphibian populations. We consider two models of crayfish removal that each exhibit a bifurcation when varying the parameter that represents trapping effectiveness. This tipping point represents the minimum effectiveness required to eliminate a local crayfish population, as seen in the new activity. We also present insights into the coding of the new activity and the mathematics involved. Slopes is available for iOS and Android and includes the new bifurcations activity.
Faculty Mentor
Timothy Lucas
Funding Source or Research Program
Academic Year Undergraduate Research Initiative, Summer Undergraduate Research Program
Location
Waves Cafeteria
Start Date
10-4-2026 1:00 PM
End Date
10-4-2026 2:00 PM
Included in
Tipping Points in Crayfish Management: Exploring Population Dynamics with the Bifurcations Activity in Slopes
Waves Cafeteria
Slopes is an interactive environment for exploring numerical methods and graphical solutions to ordinary differential equations. The app launched with five activities for exploration: slopefields, phase planes, oscillations, solutions to systems, and numerical methods for approximation. Bifurcations is a new sixth activity that we designed to investigate changes in the long term behavior of solutions to autonomous differential equations. This activity displays a slopefield and implements the ability to add solutions, but also introduces two new views that show how varying a single parameter impacts the values and stability of equilibrium solutions. We demonstrate the value of the new bifurcations activity by exploring a population model of invasive crayfish. Conservation groups have attempted to trap and manually remove crayfish because they have decimated local amphibian populations. We consider two models of crayfish removal that each exhibit a bifurcation when varying the parameter that represents trapping effectiveness. This tipping point represents the minimum effectiveness required to eliminate a local crayfish population, as seen in the new activity. We also present insights into the coding of the new activity and the mathematics involved. Slopes is available for iOS and Android and includes the new bifurcations activity.