Product ID: Teaching Modeling

Supplementary Print

Undergraduate

**Introduction**

The world is facing a wide range of ecological challenges, including

species extinction and the climate crisis. The mathematical concept of a bifurcation

point underlies many of these phenomena (see Wikipedia [2022])

and hence is an important topic for our students to understand. This concept

has found its way into popular culture, where it is referred to as a

“tipping point.”

The goal of the classroom exercise described below is to allow students

to cooperate in discovering for themselves the concept of a bifurcation

point. The bifurcation in question occurs in a one-parameter family of differential

equations that model harvesting fish. The parameter is the number

of fish harvested (or caught) each year. As the harvesting amount increases,

the model undergoes a transition from a stable, self-sustaining fish

population to extinction of the population. This transition occurs suddenly

at one specific value of the harvesting parameter: the so-called bifurcation

value.

For a number of different harvesting values, the students determine the

equilibrium points and draw the resulting phase line diagrams. By combining

multiple phase line diagrams, the students produce the bifurcation

diagram of the one-parameter family of differential equations. Finally, they

determine the bifurcation point: the point in the diagram at which the dynamics

undergoes a sudden change in behavior.

© 2022, COMAP, Inc.

UMAP Journal 43.2

12 Pages

- Differential Equations

- Environment & Sustainability, Global Issues ,
- Climate Change

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