Discovering Bifurcation

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.