Consider a variant of the basic disease model given by

Graph the equilibria as functions of α for values of α between 0 and 5, using a solid line when an equilibrium is stable and a dashed line when an equilibrium is unstable. The algebra for checking stability is messy, so it is only necessary to check stability at α = 5. You should find a saddle-node bifurcation (Exercise 18) at α = 4.
Exercises 17-20 show how the number and stability of equilibria can change when a parameter changes. Often, bifurcations have important biological applications, and bifurcation diagrams help in explaining how the dynamics of a system can suddenly change when a parameter changes only slightly. In each case, graph the equilibria against the parameter value, using a solid line when an equilibrium is stable and a dashed line when an equilibrium is unstable to draw the bifurcation diagram.

Transcribed Image Text:

dl di = al(1-1)-1.