Consider a version of the equation in Section 5.2, Exercise 29 that includes the parameter r,

Graph the equilibria as functions of r for values of r between 0 and 3, 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 r = 3. You should find a saddle-node bifurcation (Exercise 18) at r = 2.
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.

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= dt 1+N2