Mathematics

Problem

The one-parameter family of differential equations

\frac{dy}{dx}=y\,(y-k)\,(2-y),\qquad k\in\mathbb R,

arises in a model that depends on the real parameter

k

. (a) Determine all equilibrium solutions in terms of

k

. (b) For each of the following parameter regimes, classify every equilibrium solution as stable, unstable, or semi-stable. Provide a brief justification in each case. (i)

k<0

(ii)

0<k<2

(iii)

k>2

k

passes through the critical values

k=0

and

k=2

. Your answer should reference the behavior of

\dfrac{dy}{dx}

and include a clear verbal description of the bifurcation that occurs.

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