Consider the logistic population growth given by $\frac{d P}{d t}=r P\left(1-\frac{P}{\kappa}\right)$. (a) This equation is of the form $d P / d t=f(P)$. Plot the function $f(P)$ vs $P$, classify the equilibria and sketch several graphs of solutions in the $t-P$ plane.

(b) Find the solution if $0<P(0)<\kappa$ and check that the limiting behaviour agrees with your sketch in $(a)$.

(c) Find the solution if $P(0)=\kappa$.

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