Autonomous ODE's and Equilibrium States

The differential equations y'=ry and y' = ry(1-y/K) are both
examples of autonomous ODE's.  That is, the independent variable
t does not appear in the equation.  Consider the direction fields 
associated with these two equations (taking, for instance, r=1,
and K=2):

Needs["Graphics`PlotField`"]

PlotVectorField[{1,  y}, {t, -3, 3}, {y, -3, 3}, 
      Axes -> True, ScaleFunction -> (1&), Ticks -> None, 
      Frame -> True, AspectRatio -> 1];

The constant solution y=0 is distinguished.  Solutions that start
close to y=0 diverge rapidly away as t increases.

PlotVectorField[{1 ,  y (1 - y/2)}, {t, -3, 3}, {y, -3, 
        5}, Axes -> True, ScaleFunction -> (1&), 
      Ticks -> None, Frame -> True, AspectRatio -> 1];

\(TextForm
    \`Here\ there\ are\ two\ equilibrium\ \(solutions : \ y\)\  = 
      \(0\ and\ y = 2. \ \ The\ first\)\ \)is unstable,  
    but the second is stable .   The general solution to \(TextForm
    \`y' = \(y \((1 - y/2)\)\ \ is\ \ \ y\  = \ 
        2/\((1 - \ Ce\^\(-y\))\) . \ \ We\ now\ graph\ this\ function\)\)for C ranging from  - 5 to 5 :

\(Plot[
      Evaluate[Table\ [\ 2/\((1 - C\ Exp[\(-t\)])\)\ , {C, \(-5\), 5, 1}]], 

      		{t, \(-3\), 5}, \ AspectRatio -> 1, \ PlotRange\  -> \ {\(-3\), 5}]
      \)

\(TextForm
    \`Sometimes\ an\ equilibrium\ arises\ which\ is\ neither\ stable\ nor\ 
      unstable . 
Consider, \ for\ instance, \ 
    y = \(0\ for\ the\ ODE\ \ y' = \ \((e\^y - \ 1)\) \((y + 1)\)\^2\)\)

PlotVectorField[{1 ,  (Exp[y] - 1) ((( y + 1))^2)}, {t, 
        -4, 4}, {y, -4, 4}, Axes -> True, 
      ScaleFunction -> (1&), Ticks -> None, Frame -> True, 
      AspectRatio -> 1,  PlotPoints  ->  25];

Solutions starting near y=0 tend to converge to or diverge from
y=0 only fairly slowly.