Code:

myLambda = 4;
sol1 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /.
\[Lambda] -> myLambda, y[0] == 4}, y, {x, 0, 5}];
sol2 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /.
\[Lambda] -> myLambda, y[0] == 0.82}, y, {x, 0, 5}];
sol3 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /.
\[Lambda] -> myLambda, y[0] == -4}, y, {x, 0, 5}];
sol4 = NDSolve[{Derivative[1][y][x] == -y[x]^3 + y[x]^2 + y[x]*\[Lambda] - \[Lambda] /.
\[Lambda] -> myLambda, y[0] == 1.1}, y, {x, 0, 5}];
Show[{Plot[Evaluate[y[x] /. First[sol1]], {x, 0, 5},
PlotRange -> {{0, 5}, {-5, 5}}], Plot[Evaluate[y[x] /. First[sol2]],
{x, 0, 5}, PlotRange -> {{0, 5}, {-5, 5}}],
Plot[Evaluate[y[x] /. First[sol3]], {x, 0, 5},
PlotRange -> {{0, 5}, {-5, 5}}], Plot[Evaluate[y[x] /. First[sol4]],
{x, 0, 5}, PlotRange -> {{0, 5}, {-5, 5}}],
Graphics[{Dashed, Line[{{-5, 1}, {5, 1}}]}],
Graphics[{Dashed, Line[{{-5, Sqrt[myLambda]}, {5, Sqrt[myLambda]}}]}],
Graphics[{Dashed, Line[{{-5, -Sqrt[myLambda]}, {5, -Sqrt[myLambda]}}]}]}]