Table 7 Global behavior of solutions when ${\mathit{E}}_{\mathbf{1}}$ and ${\mathit{E}}_{\mathbf{2}}$ are pairs of perpendicular lines

${\beta }_1-A_1<0$

${\gamma }_2-A_2<0$

G.A.S.

Basin of attraction:

${[0,\mathrm{\infty })}^2$

–

Every soln. converges to ${\mathcal{E}}_0$.

${\beta }_1-A_1<0$

${\gamma }_2-A_2>0$

Repeller

–

Every soln. except (0,0) tends to (0,∞).

${\beta }_1-A_1>0$

${\gamma }_2-A_2<0$

Repeller

–

Every soln. except (0,0) tends to (∞,0).

${\beta }_1-A_1>0$

${\gamma }_2-A_2>0$

Repeller

Saddle

Stable manifold:

An increasing curve $\mathcal{C}$

Every soln. not on $\mathcal{C}$ except (0,0) tends to (0,∞) or (∞,0).

${\beta }_1-A_1=0$, or

${\gamma }_2-A_2=0$

There exist infinitely many equilibria along the x- or y-axis.