Table 11 Global dynamics for ${\mathit{\beta }}_{\mathbf{1}}\mathbf{-}{\mathit{A}}_{\mathbf{1}}\mathbf{>}\mathbf{0}$ and ${\mathit{\gamma }}_{\mathbf{2}}\mathbf{-}{\mathit{A}}_{\mathbf{2}}\mathbf{>}\mathbf{0}$ when ${\mathit{E}}_{\mathbf{1}}$ and ${\mathit{E}}_{\mathbf{2}}$ are pairs of transversal lines

(a)

$B_2({\beta }_1-A_1)\le B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)<C_1({\gamma }_2-A_2)$

Repeller

Saddle

Its stable manifold:

Positive x-axis

L.A.S.

Basin of attraction:

${(0,\mathrm{\infty })}^2$ and positive y-axis

–

(b)

$B_2({\beta }_1-A_1)<B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)=C_1({\gamma }_2-A_2)$

Repeller

Saddle

Its stable manifold:

An increasing curve $\mathcal{C}$

Basin of attraction:

Region below $\mathcal{C}$

L.A.S.

Basin of attraction:

Region above $\mathcal{C}$

–

(c)

$B_2({\beta }_1-A_1)\ge B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)>C_1({\gamma }_2-A_2)$

Repeller

L.A.S.

Basin of attraction:

${(0,\mathrm{\infty })}^2$ and positive x-axis

Saddle

Its stable manifold:

Positive y-axis

–

(d)

$B_2({\beta }_1-A_1)>B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)=C_1({\gamma }_2-A_2)$

Repeller

L.A.S.

Basin of attraction:

Region below an increasing curve $\mathcal{C}$

Saddle

Its stable manifold:

The increasing curve $\mathcal{C}$

Basin of attraction:

Region above $\mathcal{C}$

–

(e)

$B_2({\beta }_1-A_1)<B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)>C_1({\gamma }_2-A_2)$

Repeller

Saddle

Its stable manifold:

Positive x-axis

Saddle

Its stable manifold:

Positive y-axis

L.A.S.

Basin of attraction:

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

(f)

$B_2({\beta }_1-A_1)>B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)<C_1({\gamma }_2-A_2)$

Repeller

L.A.S.

Basin of attraction:

Region below an increasing curve $\mathcal{C}$

L.A.S.

Basin of attraction:

Region above $\mathcal{C}$

Saddle

Its stable manifold:

The increasing curve $\mathcal{C}$

(g)

$B_2({\beta }_1-A_1)=B_1({\gamma }_2-A_2)$

$C_2({\beta }_1-A_1)=C_1({\gamma }_2-A_2)$

⟶

Infinitely

Many

Equilibria