The roles of conic sections and elliptic curves in the global dynamics of a class of planar systems of rational difference equations

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 3 Global dynamics for $α_{1} = 0$ and $α_{2} = 0$ when $E_{1}$ and $E_{2}$ are pairs of parallel lines

Parameter region	$E_{0}$	$E_{1}$	$E_{2}$	$E_{3}$
$β_{1} - A_{1} < 0$ $γ_{2} - A_{2} < 0$	G.A.S. Basin of attraction: ${[0, \infty)}^{2}$	–	–	–
$β_{1} - A_{1} > 0$ $γ_{2} - A_{2} \leq 0$	Saddle Its stable manifold: Positive y-axis	L.A.S. Basin of attraction: ${(0, \infty)}^{2}$ and positive x-axis	–	–
$β_{1} - A_{1} \leq 0$ $γ_{2} - A_{2} > 0$	Saddle Its stable manifold: Positive x-axis	–	L.A.S. Basin of attraction: ${(0, \infty)}^{2}$ and positive y-axis	–
$β_{1} - A_{1} > 0$ $γ_{2} - A_{2} > 0$	Repeller	Saddle Its stable manifold: Positive x-axis	Saddle Its stable manifold: Positive y-axis	L.A.S. Basin of attraction: ${(0, \infty)}^{2}$