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 6 Positions of $K_{i}$ and $L_{i}$ in the set $B$ for various parameter regions

Case	$B_{i} γ_{i} - C_{i} β_{i}$ , i = 1,2	$B_{i} α_{i} - A_{i} β_{i}$ , i = 1,2	$C_{i} α_{i} - A_{i} γ_{i}$ , i = 1,2	Slope of $E_{1}$	Slope of $E_{2}$	The set $B$
(i)	=0	>0	>0	−	+	$\begin{matrix} f_{i} (↓, ↓) \end{matrix}$
(ii)	=0	<0	<0	+	−	$\begin{matrix} f_{i} (↑, ↑) \end{matrix}$
(iii)	>0	≥0	≥0	+ or −	−	$\underset{K_{i}}{\begin{matrix} f_{i} (↓, ↓) & f_{i} (↓, ↑) \end{matrix}}$
(iv)	>0	≥0	<0	+	−	$\begin{matrix} f_{i} (↓, ↑) \end{matrix}$
(v)	>0	<0	<0	+	+ or −	$\begin{matrix} f_{i} (↓, ↑) \\ f_{i} (↑, ↑) \end{matrix} L_{i}$
(vi)	<0	≥0	≥0	−	+ or −	$\begin{matrix} f_{i} (↑, ↓) \\ f_{i} (↓, ↓) \end{matrix} L$
(vii)	<0	<0	≥0	−	+	$\begin{matrix} f_{i} (↑, ↓) \end{matrix}$
(viii)	<0	<0	<0	+ or −	+	$\underset{K_{i}}{\begin{matrix} f_{i} (↑, ↑) & f_{i} (↑, ↓) \end{matrix}}$