The roles of conic sections and elliptic curves in the global dynamics of a class of planar systems of rational difference equations
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Abstract
Consider the class of planar systems of firstorder rational difference equations
where $\mathcal{R}=\{(x,y)\in {[0,\mathrm{\infty})}^{2}:{A}_{i}+{B}_{i}x+{C}_{i}y\ne 0,i=1,2\}$, and the parameters are nonnegative and such that both terms in the righthand side of (1′) are nonlinear. In this paper, we prove the following discretized PoincaréBendixson theorem for the class of systems (1′).
 (i)
If both equilibrium curves of (1') are reducible conics, then every solution converges to one of up to four equilibria.
 (ii)
If exactly one equilibrium curve of (1') is a reducible conic, then every solution converges to one of up to two equilibria.
 (iii)
If both equilibrium curves of (1') are irreducible conics, then every solution converges to one of up to three equilibria or to a unique minimal periodtwo solution which occurs as the intersection of two elliptic curves.
In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to $(0,\mathrm{\infty})$ or $(\mathrm{\infty},0)$.
MSC:39A05, 39A11.
Keywords
difference equation rational global behavior equilibrium orbit globally attracting coordinatewise monotone equilibrium curve reducible conic irreducible conic minimal periodtwo solution1 Introduction and main theorem
where $\mathcal{R}=\{(x,y)\in {[0,\mathrm{\infty})}^{2}:{A}_{i}x+{B}_{i}y+{C}_{i}\ne 0,i=1,2\}$, and the parameters are nonnegative and such that both terms in the righthand side of (1) are nonlinear. The class of systems (1) has been widely studied in recent years when the RHS is both linear and nonlinear. For example, general solutions of planar linear discrete systems with constant coefficients and weak delays were studied by Diblík and Halfarová in [1] and [2]. Global behavior of solutions and basins of attraction of equilibria for special nonlinear cases of system (1) called competitive and anticompetitive systems were studied by authors such as Basu, Merino and Kulenović in [3] and [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Patterns of boundedness of nonlinear cases of system (1) were studied by Ladas et al. in [15, 16, 17, 18, 19]. More general results for system (1) as well its lower and higherdimensional counterparts were obtained by, for example, Basu and Merino in [20], by Stević, Diblík et al. in [21, 22, 23, 24], and by Ladas et al. in [25].
The class of systems (1) was proposed in all its generality by Camouzis et al. in [26]. A number of open problems regarding (1) were also mentioned in the latter. Our goal in this paper is to give a complete qualitative description of the global behavior of solutions to all systems (1) whose maps are bounded and thus provide answers to many of the open problems in [26]. For example, we present the global dynamics of the system labeled $(14,38)$ in open problem 1 and the competitive system labeled $(15,29)$ in open problem 3 in [26]. We also give the global analysis of the following 22 systems in open problem 4 which may be competitive in some range of its parameters but nowhere cooperative: $(15,l)$ and $(30,l)$ with $l\in \{35,36,43,45,47,49\}$, $(36,38)$, $(36,43)$, and $(38,l)$, $(43,l)$ with $l\in \{43,45,47,49\}$. The eight systems $(36,36)$, $(36,45)$, $(36,47)$, $(36,49)$, $(45,45)$, $(45,47)$, $(45,49)$ and $(49,49)$ from open problem 5, which may be competitive in a certain region of parameters, cooperative in another region of parameters and neither competitive nor cooperative in a third region of parameters, are also analyzed in this paper.
We also look at the four systems $(34,36)$, $(34,45)$, $(34,49)$ and $(46,49)$ from open problem 6 which may be cooperative in some range of parameters but nowhere competitive. In addition, we present the global dynamics of a number of cases of system (1) from open problem 7 which are neither competitive nor cooperative in any parameter region along with many additional cases that were not mentioned in [26], namely, cases $(k,l)$ with $k>l$. In all, we give the global dynamics of all 416 cases of nonlinear system (1) for which both members of the system are bounded along with 36 cases for which one or more members of the system are unbounded. We also show that for all of these cases, for which there exists a unique nonnegative equilibrium and no minimal periodtwo solutions, local stability of the equilibrium implies global attractivity. Thus we provide the answer to open problem 2.3 in [27] for the cases mentioned above.
which can be obtained from (1) by setting ${\alpha}_{1}={\gamma}_{1}=0$ and normalizing the other parameters. It was studied in detail by Liu and Elaydi [31], Cushing et al. [32], and Kulenović and Merino [33]. This system has the nice property that its equilibria have relatively simple algebraic formulas. Hence their local stability characters can be analyzed using standard linearization techniques. Moreover, this system is competitive (see [34, 35, 36]). So, it is somewhat easier to analyze global behavior of its solutions.
The technique is based on the analysis of slopes of equilibrium curves of the system which are defined as follows. If $T(x,y):=({T}_{1}(x,y),{T}_{2}(x,y))$ is a map associated to the system, then the two equilibrium curves of the system are respectively given by the formulas ${T}_{1}(x,y)=x$ and ${T}_{2}(x,y)=y$. Thus these curves are analogous to nullclines in differential equations and their intersection points are precisely the equilibria of the system. This method was then used to establish a connection between the number of equilibria of the system and their local stability. The authors were then able to use this result along with the results proved by Kulenović and Merino in [33] to give a complete qualitative description of the global dynamics of (LG1). Also in [20], Merino and the author introduced another new method to analyze global behavior of solutions to two classes of secondorder rational difference equations which are not competitive. The goal of this paper is to apply these two new techniques to analyze global behavior of solutions to the more general family of firstorder planar systems of rational difference equations (1) with nonnegative parameters. In particular, a geometrical criterion is presented to classify a large number of cases of system (E) into subclasses exhibiting similar global dynamics. Let $\mathcal{P}\subset {\mathbb{R}}^{12}$ be the set of nonnegative parameters $({\alpha}_{1},{\beta}_{1},\dots )$ such that the RHS terms in system (1) are nonlinear. The main theorem of this paper is as follows.
 (i)If both equilibrium curves of (1) are reducible conics, that is, if
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})=0$, and
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})=0$,
then system (1) has at least one and at most four equilibria. Every solution converges to an equilibrium.
 i.
 (ii)If exactly one equilibrium curve of (1) is a reducible conic, that is, if either
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})=0$, or
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})=0$,
then system (1) has at least one and at most two equilibria. Every solution converges to an equilibrium.
 i.
 (iii)If both equilibrium curves of (1) are irreducible conics, that is, if
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})\ne 0$, and
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})\ne 0$,
then system (1) has at least one and at most three equilibria. Every solution converges to an equilibrium or to a unique minimal periodtwo solution which occurs as the intersection of two elliptic curves.
 i.
Moreover, if both equilibrium curves of (1) are reducible conics and the map associated to system (1) is unbounded, then every solution converges to one of up to infinitely many equilibria or to $(0,\mathrm{\infty})$ or $(\mathrm{\infty},0)$.
 (a)
$\{{K}_{1},{K}_{2}\}\cap [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]=\varphi $ and $\{{L}_{1},{L}_{2}\}\cap [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]=\varphi $,
 (b)
Either ${K}_{2}\in [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$ or ${L}_{1}\in [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$, and ${K}_{1}\notin [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$, ${L}_{2}\notin [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$,
 (c)
Either ${K}_{1}\in [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$ or ${L}_{2}\in [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$, and ${K}_{2}\notin [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$, ${L}_{1}\notin [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$,
 (d)
${K}_{2},{L}_{1}\in [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$ or ${K}_{1},{L}_{2}\in [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$,
 (e)
${K}_{1},{K}_{2}\in [{\mathcal{L}}_{1},{\mathcal{U}}_{1}]$ or ${L}_{1},{L}_{2}\in [{\mathcal{L}}_{2},{\mathcal{U}}_{2}]$.
 (i)
${\mathcal{B}}^{\ast}=(\overline{x},\overline{y})$.
 (ii)
There exist equilibria $({\overline{x}}_{1},{\overline{y}}_{1}){\le}_{\mathrm{se}}({\overline{x}}_{2},{\overline{y}}_{2}){\le}_{\mathrm{se}}({\overline{x}}_{3},{\overline{y}}_{3})$ such that $({\overline{x}}_{1},{\overline{y}}_{1})$ and $({\overline{x}}_{3},{\overline{y}}_{3})$ lie at the northwest and southeast corners of ${\mathcal{B}}^{\ast}$, respectively, and $({\overline{x}}_{2},{\overline{y}}_{2})$ lies in its interior.
 (iii)
There exist minimal periodtwo solutions $(p,q){\le}_{\mathrm{se}}(\overline{x},\overline{y}){\le}_{\mathrm{se}}(r,s)$ such that $(p,q)$ and $(r,s)$ lie at the northwest and southeast corners of ${\mathcal{B}}^{\ast}$, respectively, and $(\overline{x},\overline{y})$ lies in its interior.
In case (i), it is clear that the unique equilibrium $(\overline{x},\overline{y})$ is globally attracting. In case (ii), we show that the local stability of the equilibria is determined by the slopes of the equilibrium curves at these equilibria. In case (iii), we prove that system (1) has a unique minimal periodtwo solution by looking at intersections of certain elliptic curves. We then use these results to give global stability results for the two cases.
This paper is organized as follows. In Section 2, we look at the admissible parameter regions and initial conditions for system (1). In Section 3, we define the notions of southeast order, competitive maps and equilibrium curves of system (1). In Section 4, we look at explicit formulas for the cases of system (1) for which the associated map $T(x,y)$ is bounded. In Section 5, we look at regions of coordinatewise monotonicity for the map $T(x,y)$. Sections 6 and 7 respectively deal with the case where both equilibrium curves of system (1) are reducible conics and the case where exactly one of them is a reducible conic. Sections 8.18.4 respectively deal with the number of nonnegative equilibria, local stability of equilibria, existence and uniqueness of minimal periodtwo solutions, and global behavior of solutions of system (1) when both equilibrium curves are irreducible conics.
2 Parameter regions and initial conditions
The reasons for these inequalities are as follows. If ${B}_{i}={C}_{i}=0$ for $i\in \{1,2\}$, then at least one of the members of system (1) becomes linear. Since we are interested in studying nonlinear rational systems of difference equations belonging to class (1), we will ignore these cases. Next, note that if ${\alpha}_{i}+{\beta}_{i}+{\gamma}_{i}=0$ for $i=1$ or 2, then at least one of the members of system (1) becomes trivial causing the latter to reduce to a difference equation. Since we are interested in studying systems of difference equations belonging to class (1), we will ignore these cases as well. Similarly, if ${A}_{i}={B}_{i}={\alpha}_{i}={\beta}_{i}=0$ or ${A}_{i}={C}_{i}={\alpha}_{i}={\gamma}_{i}=0$ for $i\in \{1,2\}$, then at least one of the members of system (1) becomes constant, and we have the same situation as before, which we want to avoid.
Regions $\mathcal{R}$ of initial conditions
Parameter condition  $\mathcal{R}$ 

${A}_{1}>0$, ${A}_{2}>0$  [0,∞)×[0,∞) 
(${A}_{1}={B}_{1}=0$, ${A}_{2}\ne 0$) or (${A}_{2}={B}_{2}=0$, ${A}_{1}\ne 0$)  [0,∞)×(0,∞) 
(${A}_{1}={C}_{1}=0$, ${A}_{2}\ne 0$) or (${A}_{2}={C}_{2}=0$, ${A}_{1}\ne 0$)  (0,∞)×[0,∞) 
${A}_{1}={B}_{1}=0$, ${A}_{2}={B}_{2}=0$  [0,∞)×(0,∞) 
${A}_{1}={C}_{1}=0$, ${A}_{2}={C}_{2}=0$  (0,∞)×[0,∞) 
${A}_{1}={C}_{1}=0$, ${A}_{2}={B}_{2}=0$  [0,∞)×[0,∞)∖(0,0) 
${A}_{1}={B}_{1}=0$, ${A}_{2}={C}_{2}=0$  [0,∞)×[0,∞) 
(${A}_{1}=0$, ${B}_{1}\ne 0$, ${C}_{1}\ne 0$) or (${A}_{2}=0$, ${B}_{2}\ne 0$, ${C}_{2}\ne 0$)  [0,∞)×[0,∞)∖(0,0) 
3 Important definitions
Definition 3 A continuous map $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ is said to be competitive if it is monotone with respect to the southeast ordering ${\le}_{\mathrm{se}}$.
Remark One can easily check that the Jacobian of a competitive map satisfies the sign structure $\left(\begin{array}{cc}+& \\ & +\end{array}\right)$.
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})\ne 0$,
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})\ne 0$.
Moreover, since ${C}_{1}\ge 0$ and ${B}_{2}\ge 0$, ${E}_{1}$ and ${E}_{2}$ cannot be ellipses. In this paper, we consider three separate cases, namely, the cases where (i) both ${E}_{1}$ and ${E}_{2}$ are reducible conics, (ii) exactly one of ${E}_{1}$ and ${E}_{2}$ is a reducible conic, and (iii) both ${E}_{1}$ and ${E}_{2}$ are irreducible conics.
4 Bounded cases of system (1)
In this section, we look at bounded cases of system (1), that is, special cases of system (1) for which all solutions with nonnegative/positive initial conditions are bounded. These cases have the property that their associated maps are bounded. They are obtained by setting one or more of the twelve nonnegative parameters ${\alpha}_{1}$, ${\beta}_{1}$, ${\gamma}_{1}$, ${A}_{1}$, ${B}_{1}$, ${C}_{1}$, ${\alpha}_{2}$, ${\beta}_{2}$, ${\gamma}_{2}$, ${A}_{2}$, ${B}_{2}$ and ${C}_{2}$ to zero in system (1) and have been studied in great detail by Ladas et al. in, for example, [27, 37] and [38], to name a few. For a more complete list of important work done in analyzing the boundedness of a large number of special cases of system (1) by Ladas et al., the reader is referred to references [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 25, 39, 40, 41, 42]. Such systems have been referred to as having boundedness characterization (B, B) in these papers. In particular, explicit formulas for many of these systems were given in Appendices 1 and 2 of reference [37].
 (a)
${T}_{1}({x}_{n},{y}_{n})$ and ${T}_{2}({x}_{n},{y}_{n})$ are given by one of the formulas in the righthand column of Table 2.
 (b)${T}_{1}({x}_{n},{y}_{n})$ is given by one of the formulas in the righthand column of Table 2 and ${T}_{2}({x}_{n},{y}_{n})$ is given by one of the following formulas:$\begin{array}{r}\frac{{\beta}_{2}{x}_{n}}{{A}_{2}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{2}+{\beta}_{2}{x}_{n}}{{A}_{2}},\phantom{\rule{2em}{0ex}}\frac{{\beta}_{2}{x}_{n}}{{A}_{2}+{C}_{2}{y}_{n}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{2}+{\beta}_{2}{x}_{n}}{{A}_{2}+{C}_{2}{y}_{n}},\\ \frac{{\beta}_{2}{x}_{n}+{\gamma}_{2}{y}_{n}}{{A}_{2}+{C}_{2}{y}_{n}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{2}+{\beta}_{2}{x}_{n}+{\gamma}_{2}{y}_{n}}{{A}_{2}+{C}_{2}{y}_{n}}.\end{array}$(6)
 (c)${T}_{2}({x}_{n},{y}_{n})$ is given by one of the formulas in the righthand column of Table 2 and ${T}_{1}({x}_{n},{y}_{n})$ is given by one of the following formulas:$\begin{array}{r}\frac{{\gamma}_{1}{y}_{n}}{{A}_{1}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{1}+{\gamma}_{1}{y}_{n}}{{A}_{1}},\phantom{\rule{2em}{0ex}}\frac{{\gamma}_{1}{y}_{n}}{{A}_{1}+{B}_{1}{x}_{n}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{1}+{\gamma}_{1}{y}_{n}}{{A}_{1}+{B}_{1}{x}_{n}},\\ \frac{{\beta}_{1}{x}_{n}+{\gamma}_{1}{y}_{n}}{{A}_{1}+{B}_{1}{x}_{n}},\phantom{\rule{2em}{0ex}}\frac{{\alpha}_{1}+{\beta}_{1}{x}_{n}+{\gamma}_{1}{y}_{n}}{{A}_{1}+{B}_{1}{x}_{n}}.\end{array}$(7)
Some formulas for ${\mathit{T}}_{\mathbf{1}}\mathbf{(}{\mathit{x}}_{\mathit{n}}\mathbf{,}{\mathit{y}}_{\mathit{n}}\mathbf{)}$ and ${\mathit{T}}_{\mathbf{2}}\mathbf{(}{\mathit{x}}_{\mathit{n}}\mathbf{,}{\mathit{y}}_{\mathit{n}}\mathbf{)}$ for which system ( 1 ) is bounded
Number of terms in the denominator of ${\mathit{T}}_{\mathit{i}}\mathbf{(}{\mathit{x}}_{\mathit{n}}\mathbf{,}{\mathit{y}}_{\mathit{n}}\mathbf{)}$, i = 1,2  Formula for the denominator of ${\mathit{T}}_{\mathit{i}}\mathbf{(}{\mathit{x}}_{\mathit{n}}\mathbf{,}{\mathit{y}}_{\mathit{n}}\mathbf{)}$, i = 1,2  Formula for ${\mathit{T}}_{\mathit{i}}\mathbf{(}{\mathit{x}}_{\mathit{n}}\mathbf{,}{\mathit{y}}_{\mathit{n}}\mathbf{)}$, i = 1,2, for which system (1) is bounded  

1.  Three  ${A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}$  $\frac{{\alpha}_{i}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\beta}_{i}{x}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\gamma}_{i}{y}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\alpha}_{i}+{\beta}_{i}{x}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\alpha}_{i}+{\gamma}_{i}{y}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\beta}_{i}{x}_{n}+{\gamma}_{i}{y}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\alpha}_{i}+{\beta}_{i}{x}_{n}+{\gamma}_{i}{y}_{n}}{{A}_{i}+{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$ 
2.  Two  ${B}_{i}{x}_{n}+{C}_{i}{y}_{n}$  $\frac{{\beta}_{i}{x}_{n}+{\gamma}_{i}{y}_{n}}{{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\beta}_{i}{x}_{n}}{{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$, $\frac{{\gamma}_{i}{y}_{n}}{{B}_{i}{x}_{n}+{C}_{i}{y}_{n}}$ 
${A}_{i}+{B}_{i}{x}_{n}$  $\frac{{\alpha}_{i}}{{A}_{i}+{B}_{i}{x}_{n}}$, $\frac{{\beta}_{i}{x}_{n}}{{A}_{i}+{B}_{i}{x}_{n}}$, $\frac{{\alpha}_{i}+{\beta}_{i}{x}_{n}}{{A}_{i}+{B}_{i}{x}_{n}}$  
${A}_{i}+{C}_{i}{y}_{n}$  $\frac{{\alpha}_{i}}{{A}_{i}+{C}_{i}{y}_{n}}$, $\frac{{\gamma}_{i}{y}_{n}}{{A}_{i}+{C}_{i}{y}_{n}}$, $\frac{{\alpha}_{i}+{\gamma}_{i}{y}_{n}}{{A}_{i}+{C}_{i}{y}_{n}}$  
3.  One  ${A}_{i}$  ${\alpha}_{i}/{A}_{i}$ 
${B}_{i}{x}_{n}$  ${\beta}_{i}/{B}_{i}$  
${C}_{i}{y}_{n}$  ${\gamma}_{i}/{C}_{i}$ 
Thus there are at least 589 bounded cases of system (1) of which 564 cases are nonlinear.
The bounds for the last case in (6) can be found in a similar manner. The formulas in (7) are almost identical to the formulas in (6) with ${A}_{2}$, ${\beta}_{2}$ and ${x}_{n}$ respectively replaced by ${A}_{1}$, ${\gamma}_{1}$ and ${y}_{n}$. Hence their lower and upper bounds ${\mathcal{L}}_{2}$ and ${\mathcal{U}}_{2}$ can be found in a similar fashion as in (6). It follows from the previous discussion that there are $7+3+3+3+3=19$ bounded formulas for ${T}_{1}({x}_{n},{y}_{n})$ and another 19 bounded formulas for ${T}_{2}({x}_{n},{y}_{n})$ in cases (i)(iv) of Table 2 of part (a). In all, there are $19\times 19=361$ bounded cases of system 1 in part (a) and $19\times 6=114$ bounded cases each in parts (b) and (c). This gives a total of $361+2(114)=589$ bounded cases of system 1 from parts (a), (b) and (c). Moreover, there are $3\times 3=9$ ways to pair ${T}_{1}({x}_{n},{y}_{n})$ and ${T}_{2}({x}_{n},{y}_{n})$ so that both of them are constant in the RHS of (5): three choices for ${T}_{1}({x}_{n},{y}_{n})$ from Table 2 case 3 when $i=1$ combined with three choices for ${T}_{2}({x}_{n},{y}_{n})$ from Table 2 case 3 when $i=2$. In addition, the first two formulas in both parts (b) and (c) of the theorem are linear. They can be combined to give $2\times 2=4$ cases where ${T}_{1}({x}_{n},{y}_{n})$ and ${T}_{2}({x}_{n},{y}_{n})$ are both linear in the RHS of (5). Finally, there are $2\times 3=6$ ways each to respectively combine the two linear formulas in parts (b) and (c) with those in Table 2 case 3 so that the RHS of (5) is a combination of a linear formula and a constant formula. This gives a total of $6+6=12$ cases. To conclude, there are $9+4+12=25$ linear or constant cases out of the 589 bounded cases we originally identified above, which leaves us with $58925=564$ bounded nonlinear cases of system (1). □
The goal of this paper is to give a complete qualitative description of the global behavior of solutions to all bounded nonlinear cases of system (1) including the 564 bounded nonlinear cases mentioned in Theorem 2 above.
5 Regions of coordinatewise monotonicity for the map T
When both equilibrium curves are irreducible conics, the map $T(x,y)$ associated to bounded system (1) is not coordinatewise monotone throughout its bounded domain of definition. In this subsection, we will identify regions of coordinatewise monotonicity of the map $T(x,y)$. These regions will play a crucial role in determining the global behavior of solutions to system (1) when both equilibrium curves are irreducible conics.
 (i)
If ${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}=0$, then the partial derivatives of the functions ${T}_{1}(x,y)$ are continuous on ${(0,\mathrm{\infty})}^{2}$ and have constant sign on the set ℬ.
 (ii)
If ${B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2}=0$, then the partial derivatives of the functions ${T}_{2}(x,y)$ are continuous on ${(0,\mathrm{\infty})}^{2}$ and have constant sign on the set ℬ.
Proof We give the proof of part (i). The proof of part (ii) is similar and we skip it. Note that by hypotheses (2), ${B}_{1}+{C}_{1}>0$. First, suppose ${B}_{1}\ne 0$ and ${C}_{1}\ne 0$. Solving for ${\gamma}_{1}$ in ${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}=0$ and substituting in $\frac{\partial}{\partial x}{T}_{1}(x,y)$ and $\frac{\partial}{\partial y}{T}_{1}(x,y)$, we get that $\frac{\partial}{\partial x}{T}_{1}(x,y)=\frac{{B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}}{{({A}_{1}+{B}_{1}x+{C}_{1}y)}^{2}}$ and $\frac{\partial}{\partial y}{T}_{1}(x,y)=\frac{{C}_{1}({B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1})}{{B}_{1}{({A}_{1}+{B}_{1}x+{C}_{1}y)}^{2}}$. When ${B}_{1}=0$ and ${C}_{1}\ne 0$, the hypothesis implies that ${\beta}_{1}=0$. In this case, ${D}_{1}{T}_{1}(x,y)=0$ and ${D}_{2}{T}_{1}(x,y)=\frac{{C}_{1}({B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1})}{{B}_{1}{({A}_{1}+{C}_{1}y)}^{2}}$. Finally, when ${B}_{1}\ne 0$ and ${C}_{1}=0$, one must have ${\gamma}_{1}=0$ and hence ${D}_{1}{T}_{1}(x,y)=\frac{{B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}}{{({A}_{1}+{B}_{1}x)}^{2}}$ and ${D}_{2}{T}_{1}(x,y)=0$. Clearly, in all three cases the partial derivatives of ${T}_{1}(x,y)$ have constant sign on the set ℬ. □
We will need the following elementary result, which is given here without a proof.
 i.
${D}_{1}{T}_{i}(x,y)=0$ if and only if $y=\frac{{B}_{i}{\alpha}_{i}{A}_{i}{\beta}_{i}}{{B}_{i}{\gamma}_{i}{C}_{i}{\beta}_{i}}$, and ${D}_{1}{T}_{i}(x,y)>0$ if and only if $({C}_{i}{\beta}_{i}{B}_{i}{\gamma}_{i})y>{B}_{i}{\alpha}_{i}{A}_{i}{\beta}_{i}$.
 ii.
${D}_{2}{T}_{i}(x,y)=0$ if and only if $x=\frac{{C}_{i}{\alpha}_{i}{A}_{i}{\gamma}_{i}}{{B}_{i}{\gamma}_{i}{C}_{i}{\beta}_{i}}$, and ${D}_{2}{T}_{i}(x,y)>0$ if and only if $({B}_{i}{\gamma}_{i}{C}_{i}{\beta}_{i})x>{C}_{i}{\alpha}_{i}{A}_{i}{\gamma}_{i}$.
For the rest of this paper, we will need to refer to the relative positions of ${K}_{i}$ and ${L}_{i}$ where the partial derivatives of ${T}_{i}(x,y)$ change sign for $i=1,2$. The explicit formulas for ${K}_{i}$ and ${L}_{i}$ for $i=1,2$ are given in the following definition. Their relative positions according to different parameter regions are shown in the Appendix for convenience.
 (i)
${K}_{1}\in {[0,\mathrm{\infty})}^{2}$ if and only if ${L}_{1}\notin {[0,\mathrm{\infty})}^{2}$;
 (ii)
${K}_{2}\in {[0,\mathrm{\infty})}^{2}$ if and only if ${L}_{2}\notin {[0,\mathrm{\infty})}^{2}$.
 (a)
${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}>0$, ${B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}<0$, ${C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1}\ge 0$;
 (b)
${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}<0$, ${B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}\ge 0$, ${C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1}<0$.
giving a contradiction. □
6 When both ${E}_{1}$ and ${E}_{2}$ are reducible conics
 (i)In cases (a), ${E}_{1}$ and ${E}_{2}$ each belong to a pair of parallel lines. The corresponding members of system (1) have the forms${x}_{n+1}=\frac{{\alpha}_{1}+{\beta}_{1}{x}_{n}}{{A}_{1}+{B}_{1}{x}_{n}},\phantom{\rule{2em}{0ex}}{y}_{n+1}=\frac{{\alpha}_{2}+{\gamma}_{2}{y}_{n}}{{A}_{2}+{C}_{2}{y}_{n}},\phantom{\rule{1em}{0ex}}\text{where}\begin{array}{l}{C}_{1}={\gamma}_{1}=0\\ {B}_{2}={\beta}_{2}=0\end{array}\}.$
 (ii)In cases (b), ${E}_{1}$ and ${E}_{2}$ each belong to a pair of perpendicular lines. The corresponding members of system (1) look like${x}_{n+1}=\frac{{\beta}_{1}{x}_{n}}{{A}_{1}+{C}_{1}{y}_{n}},\phantom{\rule{2em}{0ex}}{y}_{n+1}=\frac{{\gamma}_{2}{y}_{n}}{{A}_{2}+{B}_{2}{x}_{n}},\phantom{\rule{1em}{0ex}}\text{where}\begin{array}{l}{C}_{1}0,{\alpha}_{1}={\gamma}_{1}=0,{B}_{1}=0\\ {B}_{2}0,{\alpha}_{2}={\beta}_{2}=0,{C}_{2}=0\end{array}\}.$
 (iii)In cases (c), ${E}_{1}$ and ${E}_{2}$ belong to a pair of nonperpendicular transversal lines each. The corresponding members of system (1) have the forms$\begin{array}{r}{x}_{n+1}=\frac{{\beta}_{1}{x}_{n}}{{A}_{1}+{B}_{1}{x}_{n}+{C}_{1}{y}_{n}},\phantom{\rule{2em}{0ex}}{y}_{n+1}=\frac{{\gamma}_{2}{y}_{n}}{{A}_{2}+{B}_{2}{x}_{n}+{C}_{2}{y}_{n}},\\ \phantom{\rule{1em}{0ex}}\text{where}\begin{array}{l}{C}_{1}0,{\alpha}_{1}={\gamma}_{1}=0,{B}_{1}0\\ {B}_{2}0,{\alpha}_{2}={\beta}_{2}=0,{C}_{2}0\end{array}\}.\end{array}$
Note that the first equation in (i) involving ${x}_{n+1}$ actually consists of six separate equations corresponding to three cases each for ${A}_{i}\ne 0$ and ${A}_{i}=0$. These three cases are: (a) ${\alpha}_{1}=0$, ${\beta}_{1}\ne 0$, (b) ${\alpha}_{1}\ne 0$, ${\beta}_{1}=0$ and (c) ${\alpha}_{1}\ne 0$, ${\beta}_{1}\ne 0$. The same is true for the second equation in (i) involving ${y}_{n+1}$. Similarly, the two equations in (ii) each consist of two separate equations, namely, the one with ${A}_{i}\ne 0$ and the one with ${A}_{i}=0$ for $i=1,2$. The same is true of (iii).
Thus this section establishes global behavior of solutions of system (1) when its members are combinations of any of the $6+2+2=10$ forms for ${x}_{n+1}$ with any of the ten forms for ${y}_{n+1}$ given in (i)(iii) of the last remark. This gives rise to 100 explicit planar systems of firstorder rational difference equations with positive parameters. It is a direct consequence of Table 2 in Theorem 2 that the equations in (i) and (iii) are bounded while the equations in (ii) are unbounded. Thus there are a total of $(6+2)\times (6+2)=64$ bounded systems out of the 100 systems. Moreover, if both members of (1) have the forms given in (iii) and, in addition, ${A}_{1}>0$ and ${A}_{2}>0$, then the resulting system is the wellknown LeslieGower model from theoretical ecology whose global dynamics was analyzed by Cushing et al. in [32]. The main theorem of this section is the following.
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})=0$, and
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})=0$,
then it has at least one and at most four equilibria. Every solution converges to an equilibrium.
We discuss the proof of Theorem 3 in Section 6.2. But first we establish the number of nonnegative equilibria of system (1) when both its equilibrium curves are reducible conics.
6.1 Number of nonnegative equilibria
The main theorem of this subsection is the following.
 (a)
If there exists one equilibrium, then it must be $(0,0)$ or an interior equilibrium.
 (b)
If there exist two equilibria, then they must include an axis equilibrium.
 (c)
If there exist three equilibria, then they must consist of $(0,0)$ and an equilibrium on each axis.
 (d)
If there exist four equilibria, then they must consist of $(0,0)$, an equilibrium on each axis and an interior equilibrium.
 (a)
${B}_{1}{x}^{2}+({A}_{1}{\beta}_{1})x{\alpha}_{1}=0$, where ${C}_{1}={\gamma}_{1}=0$,
 (b)
$x({C}_{1}y+{A}_{1}{\beta}_{1})=0$, where ${C}_{1}>0$, ${\alpha}_{1}={\gamma}_{1}=0$, ${B}_{1}=0$,
 (c)
$x({B}_{1}x+{C}_{1}y+{A}_{1}{\beta}_{1})=0$, where ${C}_{1}>0$, ${\alpha}_{1}={\gamma}_{1}=0$, ${B}_{1}>0$.
Next we discuss the global behavior of solutions to system (1) when it satisfies the hypotheses of Theorem 3.
6.2 Global behavior of solutions
In this section, we present the proof of Theorem 3. In order to do so in a manageable way, we break up the statement of Theorem 3 into six smaller theorems based upon whether the equilibrium curves of system (1) consist of two parallel lines, two perpendicular lines, two transversal lines or some mix of the three (refer to cases (i)(iii) at the start of Section 6). In particular, we give the explicit proof for the case where both equilibrium curves are parallel lines and state the remaining five theorems, Theorems 1418, in the Appendix at the end of this paper to avoid unnecessary repetition.
First, we present a definition and a lemma which will be required for the proof of the theorem mentioned above.
then $(\overline{x},\overline{y})$ is globally asymptotically stable.
Hence we have ${T}^{n}(x,y)\to (\overline{x},\overline{y})$ in both these cases. □
Our next theorem gives the global behavior of solutions when both equilibrium curves ${E}_{1}$ and ${E}_{2}$ of system (1) are pairs of parallel lines. It is as follows.
 (i)
If ${\alpha}_{1}\ne 0$ and ${\alpha}_{2}\ne 0$, then the unique equilibrium ${\mathcal{E}}_{3}$ is globally asymptotically stable.
 (ii)If ${\alpha}_{1}=0$ and ${\alpha}_{2}\ne 0$, then

If${\beta}_{1}{A}_{1}\le 0$, then the unique equilibrium${\mathcal{E}}_{3}$is globally asymptotically stable.

If${\beta}_{1}{A}_{1}>0$, then${\mathcal{E}}_{2}$is a saddle point with the nonnegative yaxis as its stable manifold. ${\mathcal{E}}_{3}$is LAS and attracts all solutions with initial conditions in${(0,\mathrm{\infty})}^{2}$or on the positive xaxis.

 (iii)If ${\alpha}_{1}\ne 0$ and ${\alpha}_{2}=0$, then

If${\gamma}_{2}{A}_{2}\le 0$, then the unique equilibrium${\mathcal{E}}_{1}$is globally asymptotically stable.

If${\gamma}_{2}{A}_{2}>0$, then${\mathcal{E}}_{1}$is a saddle point with the nonnegative xaxis as its stable manifold. ${\mathcal{E}}_{3}$is LAS and attracts all solutions with initial conditions in${(0,\mathrm{\infty})}^{2}$or on the positive yaxis.

 (iv)
If ${\alpha}_{1}=0$ and ${\alpha}_{2}=0$, then the nonnegative equilibria of system (1) and their basins of attraction must satisfy Table 3.
Global dynamics for ${\mathit{\alpha}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$ and ${\mathit{\alpha}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$ when ${\mathit{E}}_{\mathbf{1}}$ and ${\mathit{E}}_{\mathbf{2}}$ are pairs of parallel lines
Parameter region  ${\mathcal{E}}_{\mathbf{0}}$  ${\mathcal{E}}_{\mathbf{1}}$  ${\mathcal{E}}_{\mathbf{2}}$  ${\mathcal{E}}_{\mathbf{3}}$ 

${\beta}_{1}{A}_{1}<0$ ${\gamma}_{2}{A}_{2}<0$  G.A.S. Basin of attraction: ${[0,\mathrm{\infty})}^{2}$  –  –  – 
${\beta}_{1}{A}_{1}>0$ ${\gamma}_{2}{A}_{2}\le 0$  Saddle Its stable manifold: Positive yaxis  L.A.S. Basin of attraction: ${(0,\mathrm{\infty})}^{2}$ and positive xaxis  –  – 
${\beta}_{1}{A}_{1}\le 0$ ${\gamma}_{2}{A}_{2}>0$  Saddle Its stable manifold: Positive xaxis  –  L.A.S. Basin of attraction: ${(0,\mathrm{\infty})}^{2}$ and positive yaxis  – 
${\beta}_{1}{A}_{1}>0$ ${\gamma}_{2}{A}_{2}>0$  Repeller  Saddle Its stable manifold: Positive xaxis  Saddle Its stable manifold: Positive yaxis  L.A.S. Basin of attraction: ${(0,\mathrm{\infty})}^{2}$ 
Hence we have ${T}^{2n}(0,0)\to {\mathcal{E}}_{2}$ and ${T}^{2n}(0,{\mathcal{U}}_{2})\to {\mathcal{E}}_{2}$. As a result, ${T}^{2n}(0,y)\to {\mathcal{E}}_{2}$ for $0<y<{\mathcal{U}}_{2}$. Thus ${\mathcal{E}}_{2}$ is a saddle equilibrium with the nonnegative yaxis as its stable manifold.
If ${\beta}_{1}{A}_{1}\le 0$, then ${\stackrel{\u02c6}{\ell}}_{2}\notin {(0,\mathrm{\infty})}^{2}$ and hence ${\mathcal{E}}_{2}$ is the only equilibrium in ${[0,\mathrm{\infty})}^{2}$. Note that in this case, ${Q}_{1}({\mathcal{E}}_{2})=\mathcal{R}(,)$ and ${Q}_{4}({\mathcal{E}}_{2})=\mathcal{R}(,+)$. Hence, by Lemma 4, ${\mathcal{E}}_{2}$ attracts all solutions with initial conditions in ${(0,\mathrm{\infty})}^{2}$. The proof of global attractivity of ${\mathcal{E}}_{2}$ for all solutions with initial conditions on the nonnegative yaxis is similar to the previous case. Finally, note that all solutions with initial conditions on the positive xaxis enter the region ${(0,\mathrm{\infty})}^{2}$ under a single application of the map T.
If ${\beta}_{1}{A}_{1}\le 0$ and ${\gamma}_{2}{A}_{2}\le 0$, then ${\ell}_{2},{\stackrel{\u02c6}{\ell}}_{2}\not\subset {(0,\mathrm{\infty})}^{2}$ and the unique equilibrium ${\mathcal{E}}_{0}=(0,0)$ is globally asymptotically stable by Lemma 4.
If ${\beta}_{1}{A}_{1}\le 0$ and ${\gamma}_{2}{A}_{2}>0$, then ${\ell}_{2}\not\subset {(0,\mathrm{\infty})}^{2}$ and ${\stackrel{\u02c6}{\ell}}_{2}\subset {(0,\mathrm{\infty})}^{2}$. Hence ${\mathcal{E}}_{0}$ and ${\mathcal{E}}_{2}=(0,\frac{{\gamma}_{2}{A}_{2}}{{C}_{2}})$ are the only equilibria present. Note that in this case, ${Q}_{1}({\mathcal{E}}_{2})=\mathcal{R}(,)$ and ${Q}_{4}({\mathcal{E}}_{2})=\mathcal{R}(,+)$. Also, the dynamics of solutions with initial conditions along the positive x and yaxes can be determined in the same way as in the proof of the case ${\alpha}_{1}=0$ and ${\alpha}_{2}\ne 0$. The result follows from this and Lemma 4.
If ${\beta}_{1}{A}_{1}>0$ and ${\gamma}_{2}{A}_{2}\le 0$, then ${\ell}_{2}\subset {(0,\mathrm{\infty})}^{2}$ and ${\stackrel{\u02c6}{\ell}}_{2}\not\subset {(0,\mathrm{\infty})}^{2}$. Hence the only equilibria present are ${\mathcal{E}}_{0}$ and ${\mathcal{E}}_{1}=(\frac{{\beta}_{1}{A}_{1}}{{B}_{1}},0)$. This case is symmetric to the previous case and has an almost identical proof.
7 When exactly one of ${E}_{1}$ and ${E}_{2}$ is an irreducible conic
In this section, we look at the case where exactly one of the equilibrium curves ${E}_{1}$ and ${E}_{2}$ of system (1) is an irreducible conic and the map T associated to system (1) is bounded. Note that this case corresponds to ${E}_{1}$ and ${E}_{2}$ being combinations of pairs of parallel lines, pairs of transversal nonperpendicular lines, parabolas and hyperbolas. The cases where ${E}_{1}$ or ${E}_{2}$ is a pair of perpendicular lines are unbounded and hence not of interest to us in this paper. Thus there are $2\times (3+2)\times (195)=140$ bounded members and the rest are unbounded. The next theorem is the main theorem of this section and is as follows.
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})=0$, or
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})=0$,
then system (1) has at least one and at most two equilibria. Every solution converges to an equilibrium.
The proof of the number of equilibria is given in the next theorem. To see that every solution converges to an equilibrium, observe that in this case, exactly one member of system (1) has one of the formulas given in (i)(iii) of the previous section. Hence exactly one of the coordinates of the map $T(x,y)$ is monotone. Thus one can use a mix of the techniques already introduced in the previous section for reducible conics along with some new techniques that will be introduced in the next section for irreducible conics to prove global convergence results for this case. We skip the proofs to avoid unnecessary repetition.
 (a)
If there exists one equilibrium, then it may be an axis equilibrium or an interior equilibrium.
 (b)
If there exist two equilibria, then they must include an axis equilibrium and an interior equilibrium.
 (c)
The set of equilibrium points must be linearly ordered by ${\u2aaf}_{\mathrm{ne}}$.
 (a)
${B}_{1}{x}^{2}+({A}_{1}{\beta}_{1})x{\gamma}_{1}y{\alpha}_{1}=0$, where ${C}_{1}=0$, ${\gamma}_{1}>0$;
 (b)
${B}_{1}{x}^{2}+{C}_{1}xy+({A}_{1}{\beta}_{1})x{\gamma}_{1}y{\alpha}_{1}=0$, where ${C}_{1}>0$, ${\alpha}_{1}+{\gamma}_{1}>0$.
8 When both ${E}_{1}$ and ${E}_{2}$ are irreducible conics
The main theorem of this section is the following.
 i.
${C}_{1}({C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1})+{\gamma}_{1}({C}_{1}{\beta}_{1}{B}_{1}{\gamma}_{1})\ne 0$, and
 ii.
${B}_{2}({B}_{2}{\alpha}_{2}{A}_{2}{\beta}_{2})+{\beta}_{2}({B}_{2}{\gamma}_{2}{C}_{2}{\beta}_{2})\ne 0$,
then system (1) has at least one and at most three equilibria. Every solution converges to an equilibrium or to a unique minimal periodtwo solution which occurs as the intersection of two elliptic curves.
We present the proof of Theorem 8 at the end of Section 8.4. But first we present the number of nonnegative equilibria, local stability of equilibria, existence and uniqueness of minimal periodtwo solutions, and the global behavior of solutions to system (1) in Sections 8.18.4, respectively.
8.1 Number of nonnegative equilibria
We start this section by presenting a lemma which will help us establish bounds on the number of nonnegative equilibria of system (1) when both its equilibrium curves are irreducible conics.
 (i)
If ${C}_{1}=0$ and ${B}_{2}=0$, then the graphs of ${E}_{1}$ and ${E}_{2}$ are parabolas with positive slopes in ℬ.
 (ii)
If ${C}_{1}>0$ or ${B}_{2}>0$, then the graphs of ${E}_{1}$ and ${E}_{2}$ are respectively hyperbolas whose slopes in ℬ have signs as given in the last two columns of Table 4. The expression ‘+ or −’ implies an exclusive or.
Signs of slopes of ${\mathit{E}}_{\mathbf{1}}$ and ${\mathit{E}}_{\mathbf{2}}$ in $\mathcal{B}$ when ${\mathit{C}}_{\mathbf{1}}\mathbf{>}\mathbf{0}$ or ${\mathit{B}}_{\mathbf{2}}\mathbf{>}\mathbf{0}$
${\mathit{B}}_{\mathit{i}}{\mathit{\gamma}}_{\mathit{i}}\mathbf{}{\mathit{C}}_{\mathit{i}}{\mathit{\beta}}_{\mathit{i}}$, i = 1,2  ${\mathit{B}}_{\mathit{i}}{\mathit{\alpha}}_{\mathit{i}}\mathbf{}{\mathit{A}}_{\mathit{i}}{\mathit{\beta}}_{\mathit{i}}$, i = 1,2  ${\mathit{C}}_{\mathit{i}}{\mathit{\alpha}}_{\mathit{i}}\mathbf{}{\mathit{A}}_{\mathit{i}}{\mathit{\gamma}}_{\mathit{i}}$, i = 1,2  Slope of ${\mathit{E}}_{\mathbf{1}}$  Slope of ${\mathit{E}}_{\mathbf{2}}$  

(i)  =0  >0  >0  −  + 
(ii)  =0  <0  <0  +  − 
(iii)  >0  ≥0  ≥0  + or −  − 
(iv)  >0  ≥0  <0  +  − 
(v)  >0  <0  <0  +  + or − 
(vi)  <0  ≥0  ≥0  −  + or − 
(vii)  <0  <0  ≥0  −  + 
(viii)  <0  <0  <0  + or −  + 
 (a)
${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}>0$, ${B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}\ge 0$, ${C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1}\ge 0$,
 (b)
${B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}<0$, ${B}_{1}{\alpha}_{1}{A}_{1}{\beta}_{1}<0$, ${C}_{1}{\alpha}_{1}{A}_{1}{\gamma}_{1}<0$.
Since an equilibrium $(\overline{x},\overline{y})$ of system (1) is a fixed point that lies on the curve ${E}_{1}$, it follows that $\overline{x}$ must satisfy ${K}_{1}<\overline{x}<\frac{{\gamma}_{1}}{{C}_{1}}$. Hence ${E}_{1}$ must lie in the region $x<\frac{{\gamma}_{1}}{{C}_{1}}$ and must be an increasing function of x. One can similarly argue that if ${K}_{1}>\frac{{\gamma}_{1}}{{C}_{1}}$, then ${E}_{1}$ must be a decreasing function of x. Note that the case ${K}_{1}=\frac{{\gamma}_{1}}{{C}_{1}}$ cannot exist. Indeed, if it did, then the previous analysis would imply that the equilibrium $(\overline{x},\overline{y})$ must lie on the line $x={K}_{1}=\frac{{\gamma}_{1}}{{C}_{1}}$. But this is impossible since this line is a vertical asymptote for the curve ${E}_{1}$ which contains the point $(\overline{x},\overline{y})$. In case (viii), one can use a similar proof to show that if ${K}_{1}<\frac{{\gamma}_{1}}{{C}_{1}}$, then ${E}_{1}$ is a decreasing function of x and if ${K}_{1}>\frac{{\gamma}_{1}}{{C}_{1}}$, then ${E}_{1}$ is an increasing function of x. □
 i.
The graph of ${E}_{1}$ is a decreasing function of a single variable in ℬ if and only if $\frac{\partial}{\partial y}{T}_{1}(x,y)<0$.
 ii.
The graph of ${E}_{2}$ is a decreasing function of a single variable in ℬ if and only if $\frac{\partial}{\partial x}{T}_{2}(x,y)<0$.
The next theorem establishes bounds on the number of nonnegative equilibria of system (1).
 (a)
If ${E}_{1}$ or ${E}_{2}$ is a parabola, then either there exists a unique interior equilibrium or there exist two equilibria, namely, $(0,0)$ and an interior equilibrium which are linearly ordered by ${\u2aaf}_{\mathrm{ne}}$.
 (b)
If both ${E}_{1}$ and ${E}_{2}$ are hyperbolas, then there exist between one and three equilibria all of which are interior equilibria linearly ordered by ${\le}_{\mathrm{se}}$.
Clearly, ${E}_{1}$ has a vertical asymptote $x=\frac{{\gamma}_{1}}{{C}_{1}}$ and an oblique asymptote $y=\frac{{B}_{1}}{{C}_{1}}x\frac{{A}_{1}{C}_{1}+{B}_{1}{\gamma}_{1}{C}_{1}{\beta}_{1}}{{C}_{1}^{2}}$ with a negative slope. It also has xintercepts of opposite signs when ${\alpha}_{1}>0$ and a zero xintercept when ${\alpha}_{1}=0$. It follows from this that the branch of ${E}_{1}$ which lies in ${[0,\mathrm{\infty})}^{2}$ must lie either in the region $x<\frac{{\gamma}_{1}}{{C}_{1}}$ or in the region $x>\frac{{\gamma}_{1}}{{C}_{1}}$ but not both. Clearly, it must be increasing in the former case and decreasing in the latter case. Similarly, if ${E}_{2}$ is a parabola, then one can show that it must lie either in the region $y<\frac{{\beta}_{2}}{{B}_{2}}$ or in the region $y>\frac{{\beta}_{2}}{{B}_{2}}$ but not both. Also, it must be increasing in the former case and decreasing in the latter case. It follows from this that if ${E}_{1}$ is a parabola and ${E}_{2}$ is a hyperbola or vice versa, then the two must intersect in at most two points in ${[0,\mathrm{\infty})}^{2}$ including $(0,0)$ and an interior point. Moreover, if both ${E}_{1}$ and ${E}_{2}$ are hyperbolas such that one or both of them are increasing in ${[0,\mathrm{\infty})}^{2}$, then the opposite signs of their slopes/concavities guarantee that they must intersect in at most two points in ${[0,\mathrm{\infty})}^{2}$ including $(0,0)$ and an interior point.
8.2 Local stability of equilibria
In this section, we establish local stability results for the nonnegative equilibria of system (1) when both of its equilibrium curves ${E}_{1}$ and ${E}_{2}$ are irreducible conics. In particular, we show that the local stability of the equilibria is determined by the slopes of ${E}_{1}$ and ${E}_{2}$ at these equilibria. In Theorem 9, we present local stability results when both ${E}_{1}$ and ${E}_{2}$ have negative slopes, and in Theorem 10, we do the same when at least one of them has a positive slope. We start out by giving a preliminary result on the equilibrium curves (sets) of system (1). It is a generalization of Theorem 1 in [3] and has weaker hypotheses than the latter. It also extends the latter to include the complex eigenvalues case and will be useful for proving Theorems 9 and 10.
 i.
There exists a neighborhood $I\subset \mathbb{R}$ of $\overline{x}$ and $J\subset \mathbb{R}$ of $\overline{y}$ such that the sets ${E}_{1}\cap (I\times J)$ and ${E}_{2}\cap (I\times J)$ are the graphs of class ${C}^{p}$ functions ${y}_{1}(x)$ and ${y}_{2}(x)$ for $x\in I$.
 ii.The eigenvalues ${\lambda}_{1}$ and ${\lambda}_{2}$ of the Jacobian matrix of T at $(\overline{x},\overline{y})$ satisfy:
 (a)
If ${\lambda}_{1}$, ${\lambda}_{2}$ are real and equal, then $1<{\lambda}_{1},{\lambda}_{2}<1$.
 (b)If ${\lambda}_{1}$, ${\lambda}_{2}$ are real and distinct with ${\lambda}_{2}<{\lambda}_{1}$, then $1<{\lambda}_{1}$ and ${\lambda}_{2}<1$. Furthermore, $b\ne 0$ and$sign(1+{\lambda}_{2})=sign(1+a+d+adbc)$(14)
and$sign(1{\lambda}_{1})=\{\begin{array}{ll}sign({y}_{1}^{\prime}(\overline{x}){y}_{2}^{\prime}(\overline{x}))& \mathit{\text{if}}b0,\\ sign({y}_{1}^{\prime}(\overline{x}){y}_{2}^{\prime}(\overline{x}))& \mathit{\text{if}}b0.\end{array}$(15) (c)If ${\lambda}_{1}$ and ${\lambda}_{2}$ are complex numbers, then${\overline{\lambda}}_{1}={\lambda}_{2}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}1<{\lambda}_{1}={\lambda}_{2}<1.$(16)
 (a)
 i.The existence of I and J and of smooth functions ${y}_{1}(x)$ and ${y}_{2}(x)$ defined in I as in the statement of the theorem is guaranteed by the hypotheses and the implicit function theorem. Moreover, when ${f}_{y}(x,y)\ne 0$, one has${y}_{1}^{\prime}(x)=\frac{1{f}_{x}(x,y)}{{f}_{y}(x,y)}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}_{2}^{\prime}(x)=\frac{{g}_{x}(x,y)}{1{g}_{y}(x,y)},\phantom{\rule{1em}{0ex}}x\in I.$(17)
Note that ${f}_{y}(x,y)\ne 0$ since otherwise one would have ${f}_{x}(x,y)=1$ in (17) upon rewriting the first expression as ${f}_{y}(x,y){y}_{1}^{\prime}(x)=1{f}_{x}(x,y)$ and thus $a:={f}_{x}(\overline{x},\overline{y})=1$, contradicting one of the hypotheses of the theorem.
 ii.The characteristic polynomial of the Jacobian of T,$p(\lambda )={\lambda}^{2}(a+d)\lambda +(adbc),$(18)has ${\lambda}_{1}$ and ${\lambda}_{2}$ as its roots. If ${\lambda}_{1}={\lambda}_{2}=\lambda $, then the hypotheses $1<a<1$ and $1<d<1$ and the sumofroots relation for quadratic functions applied to (18) imply$2<2\lambda =a+d<2\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}1<\lambda <1,$which proves (a). Now, suppose ${\lambda}_{1}$, ${\lambda}_{2}$ are real and distinct with ${\lambda}_{2}<{\lambda}_{1}$. Since $2<a+d={\lambda}_{1}+{\lambda}_{2}<2$, the larger root ${\lambda}_{1}$ must satisfy $1<{\lambda}_{1}$ and the smaller root ${\lambda}_{2}$ must satisfy ${\lambda}_{2}<1$. Moreover, the remark following (17) in part i gives that $b:={f}_{y}(\overline{x},\overline{y})\ne 0$. To see the proof of (14), note that in (18), we have $p(1)=1+(a+d)+adbc=(1{\lambda}_{1})(1{\lambda}_{2})$. Since $1<{\lambda}_{1}$ from above, it follows that $p(1)>0$ if and only if $1{\lambda}_{2}<0$, that is, if and only if $1+{\lambda}_{2}>0$. Next note that from (17), we have$\begin{array}{rl}{y}_{1}^{\prime}(\overline{x}){y}_{2}^{\prime}(\overline{x})& =\frac{1a}{b}\frac{c}{1d}=\frac{1(a+d)+adbc}{b(1d)}\\ =\frac{p(1)}{b(1d)}=\frac{(1{\lambda}_{1})(1{\lambda}_{2})}{b(1d)}.\end{array}$(19)
□
Corollary 2 If $({y}_{1}^{\prime}(\overline{x}){y}_{2}^{\prime}(\overline{x}))b>0$, then system (1) cannot possess any repelling fixed points.
The next lemma gives a connection between the slopes of equilibrium curves ${E}_{1}$, ${E}_{2}$ in the invariant attracting box ℬ and the signs of entries of the Jacobian in (21) evaluated at an equilibrium point of (1).
Lemma 6 The map T satisfies the hypotheses of Theorem 10.
which are clearly positive. It follows that $1<a<1$ and $1<d<1$. □
 (i)
System (1) has at least one and at most three equilibria in ${(0,\mathrm{\infty})}^{2}$. The set of equilibrium points is linearly ordered by ${\le}_{\mathrm{se}}$.
 (ii)
If system (1) has exactly one equilibrium in ${(0,\mathrm{\infty})}^{2}$, then it is locally asymptotically stable. If $(0,0)$ is an equilibrium, then it is a repeller.
 (iii)
If system (1) has three distinct equilibria in ${(0,\mathrm{\infty})}^{2}$, say $({\overline{x}}_{\ell},{\overline{y}}_{\ell})$, $l=1,\dots ,3$, with $({\overline{x}}_{1},{\overline{y}}_{1}){\le}_{\mathrm{se}}({\overline{x}}_{2},{\overline{y}}_{2}){\le}_{\mathrm{se}}({\overline{x}}_{3},{\overline{y}}_{3})$, then $({\overline{x}}_{1},{\overline{y}}_{1})$ and $({\overline{x}}_{3},{\overline{y}}_{3})$ are locally asymptotically stable, while $({\overline{x}}_{2},{\overline{y}}_{2})$ is a saddle point.
 (iv)
If there exist exactly two equilibria in ${(0,\mathrm{\infty})}^{2}$, then one is locally asymptotically stable and the other is a nonhyperbolic equilibrium.
which is positive by the inequality $\overline{y}>\frac{{\beta}_{2}}{{B}_{2}}$ since $(\overline{x},\overline{y})$ lies on the decreasing curve ${E}_{2}$ with a horizontal asymptote at $y=\frac{{\beta}_{2}}{{B}_{2}}$. Hence, in (14), ${\lambda}_{2}>1$. It follows from Theorem 10 part ii.(b) that $1<{\lambda}_{2}<1$.
 (i)
This is direct consequence of Theorem 9.
 (ii)
Solving for y and x respectively in the equations defining ${E}_{1}$ and ${E}_{2}$ in (4) gives that the vertical asymptote of ${E}_{1}$ is $x=\frac{{\gamma}_{1}}{{C}_{1}}$ and the horizontal asymptote of ${E}_{2}$ is $y=\frac{{\beta}_{2}}{{B}_{2}}$. The asymptotes guarantee that in order to have exactly one intersection point $(\overline{x},\overline{y})$ in ${[0,\mathrm{\infty})}^{2}$, the slopes of the functions ${y}_{1}(x)$ and ${y}_{2}(x)$ of ${E}_{1}$ and ${E}_{2}$, respectively, must satisfy the relation ${y}_{1}^{\prime}$