##
**Rational systems in the plane.**
*(English)*
Zbl 1169.39010

Given the rational recurrence system
\[
\begin{aligned} x_{n+1}&=\frac{\alpha _{1}+\beta _{1}x_{n}+\gamma _{1}y_{n}}{ A_{1}+B_{1}x_{n}+C_{1}y_{n}} \\ y_{n+1}&=\frac{\alpha _{2}+\beta _{2}x_{n}+\gamma _{2}y_{n}}{A_{2}+B_{2}x_{n}+C_{2}y_{n}} \end{aligned}\tag{1}
\]

where \(n\in \{0,1,2,\dots\},\) and where the parameters are nonnegative and the initial values \(x_{0},y_{0}\) are nonnegative such that the denominators are always positive, there are \(2401=49\times 49\) special cases (depending on the parameters). For example, one special case of the first equation is

\[ x_{n+1}=\frac{\alpha _{1}}{y_{n}} \]

and this case is marked as type \(k=3.\) In this manner, all special cases can be marked by \((k,l)\) where \(k,l\in \{1,\dots,49\}\)

Some of these special cases can be transformed into scalar equations, such as the Riccati difference equation

\[ z_{n+1}=\frac{\alpha +\beta z_{n}}{A+Bz_{n}},\quad n=0,1,2,\dots. \]

Hence their properties can be inferred from these scalar rational difference equations.

Some other special cases can be classified as competitive and/or cooperative systems. More precisely, consider a recurrence system of the form \[ \begin{aligned} x_{n+1}&=f(x_{n},y_{n}), \\ y_{n+1}&=g(x_{n},y_{n}), \end{aligned}\tag{2} \]

where \(n\in \{0,1,2,\dots\},\) \(I,J\) are real intervals, and \(f:I\times J\rightarrow I,\) \(g:I\times J\rightarrow J\) and \((x_{0},y_{0})\in I\times J\) . If \(f\) is increasing in the first variable and decreasing in the second, while \(g\) is decreasing in the first variable and increasing in the second, then the recurrence system is said to be competitive. If \(f\) and \(g\) are increasing in the first and in the second variables, then the system is said to be cooperative. For competitive and cooperative systems, some monotonic properties of their solutions are known. From these monotonic properties, stability properties of solutions can then be derived.

However, much is not known about the qualitative properties of solutions of (1). These include the boundedness character of solutions, the local stability of equilibrium points, the existence of periodic solutions. Indeed, a list of ‘open problems’ or rather, open projects, is posed in this paper.

It would, however, be nice if concrete everyday real models can be built to provide motivation for the study of (1).

where \(n\in \{0,1,2,\dots\},\) and where the parameters are nonnegative and the initial values \(x_{0},y_{0}\) are nonnegative such that the denominators are always positive, there are \(2401=49\times 49\) special cases (depending on the parameters). For example, one special case of the first equation is

\[ x_{n+1}=\frac{\alpha _{1}}{y_{n}} \]

and this case is marked as type \(k=3.\) In this manner, all special cases can be marked by \((k,l)\) where \(k,l\in \{1,\dots,49\}\)

Some of these special cases can be transformed into scalar equations, such as the Riccati difference equation

\[ z_{n+1}=\frac{\alpha +\beta z_{n}}{A+Bz_{n}},\quad n=0,1,2,\dots. \]

Hence their properties can be inferred from these scalar rational difference equations.

Some other special cases can be classified as competitive and/or cooperative systems. More precisely, consider a recurrence system of the form \[ \begin{aligned} x_{n+1}&=f(x_{n},y_{n}), \\ y_{n+1}&=g(x_{n},y_{n}), \end{aligned}\tag{2} \]

where \(n\in \{0,1,2,\dots\},\) \(I,J\) are real intervals, and \(f:I\times J\rightarrow I,\) \(g:I\times J\rightarrow J\) and \((x_{0},y_{0})\in I\times J\) . If \(f\) is increasing in the first variable and decreasing in the second, while \(g\) is decreasing in the first variable and increasing in the second, then the recurrence system is said to be competitive. If \(f\) and \(g\) are increasing in the first and in the second variables, then the system is said to be cooperative. For competitive and cooperative systems, some monotonic properties of their solutions are known. From these monotonic properties, stability properties of solutions can then be derived.

However, much is not known about the qualitative properties of solutions of (1). These include the boundedness character of solutions, the local stability of equilibrium points, the existence of periodic solutions. Indeed, a list of ‘open problems’ or rather, open projects, is posed in this paper.

It would, however, be nice if concrete everyday real models can be built to provide motivation for the study of (1).

Reviewer: Sui Sun Cheng (Hsinchu)

### MSC:

39A20 | Multiplicative and other generalized difference equations |

### Keywords:

rational difference system; competitive system; cooperative system; monotonicity; Riccati difference equation; recurrence system
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\textit{E. Camouzis} et al., J. Difference Equ. Appl. 15, No. 3, 303--323 (2009; Zbl 1169.39010)

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### References:

[1] | DOI: 10.1080/10236190701388492 · Zbl 1131.39005 |

[2] | DOI: 10.1080/10236190701761482 · Zbl 1138.39002 |

[3] | LadasG., On the dynamics of a rational difference equation, part 1, (2008b) |

[4] | LadasG., On the dynamics of a rational difference equation, part 2, (2008c) |

[5] | LadasG., On the dynamics of the rational difference equation, (2008d) |

[6] | DOI: 10.1080/10236190410001728104 · Zbl 1070.39025 |

[7] | DOI: 10.1016/j.jmaa.2004.06.035 · Zbl 1070.39024 |

[8] | Burgić Dž., Adv. Dyn. Syst. Appl. 3 (2008) |

[9] | Camouzis E., Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures (2008) · Zbl 1129.39002 |

[10] | LadasG., On the dynamics of the rational difference equation, (2008b) |

[11] | LadasG., On the dynamics of the rational difference equation, (2008c) |

[12] | DOI: 10.1016/S0898-1221(01)00326-1 · Zbl 1001.39017 |

[13] | DOI: 10.1016/S0362-546X(02)00294-8 · Zbl 1019.39006 |

[14] | DOI: 10.1080/10236190412331334464 |

[15] | DOI: 10.1080/10236190410001652739 · Zbl 1071.39005 |

[16] | DOI: 10.1080/17513750701610010 · Zbl 1284.92108 |

[17] | Dancer E., J. Reine Angew Math. 419 pp 125– (1991) |

[18] | DOI: 10.1007/BF00160333 · Zbl 0735.92023 |

[19] | DOI: 10.1016/0362-546X(91)90163-U · Zbl 0724.92024 |

[20] | DOI: 10.1016/0022-247X(92)90167-C · Zbl 0778.93012 |

[21] | Grove E.A., Periodicities in Nonlinear Difference Equations (2005) · Zbl 1078.39009 |

[22] | Hirsch M., Handbook of Differential Equations 2 (2005) |

[23] | DOI: 10.1080/10236190412331335445 · Zbl 1080.37016 |

[24] | DOI: 10.1090/S0002-9947-96-01724-2 · Zbl 0860.47033 |

[25] | Kocic V.L., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001 |

[26] | DOI: 10.1201/9781420035384 |

[27] | M.R.S. Kulenović and O. Merino, Global bifurcations for competitive systems in the plane, (to appear) · Zbl 1175.37058 |

[28] | MerinoO., Invariant manifolds for competitive systems in the plane, (to appear) |

[29] | Merino O., Discrete Dynamical Systems and Difference Equations with Mathematica (2002) · Zbl 1001.37001 |

[30] | DOI: 10.3934/dcdsb.2006.6.97 · Zbl 1092.37014 |

[31] | DOI: 10.3934/dcdsb.2006.6.1141 · Zbl 1116.37030 |

[32] | Kulenović M.R.S., Rad. Mat. 11 pp 59– (2002) |

[33] | Nurkanović M., J. Inequal. Appl. pp 127– (2005) |

[34] | Nurkanović M., Adv. Differ. Equ. 3 pp 1– (2006) |

[35] | DOI: 10.1007/BF00276900 · Zbl 0474.92015 |

[36] | Selgrade J.F., J. Math. Biol. 25 pp 477– (1987) · Zbl 0634.92008 |

[37] | DOI: 10.1007/BF00307854 · Zbl 0344.92009 |

[38] | DOI: 10.1137/0517075 · Zbl 0606.47056 |

[39] | DOI: 10.1016/0022-0396(86)90086-0 · Zbl 0596.34013 |

[40] | DOI: 10.1080/10236199708808108 · Zbl 0907.39004 |

[41] | E.C. Zeeman, Geometric unfolding of a difference equation, Preprint, (1996), (Hertford College, Oxford) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.