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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).

MSC:

39A20 Multiplicative and other generalized difference equations
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