Dynamics of second order rational difference equations. With open problems and conjectures. (English) Zbl 0981.39011

Boca Raton, FL: Chapman & Hall/CRC. xi, 218 p. (2002).
The authors present a detailed treatment concerning the dynamics of the real difference equation \[ x_{n+1}=(\alpha+\beta x_n+\gamma x_{n-1})/(A+ Bx_n+C x_{n-1}), \quad n\in \mathbb{N}_0, \] with nonnegative coefficients and nonnegative initial values \(x_{-1},x_0\). Depending on the coefficients which are strictly positive they distinguish 49 different types (including 21 cases of trivial, linear and Riccati equations, respectively) the solutions of which are investigated with respect to equilibrium points, semicycles, invariant intervals, boundedness, oscillations, periodicity, local, global and asymptotic stability, stability trichotomy and global attractivity. An introductory chapter contains the needed basic definitions and general results concerning nonlinear difference equations of order two, an appendix, corresponding generalizations to higher order equations.
A speciality of the book are the numerous open problems and conjectures stimulating further investigations. These problems concern also generalizations of the difference equations in question and in particular mathematical models of various biological systems.


39B05 General theory of functional equations and inequalities
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
92D25 Population dynamics (general)