Dynamics of a Quadratic Rational Delay Difference Equation

The study makes a significant contribution to the theoretical understanding of nonlinear difference equations, a class of recursive equations with wide applications in fields like population dynamics and economics. In [1] E. Tasdemir, et al. proved that the positive equilibrium of the nonlinear discrete equation \(x_n+1=1+p\left(\frac{x_{n-m}}{x_n^2} \right)\) is globally asymptotically stable for \(p \in \left(0, \frac{1}{2} \right)\), locally asymptotically stable for \(p \in (\frac{1}{2},\frac{3}{4}\)) and it was conjectured that for any p in the open interval (\(\frac{1}{2},\frac{3}{4}\)) the equilibrium is globally asymptotically stable. In this paper, we prove that this conjecture is true for the closed interval [\(\frac{1}{2},\frac{3}{4}\)]. In addition, it is shown that for \(p \in (\frac{3}{4}, 1\)) the behaviour of the solutions depend on the delay m. Indeed, here we show that in case m = 1, there is an unstable equilibrium and an asymptotically stable 2-periodic solution. But, in case m = 2, there is an asymptotically stable equilibrium. These results are obtained by using linearisation, a method lying on the well known Perron's stability theorem. Finally, a conjecture is posed about the behaviour of the solutions for m > 2 and \(p \in (\frac{3}{4}, 1\)). The advanced analytical methods employed showcase techniques that can guide future research and developments in the field.