Dynamics of a rational difference equation using both theoretical and computational approaches

This paper is concerned with the qualitative behavior of solutions to the difference equationxn+1=xn+pxn+qxn-k,n=0,1,2,…where the initial conditions x−k, …, x−1, x0 are non-negative, k ∈ {1, 2, 3, …}, and the parameters p, q are non-negative. Our concentration is on invariant intervals, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. It is worth to mention that this difference equation was an open problem introduced by Kulenovic and Ladas in [M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002.]. Our final comments are about informative examples.