Periodically forced Pielou's equation

We investigate the global character of solutions of the periodically forced Pielou's equationxn+1=βnxn1+xn−1,n=0,1,…, and prove that when the sequence {βn}{βn} is periodic with prime period k  , with positive values, and ∏i=0k−1βi>1, every positive solution converges to a periodic solution with prime period k.