Periodic solutions, oscillation and attractivity of discrete nonlinear delay population model

The objective of this paper is to systematically study the qualitative behavior of solutions of the nonlinear delay population model x ( n + 1 ) = x ( n ) exp ( − p ( n ) + q ( n ) r + x m ( n − ω ) ) , n = 0 , 1 , … , where p ( n ) and q ( n ) are positive periodic sequences of period ω , m , and ω are positive integers and ω > 1 . First, by using the continuation theorem in conincidence degree theory, we establish a sufficient condition for the existence of a positive ω -periodic solution x ¯ ( n ) with strictly positive components. Second, we establish some sufficient conditions for oscillation of the positive solutions about a periodic solution. Finally, we give an estimation of the lower and upper bounds of the oscillatory solutions and establish some sufficient conditions for the global attractivity of { x ¯ ( n ) } . Some illustrative examples are included to demonstrate the validity and applicability of the results.