Feedback control has no influence on the persistent property of a single species discrete model with time delays

By developing some new analysis technique, we show that the following discrete single species model with feedback control and time delays is permanent.    N(n + 1) = N(n) exp  r(n) [ 1 − N 2 (n − m) k 2 (n) − c(n)u(n − η) ] , ∆u(n) = −a(n)u(n) + b(n)N(n − σ), where N(n) is the density of species, u(n) is the control variable, ∆ is the first-order forward difference operator ∆u(n) = u(n + 1) − u(n). m, σ , η are all nonnegative integers and r(n), a(n), k(n), c(n), b(n), d(n) are bounded nonnegative sequences. Our result shows that feedback control has no influence on the persistent property of the system.