Extinction, persistence and global stability in models of population growth

First, we systematize earlier results on the global stability of discrete model An+1=λAn+F(An−m) of population growth. Second, we invent the effect of delay m when F is unimodal. New, deep and strong results are discussed in Section 4, although Theorems 3-5 (Section 3) are still freshly new. This paper may be considered as a discrete version of our earlier work on the model x˙(t)=−μx(t)+f(x(t−τ)) [D.V. Giang, Y. Lenbury, Nonlinear delay differential equations involving population growth, Math. Comput. Modelling 40 (2004) 583-590]. We are mainly using ω-limit set of persistent solution, which is discussed in more general by P. Walters [An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982].