ABSTRACT
We make some assumptions about F. Either the population size is assumed to be static or growing. When the population size is static, then F is assumed to satisfy F(P) = 1 with initial value Po= K so that the population size remains constant. When the population size is growing, 0 < Po < K, and F satisfies the following three conditions:
and hence, lime-+oo F(P,) = 1. An example of a function F satisfying the above assumptions is the logistic growth function:
following scalar difference equation:
Zt+l = ze(1 - b) + (K-ze)(1 - e-Ma) = h(ze), zo E (0, K). (6) The function h has some nice properties which makes its behavior easy to analyze. The basic reproductive number for equation (6) is
Ro= >..K. b
20 L. J. S. Allen and J. C. Kesinger
{b) if Ro > 1 and 0 < z• < zm ~ K, then the sequence {zt}~1 is monotone and limt-+oo Zt = z•.
