The net reproduction rate of a population is defined as where the bi are the birth rates

Chapter 4, Problem 4.6.23

(choose chapter or problem)

The net reproduction rate of a population is defined as where the bi are the birth rates and the sj are the survival rates for the population. (a) Explain why r can be interpreted as the average number of daughters born to a single female over her lifetime. r b1 b2s1 b3s1s2 p bns1s2 p sn (b) Show that r 1 if and only if l1 1. (This representszero population growth.) [Hint: LetShow that l is an eigenvalue of L if and only ifg(l) 1.](c) Assuming that there is a unique positive eigenvaluel1, show that r 1 if and only if the populationis decreasing and r 1 if and only if the populationis increasing.A sustainable harvesting policy is a procedure that allows acertain fraction of a population (represented by a populationdistribution vector x) to be harvested so that the populationreturns to x after one time interval (where a time interval isthe length of one age class). If h is the fraction of each ageclass that is harvested, then we can express the harvestingprocedure mathematically as follows: If we start with apopulation vector x, after one time interval we have Lx;harvesting removes hLx, leaving