By N. Keyfitz, Hal Caswell
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Additional resources for Applied Mathematical Demography, Third Edition (Statistics for Biology and Health)
The initial and ﬁnal conditions are both of zero increase; that is, the curves coincide at beginning and end. We seek the ratio of increase in the population between its initial and ﬁnal stationary conditions. 1, if they begin and end together the exponential of the area between them is the total T T increase over the time in question. 1) the ratio of increase in the population must be T exp[ 0 r(t) dt], or simply eA . This applies for any pair of monotonically descending curves that start at the same level and end at the same level.
Even more important than the selection of the time interval over which a ﬁt is made is the nature of the curve chosen. Consider the United States population from 1870 to 1970, and ﬁt to it the hyperbola (von Foerster et al. 1960) α N (t) = . te − t The hyperbola contains two constants, of which te is the time of population explosion, when N (te ) = ∞. The time te is easily calculated from observations at two dates, t1 and t2 , where the population is known to be N1 at time t1 and N2 at time t2 ; the reader may show that te = N 2 t2 − N 1 t1 .
The integral under the straight line is x+5 x Since x+5 x l(a) da = 52 (lx + lx+5 ). 4) from which the value of lx+5 /lx may be obtained by dividing numerator and denominator on the right by lx , and then solving a linear equation for lx+5 /lx to ﬁnd lx+5 1 − 55 Mx /2 . 04147. 1). 5) the probability of dying is 5 qx =1− lx+5 55 Mx . 6) 34 2. 2) the quantity x µ(a) da. We would expect the x+5 answer to emerge as x µ(a) da = 5(5 Mx + C), where it will turn out that C is a correction easily obtained on the assumption that both the population p(a) and the death rate µ(a) change linearly within the interval.
Applied Mathematical Demography, Third Edition (Statistics for Biology and Health) by N. Keyfitz, Hal Caswell