THEORETICAL

POPULATION

BIOLOGY

105-l 26 ( 1977)

11,

Comparison of Linear and Nonlinear Models Human Population Dynamics*

for

JOEL YELLIN School of Humanities and Social Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND PAUL A. SAMUELSON Department

of Economics, Massachusetts Institute Cambridge, Massachusetts 02139

of Technology,

Received June 8, 1976

In standard demographic practice, population projections are commonly based on one-sex linear models of the Lotka-Leslie type. We demonstrate here that such projections based solely on time-invariant, age-specific male fertilities and mortalities are incompatible with those based solely on female fertilities and mortalities. This incompatibility obtains even in the singular case where effective male and female fertility functions are equal, and generate equal ultimate rates of growth. In standard demographic practice, the incompatibility is initially masked, since one-sex fertility functions are generally calculated from the same initial-time data and thereby tautologically forced to initially concur; however, with the passage of any finite time, the incompatibility reasserts itself-the only exception being the uninteresting case where the system is already in the stationary age and sex distribution of balanced exponential growth. An example is adduced of a nonlinear (age-free) system whose true rate of ultimate growth is correctly bracketed by the male and female one-sex rates of ultimate growth. Analysis of more general two-sex models shows that the two one-sex growth rates, calculated for arbitrary male and female initial age distributions, need not bracket the true rate. We show, however, that models exist such that with appropriate choices of initial conditions, this bracketing will occur.

1. INTRODUCTION The standard, linear demographic model (Sharpe and Lotka, 1911; Leslie, 1945) is based on one sex only. In general, such a model provides at best a linear approximation to real, two-sex dynamics. As a linear system, the Lotka-Leslie * Research supported in part by a grant from the National No. 5 ROl-HD 09081-02).

Institutes

of Health

(NIH

105 Copyright AI1 rights

0 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0040-580!3

106

YELLIN

AND

SAMUELSON

model obeys the principle of superposition: when one adds equal increments of fertile-aged individuals of one sex, one observes equal increments of subsequent population. This suggests a manifest absurdity, since adding any increment of individuals of one sex to a world with zero of the other sex can never generate any population whatever. So long as the sex ratio happens to be nearly normal, the linear approximation may seem to serve well. This explains why Sharpe and Lotka, in their original specification, could work with a male version of the model in computing the intrinsic rate of autonomous increase. Since, however; it is easier to identify an infant’s mother than father, and since it is plausible that age-specific female fertility may be more time-invariant than age-specific male fertility, subsequent writers (Lotka, 1913; Dublin and Lotka, 1925; and Kuczynski, 1928) have tended to work with the female version of the model. Just as demographers were beginning to master the mathematics of the Lotka model, World War I mortality for males resulted in a measurable discrepancy between the calculated male and female net reproductiwe rates, as noted in Lotka (1932) calculated that during 1920-l 923 France (1929). Thus, Kuczynski had a female NRR of 0.98, suggesting an ultimate exponential decay of total population; at the same time his male NRR of 1.19, calculated from data of the same period, implied an ultimate positive exponential growth of total popu1ation.l This dramatic discrepancy was difficult to shrug off. Ultimately, Hajnal (1947), Karmel (1947), Pollard (1948), Kendall (1949), Goodman (1953, 1967) Keyfitz (1968, 1971), Pollard (1973), Das Gupta (1972, 1973, 1976), and many other writers grappled with two-sex models, either with or without explicit marriage functions, and explicit nonlinear functional relationships. In this paper we continue research on nonlinear two-sex models begun in Yellin and Samuelson (1974). We demonstrate here that, in general, the femaledominant and male-dominant versions of the one-sex model are mutually contradictory. This incompatibility tends to be masked by the fact that, in using the data of the same base year to estimate separate female and male age-specific fertilities, one forces the two models to be compatible for the base year--a spurious agreement lost with the passage of any finite time. A more subtle camouflaging of this basic incompatibility occurs also for any situation in which the true system is fortuitously at or near a stable age distribution: Each one-sex model must then parasitically mirror the true two-sex exponential rate of increase. Despite the literal inconsistency of the male and female one-sex modelswith each other and with the true nonlinear dynamics-one might hope that the two asymptotic rates of growth derived from the male and female models would ’ A similar but smaller discrepancy the United States: in 1972 and 1973, NRR was below unity; after 1973 both

between male and female NRR’s occurred for the male NRR exceeded unity while the female dipped below unity.

107

NONLINEAR POPULATION DYNAMICS

always bracker the true rate of growth, or at least do so in a wide set of circumstances. As shown below, such bracketing need not obtain. Indeed, we adduce a nonlinear case for which the male and female one-sex rates of increase must be exactly equal (by virtue of symmetry in the postulated male and female mortalities and fertility propensities), with a common value that systematically differs from the true two-sex rate of ultimate increase.

2. INCONSISTENCIES OF ONE-SEX MODELS Let B(t) be the total of births at time t, characterized by a fixed male/female sex ratio (1 - r)/r. Let B(A, A’; t) be the (age-specific) number of births to fathers of age A and mothers of age A’, at time t. Then

B(t)=[‘~m’B(A,A’;t)dAdA’,

O to , being constrained to follow two incompatible paths. One most easily sees this incompatibility in the following way. By definition, we have for all t, most particularly for t, itself, the relations of Coale (1972, p. 55): B(t,)

= jB jB B(A, A’; to) dA dA’ a OL

= (1- y)-1jeO m,(A) N&A;to> dA =Y

-1

(2.6) (2.6a)

0 s oi

m,(A’)

N,(A’;

to) dA’.

(2.6b)

However, if the equality of (2.6a) and (2.6b) must hold for arbitrary initial NM(A; t,) and N,(A’; t,) functions, we clearly have a contradiction, whether or not (2.5) is valid. z Euler (1761), in a private communication to Stissrnilch, assumed that all males and females marry exactly at age 20, all dying at age 40. At ages 22, 24, and 26, they produce one boy and one girl baby. He showed that an ultimate geometric progression of evenyear births is implied. Euler’s symmetry postulate for the two sexes thus does satisfy the identity (2.3, in the form: ~~(22) = ~~(22) = p&24) = ~~(24) = ~~(26) = ~~(26) = 1; &(22) m,(22) = ~~(22) q&22) = 1; ~~(24) ~~(24) = ~~(24) ~~~(24) = 1; ~~(26) v&26) = ~~(26) q(26) = 1.

NONLINEAR

POPULATION

109

DYNAMICS

The incompatibility of the alternative one-sex equalities of (2.6a) and (2.6b) would be glaring were it not for the custom-a reasonable one-of estimating the m,(A) and m,(A’) functions a posteriori from the same data at t = t, . as will be presently Tautologically, this does avoid the (2.6) contradiction, demonstrated. But, for any time subsequent to t, , the contradiction between the mutually incompatible one-sex models reasserts itself. satisfied, assume To demonstrate that in such a case (2.6) is tautologically empirical fertilities GzM(A) the usual procedure of estimating or “identifying” and GzF(A’) from data for B(A, A’; t), N,(A; t), N,(A; t), and y at t = t, . We then have the defining relations

T%M(A)= (1 - y) J3 B(A, A’; to) dA’/N,(A; OL

to),

(2.7a)

&(A’) = y j” B(A, A’; to) dA/N,(A’; to).

(2.7b)

OL

From (2.7), we then easily see that the identity

(1 - y)-1 s” ?&(A) N,(A; ol

to) dA = y-l s” ++(A’) NM; LI

to) dA’

’ ’ B(A, A’; to) dA dA’ = B(t,) =Sf a a Q.E.D.

is satisfied.

But now let the system autonomously develop its own post-t,, development in the usual Lotka manner (2.4b), according to the hF(A)pF(A) schedule. The data so generated cannot agree with those following from (2.4a), using the male h,(A) p&A) schedule. Of course, the tautological constraint (2.6) must hold for all t, not just for initial t, , if both one-sex models are to be valid. Specifically, (2.1) and (2.7a) plus (2.7b) imply

B(t) 5.s(I - $1 L6 #z&A) N(A; t) dA = y1 j-B &(A’) c1

N,(A’;

t) dA’,

t 3 t, .

(2.6~)

This requirement in general will be violated by each of the one-sex dominant models. In particular, (2.6~) is in disagreement with the determinate motions generated by the male-dominant relations (2.1), (2.2), (2.3a), and arbitrary initial choices N,(A; to) and N,(A’; to). Equation (2.6~) is also inconsistent with the determinate motions generated by the female-dominant relations

110

YELLIN

AND

SAMUELSON

(2.1), (2.2), (2.3b), and those same initial age distributions. Any true two-sex model will, in general, also violate (2.6~). To see explicitly that these statements are correct, we may differentiate the identity (2.6~) with respect to time, obtaining dB(t)/dt

= (1 - $1

s,” +zM(A) [BN,(A;

dPd4 x-

[P&W p&4-l

t)/at] dA

&(A) 7

NM(A;

NdA;

t) -

t) - aN;(f’ BN,(A; aA

t)

] dA.

t)] dA

V-8)

The transition from the first to the second line in (2.8) makes use of differentiation of (2.2a); similarly, use of (2.2b) has been made in the last line of (2.8). If (2.8) is an identity in t, it must certainly be valid at t = to . At t = t, , this leads to the asserted contradiction: arbitrarily assignable N,(A; t,,)/aA and BN,(A’; t,)/aA’ functions will violate (2.Q even if (2.5) should be valid. We have used up all the tautological rope available in having already estimated and can no longer avoid the asserted contradiction to the k,(A) and &(A’), simultaneous validity of the parallel one-sex models. There is, however, a fortuitous singular case where the contradiction has no leverage to reveal itself. Suppose we begin at t, with a stable age distribution for both sexes, appropriate to some exponential rate of growth, cut. The rate u might come from some model remote from the Lotka system. Or it might actually come from a Lotka female-dominant model, with a “true” m,(A’) function. In that case, we know there cannot be a true m,(A) that obtains under perturbations of the stable age distribution; however, no matter how births are allocated to fathers by age, observations for t > t, will yield a “pseudo” m,(A) function that echoes the exp(ut) we started with. This echoing will be shown to be purely formal. Assume that the initial stable age distribution is generated by a nonlinear two-sex model that differs in its qualitative essentials from either Lotka model. Despite any such differences, the initial B(A, A’; t,)-assumed to be already to a naive statistician hell-bent in the equilibrium eUt configuration-yields to estimate linear-Lotka paraineters, pseudo &,(A) and/or G+(A’) functions that each give for the intrinsic or asymptotic rate of increase, the same root, r=YM= u and Y = yF = u, in the standard equations

1= sB PM(A) a

&,,(A)crA

1 = J8 PFW bi

#z,(A’)e-‘A’

dA = dA’ =

$&I);

$F(~).

(2.9a) (2.9b)

NONLINEAR

POPULATION

111

DYNAMICS

To show this, we observe from (2.2) and (2.7) that

p,+,(A) T&(A) = B(t - A)-l s” B(A, A’; t) dA’; n

(2.10a)

j+(A) hF(A) = B(t - A)-l 1” B(A’, A; t) dA’. ct

(2.1Ob)

Also, for an initial stable age distribution

Inserting

with growth rate u.

B(A, A’; t) = B(A, A’; to) exp[u(t - t,)];

(2.lla)

B(t) = B(t,) exp[u(t - to)].

(2.11b)

(2.11) into (2. lo), we have

Substituting

p,,,,(A) fiM(A) = eUAB(t&l l” B(A, A’; t) dA’; CY

(2.12a)

pp(A) rjlF(A) = eUAB(t,)-l s” B(A’, A; t) dA’. a

(2.12b)

(2.12) into (2.9) we then have 1 = #M(r) = B(t,)-l

s” e-rA+sA dA 1’ B(A, A’; t) dA’;

(2.13a)

1 = #r(r)

sB e-rAfUA dA JN’B(A’, A; t) dA’.

(2.13b)

= B(t,)-l

a

a

a

Clearly, r = u is a root of both (2.13a) and (2.13b). Explicitly, becomes

at T

u, (2.13)

j-” j-” B(A, A’; t,) dA’ dA;

(2.14a)

1 = z/+(u) = B(t,)-l j-” j.’ B(A, A’; to) dA dA’. a OL

(2.14b)

1 = lclM(u) = B(t,)-l

OL ci

Q.E.D.

When the identity (2.5) holds, we have the stronger identity #M(r) = $r(r) in (2.9), and rM = rF regardless of initial conditions. Therefore, even though the initiaI perturbation in the sex ratio occasioned by a war will lead to alternative and incompatible subsequent dynamic extrapolations from the male and female one-sex models, so long as (2.5) holds, the incompatible models must nonetheless generate the same ultimate growth rates, rM and rF . As is well known, the superior mortality experience of females is partially compensated in real life by the excess fraction of boy babies born, 653/11/r-8

112

YELLIN

AND SAMUELSON

(1 - 7)/r w 1.05. Does natural selection perhaps work in the very long run to preserve some tolerable approximation to (2.5), as systems that depart from that identity generate sex-ratio distortions harmful to survivability3 ?

3. APPROXIMATION PROCEDURES Despite their incompatibility with each other and with nonlinear two-sex models, the alternative one-sex models have appeal as possibly giving useful approximations to the unknown true rate of increase [see Lotka (1922) Das ome limited support for this hope is provided in this section S Gupta (1976)]. by a blending theorem which follows below, and by an example of a nonlinear system in which the bounding procedure happens to be always valid. One procedure followed by demographers uneasy with exclusive reliance on one sex alone is to regard reality as some kind of a weighted average of each linear one-sex model. It can be shown that the intrinsic growth rate from such a blend is indeed an internal mean of the one-sex growth rates. Let W,,(A) and W,(A) be positive net fertility

BLENDING THEOREM. functions such that

l&(t) = SE W,(A) &(t 01

- A) dA,

(i = 0, l),

weighting

(3.1)

where Wi(A)

for

> 0

0 < a < /3 < co.

Consider the blended system

R,(t) = s”a [two Ess ’ (I which has an asymptotic root of

W,(A)

+ (1 - w) W,(A)1fL(t - 4 d-4 B,(t

- A) dA,

rate of exponential

1 = Ja W,(A)eerA a

0AN,] = WV, , NJ > 0,

(3.6~)

with B[O, NJ = 0 = B[N, asphi,

> 0

for

101;

(NI , NJ

> 0.

By inserting a trial solution, N,(t) = cieUt in (3.6), one observes that this model will have a stable exponential mode proportional to exp(u*t) provided roots (u*, k*) can be found for zd

-Y)B(l,k)-* 1

= r&V

~_ 1 =yB(l,k) k = r,(k).

6 2 (3.7)

114

YELLIN AND SAMUELSON

Since the calculated male ultimate growth rate, r,(k), is an increasing function of k and since the calculated female growth rate, r,(k), is a decreasing function of K, at any initial R different from K* one of these respective one-sex estimates, ri(k), will be above u* and one will be below u*. The bracketing property is thus seen to hold. The nonlinear model (3.6) is instructive also as a warning against one-sex approximations. Suppose, as in Samuelson [1976, (11.17)] or Keyfitz (1971), we select for B(N, , NJ the reasonable form of the harmonic mean, 2[N$ + N&l]-l. Then, by making one of the death rates very large, say 6, > 6, > 0, the system (3.7) will have no root for positive k* and hence no exponential growth mode. The sex ratio, we know, then goes asymptotically to infinity or to zero: lim,,, NJN, = 00 or 0. What then happens if we blindly calculate one-sex intrinsic growth rates at any time t ? The calculated ri(k) will bracket a range of numbers, but since there is no true u*, positive, negative, or zero, this estimated range is illusory. Furthermore, as the next sections will show, calculating one-sex growth rates cannot in general provide reliable bounds on the true rate.

4. DISCRETE TIME AND AGE MODEL To bring out essentials and avoid functionals, we may suppose that time fl, +2,... . These times are defined only in age periods 1 and 2, which may to age groups 15-30 and 3045. Thus, NM;

t) = ~ri4(4(1

N&4’;

t) = p,(K)

the needless complexities of analyzing takes on discrete values, t = . . . -1, 0, so that females and males are fertile be thought of as roughly corresponding as in (2.2),

- r) B(t - 4

(A = 1, 2h (4.1)

yB(t - A’),

where A and A’ are now discrete age variables, as t. In this notation, (2.1) becomes B(t) =

i

i:

A=1

A'=1

= B(I,

(A’ = 1, 2), measured in the same periods

B(A, A’; t)

1; t) + B(1, 2; t> + B(2, 1; t) + B(2, 2; t).

(4.2)

We now make the usual assumption that the system is scale-free, or homogeneous-first-degree. Thus, for any A and A’,

B(A,A’;t) = f-qv&; t),N,(2; t),N,(l;4,N&i a?

(4.3)

NONLINEAR

POPULATION

115

DYNAMICS

where

f A-q+ , Ax, , Ax,>Ax41= hf AA’C%, x2 9x3, x*1,

(A, A’ = I, 2)

and

II(t) = i

i f A‘qNM(l

; t), N,(2;

t), NF(l ; q, Nr(2;

t)l

A=1 A'=1

= f[N,(l;

t), Iv&;

which defines the function

f.

t), fvF(Z

t), N,(l;

(4.4)

t)l,

By homogeneity,

f rk , Ax2 1Ax,, h] = Af[x1, x2, x3) x4] 2 0. Substituting B(t) =fKl

(4.1) into (4.4) we derive - Y) PM(l) w

11, (1 - Y) P.42) w

-

- 21,

m(l) = B[fqt

(4.5)

-

I), B(t - 211,

w

t 3 t, t-2,

-

I),

rPFwB(t

-

211 (4.6)

where [Iqt -

l), B(t - 2)] 2 0.

Note that (4.6) holds 2 periods or more after the initial N,(A; bs) and N,(A’; t,) have been arbitrarily prescribed, but does not necessarily apply for t = t, or t = t, + 1, when sex ratios can still reflect the initial mortality perturbations. Defining the growth ratio of births as

R(t) = B(t)/B(t - 1), we can reduce (4.6) and (4.7) to the first-order R(t) = /3[1, R(t -

I)-‘],

relation tato+2

(4.7)

= VW - 111, where the function F[ ] is defined by (4.7). If there exists a positive root, R* = 1 + r*, of

R = T[R],

I r’[R](

< 1,

(4.8)

then either Y* [or In(1 + r*)] represents the ultimate true intrinsic growth rate of the nonlinear two-sex model. The asymptotic stable age and sex distributions can then be calculated by setting B(t) = ~(1 + I*)~-~D, and applying the relations (4.1).

YELLIN AND SAMUELSON

116

We reserve detailed treatment of the general case (4.6) for a subsequent paper. Here we wish to analyze some interesting special cases, where the funcand N,(A’), as certain weighted geometric tions f A*‘[ ] depend only on N&4) means or otherwise4. In particular, two special patterns are worth examining in detail: the diagonal case where fathers and mothers are always of the same age; and the symmetric case where male and female mortality and fertility propensities are the same.

5. THE LINEARIZABLE DIAGONAL CASE Here, by definition, for fertility, so that

pairings of male and female of dzjhnt

B(t) = fll[iv‘&;

B(t) = G,B(t - 1) + G,B(t - 2),

GA

=f""[(l

41.

t), N,(l ; t)1 + fZ2[Nh4(2; t), N&i

Using the relations (4.1) and the homogeneity we see that (5.1) becomes the linear relation

where the time-independent

ages are irrelevant

property

(5.1)

(4.3) of f**[

t b to +2,

1,

(5.2)

constants GA are defined by

- Y)$'A@>B(t -A), I'$%@)B(t - hI/B(t -A)

= f”“[(l - Y)PMW YPFWI2 0

(A = 1, 2).

(5.3)

Since (5.2) has the standard linear form, we know that the general solution is B(t) = b,R,t + b2R2, where b, and b, depend on the initial (5.2), we observe that the parameters equation

(5.4)

conditions for B. Inserting (5.4) into (R, , R,) are roots of the quadratic

R= - G,R - G, = 0.

(5.4’)

It follows easily from (5.4) that one root, call it R, , is positive and the other negative, with -R, < R, < 0 < R, . This further implies the asymptotic behavior

$+t B(t)Ryt = b, . 4 The fact that f**‘[ ] does not depend on NM(A f 1; t) or N,(A’ rt 1; t) is a serious restriction that will have to be dropped in the most general analysis of (4.6).

NONLINEAR

POPULATION

117

DYNAMICS

From (5.4’), we observe that if G, + G, = 1, then R, = 1, and the growth rate r* = 0. By making the initial sex ratios sufficiently distorted we can certainly contrive that the male and female one-sex NRR’s (and hence, the one-sex ultimate growth rates) calculated for to < 2 < to + 2 be one above and the other below unity, even when G, + G, = 1 and the true ultimate growth rate of the population is zero. However, provided there initially exist some males and females of the same age, after 2 periods at most, when t 3 t, + 2, the sex ratios will have normalized. We will then have, for all subsequent t,

hM(A) = (1 - r)f”W - Y)Pd4 YPFWII (1 - r>PMW ’ = GA/PM(A),

t>,t,+2 (5Sa)

(A = 1, 2)

and similarly,

%(A) = GA/P&~

(5Sb)

Clearly, from (5.5) the one-sex models do satisfy-for the diagonal case (5. I)-the crucial identity (2.5). Therefore, feeding into them the same initial B(t,, - 1) and B(t,, - 2) values will generate for the male and female models identical B(t) values ever afterward, and the same values for all demographic variables. Moreover, both one-sex models now satisfy precisely the same linear relation as the correct diagonal two-sex models

B(t) = &f(l)

P,(l) B(t - 1) + f&(2) P,(2) B(t - 2)

= &F(l) p&)

B@ -

s G,B(t -

1) + G&t

1) + &F(2) PF(2) B(t - 2) - 2),

(5.6)

t>t,+2

for common initial values [B(t,), B(t, + I)]. The model of Euler (1761) (mentioned in footnote 2), has the form of the diagonal case. This diagonal case is the most favorable one for one-sex approximations. For it, once the initial perturbations have had time to disappear, the one-sex models do more than predict the correct ultimate two-sex growth rate: they actually give correct transient growth patterns. Of course, the diagonal model is not realistic, since parents in real life often do have different ages.

6.

THE

SYMMETRIC

TWO-SEX

CASE

Just as the diagonal model is most favorable for one-sex approximations, our present case is in a sense the most disastrous one for would-be bracketeers.

118

YELLIN

The basic postulate

AND

SAMWLSON

here is the relation B(1,2;

t) = B(2,l;

t),

(6.1)

which implies that marriages between older women and younger men produce equal numbers of offspring as do marriages between older men and younger women. From (2.3) or (2.10), we observe that-provided we begin at a time t which has been preceded by at least 2 periods in which the normal mortality coefficients [p,(A), p,(A)] are applicable-the one-sex effective fertility coefficients are equal for the male and female case, and the identity (2.5) is satisfied. Under those circumstances the one-sex ultimate growth rates have a common value R,”

=

1 +

Now the question of bracketing calculated, as in (5.5), for arbitrary

I~*

=

1 +

YF*.

(6.2)

becomes equivalent to whether [B(t,,), B(to + I)] in

the

R,*,

PJM(-% to+ 3, N&4’;to+ 91 = [(I - Y)p,(A)qto + 2 - 4 m(4 wo + 2 - 41> is equal to the true value R* asymptotically characterizing the nonlinear system. We shall show that no matter how many periods we project the true symmetric model forward, if we do not begin in stable growth, the one-sex models will always produce a systematically biased estimate of that growth. Fortunately, the degree of bias attenuates as the chosen initial conditions approach the stable configuration. To specify an easy example of such nonbracketing, we can work through family (whose general analysis the symmetric case for the “geometric-mean” is provided in Section 8 below). With the choice of geometric-mean form for f”“‘[ 1, we have

f”“‘Kl - Y)Ph4Ww - 4%YPFW)w - 41 = [fAB(t - A)]““[f,*B(t - A’)]l’2, (A, A' = 1,2), where we have introduced the pair of positive coefficients (fr , ii). Note that (6.3) implies, in addition to the sex-symmetry constraint the additional age-symmetry relation B( I, 1; t)/B( I ) 2; t) = B(2, 1; t)/B(2,2;

t).

(6.3)

(6.1)

(6.4)

NONLINEAR

POPULATION

119

DYNAMICS

(2.1) and (4.2), we can now use (6.3) to derive the basic dynamical

Following equation

B(t) = f A=1

i

B(A, A’; t)

A'=1

(6.5) = fJi(t

-

1) + 2f;‘2f;‘aB(t

-

1)1’2 B(t - 2)“2 +f,B(t

The nonlinear system (6.5) can be converted the transformation of variables:

- 2).

into an exact linear system by

z(t) = B(t)l’2.

(6.6)

Then (6.5) becomes the linear d@erence equation z(t) = f:‘2Z(t The solution

-

1) +f’2’%(t

- 2),

t 3 t, + 2.

(6.7)

of (6.7) is again of standard form 44

=

mt

+

w4

C2P2t,

where the pi are the roots of

t)(p) = p2 - fi’“p - f;‘” = 0,

(6.9)

with -P1

as in (5.4’) above. Therefore,


t, .

(8.1)

124

YELLIN

AND

SAMUELSON

An exponential solution of (8.1) exists, of the form B(t) = b,(R*)t where p1 is the real positive root of the fourth-degree polynomial 1 = q,

1) P-2 + [q1,2)

For B(tO - l)/B(ts one-sex growth rates given an initial stable that the growth rates

+ F(2, l)] F3 + F(Z2) p-4 = P(p).

= br$

(8.2)

2) = R, sufficiently near the true R*, the calculated will, singular symmetric cases aside, bracket R*. Also, distribution with R,, = R* = 1 + r*, one may verify 1 + Y,,,,, 1 + rF , and 1 + Y*, will all coalesce.

Remark. A fourth-degree polynomial has 4 roots, (pl , p2 , p3 , p4), some of them possibly complex. It is tempting, extrapolating uncritically from linear analysis, to assert that any dynamic path for B(t) can be written in the series form B(t) = fJ,f,2t +

bzf?

t

b3pf

+

b4P24t.

(8.3)

This is nonsense for a true nonlinear system. Only p12 has genuine significance for an exact real-time solution or for an asymptotic solution. Any other solution of the form p& does not correspond to an admissible, nonnegative B(t). Also, since the principle of superposition does not hold for a nonlinear system, weighted sums of two or more such distinct exponentials will not be solutions of the nonlinear recurrence relation (&I), even if it is possible for each of them to be. Actually, along with p12 = R*, there is another relevant “root’‘-call it R,-but it is not one of the P:+~ when the dynamic system is nonlinear. This secondary growth rate is the nonpositive root of the characteristic quadratic equation of the associated linear system-the linear difference-equation system whose solutions approximate closely to the true time development of the divergences from B(t) = b(l + T*)~, when those divergences are small. That associated linear system is defined, for the general nonlinear case (4.6), as

x(t) = #B&t - 1) + B24t - 3,

(8.4)

where

pi = a f, $ F(A,A’) B(t - A)l” B(t - A’)1,‘]/iiB(t- i), [A=1A’=1 evaluated at B(t - l)/B(t - 2) = R*. The characteristic quadratic equation

(i == 1, 9,

associated with (8.4) is

0 = 1 - ,5,R-1 + ,t3zR-2 = (1 - R*R-l)(l

- R2R-‘),

-R*

Comparison of linear and nonlinear models for human population dynamics.

Population projections are usually based on 1-sex linear models of the Lotka-Leslie type. This paper explores the inconsistencies of 1-sex models and ...
1MB Sizes 0 Downloads 0 Views