Optimal Life Histories, Optimal Notation, and the Value of Reproductive Value

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Optimal Life Histories, Optimal Notation, and the Value of Reproductive ValueAuthor(s): Daniel GoodmanSource: The American Naturalist, Vol. 119, No. 6 (Jun., 1982), pp. 803-823Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/2460964 .

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Vol. 119, No. 6 The American Naturalist June 1982

OPTIMAL LIFE HISTORIES, OPTIMAL NOTATION, AND THE VALUE OF REPRODUCTIVE VALUE

DANIEL GOODMAN

Department of Biology, Montana State University, Bozeman, Montana 59717 and Scripps Institution of Oceanography, La Jolla, California 92023

Submitted May 11, 1981; Accepted November 25, 1981

There have recently appeared in print a number of confusions in the matter of life history optimization. The principal confusion has to do with the role of reproductive value in life history optimization. Ancillary confusions have to do with the form of the expression for reproductive value and with the quantities to which symbols in that expression refer in various uses.

Fisher (1930) developed the concept of reproductive value in order to evaluate the relative contributions of individuals of different ages to population growth, and conversely, to evaluate the relative importance to fitness of events at different ages. The mathematical expression itself appears implicitly in the formula for the scale factors in Lotka's (1939) renewal equation for the trajectory of a population's birth rate; and a quantity with the same properties appears in Leslie's (1945) discrete time model of the population projection process. Interestingly, Leslie developed the calculation of this quantity via operations on the projection matrix, and did not present an actual formula for reproductive value in terms of the life table elements. Subsequent authors, attempting to write a discrete time formula analogous to Fisher's integral form, almost without exception have done so incorrectly.

With the resurgence of interest in the problems of evolutionary optimization in the 1960s, the potential significance of the reproductive value concept was recog- nized in four suggestive applications. MacArthur (1960) proposed that the predation loss from a population be scaled in units of reproductive value in order to reveal the optimal age specificity of a prudent harvest. Hamilton (1966) argued that the schedule of reproductive value could indicate the selective forces operating on certain forms of mortality (but not others). Williams (1966) referred to reproductive value as a means of scaling costs and benefits of activities at a given age to fitness. MacArthur and Wilson (1967) noted the relevance of the reproductive value schedule in a discussion of the optimal age composition of a population of colonizing dispersers.

In 1974 three papers presented proofs utilizing reproductive value in weighing the relative contributions of different activities at a given age to the growth rate, or

Am. Nat. 1982. Vol. 119, pp. 803-823. ? 1982 by The University of Chicago. 0003-0147/82/1906-0004$02.00. All rights reserved.

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804 THE AMERICAN NATURALIST

fitness, conferred by the associated life table. Goodman (1974) treated the trade- off between survival and reproduction; Schaffer (1974) extended the conceptual framework to include trade-offs involving growth; and Taylor et al. (1974) developed a model capable of dealing in general with traits affecting the life history. This latter paper contained the most incisive and comprehensive proofs concern- ing optimal life histories, but perhaps because of the forbiddingly formal, mathematical presentation, it was neglected for a number of years, and more recently it has been misunderstood to the extent that it is cited notably in papers (Caswell 1980; Ricklefs 1981) which argue that reproductive value is not the key to life history optimization.

Here, I will show that reproductive value, when properly formulated, is the fundamental quantity which is maximized in every optimization of a life history. In order to remove some of the mystery surrounding the theory of life history optimization as a discipline unto itself, I will develop the proof from an elementary verbal premise and proceed via standard maximization methods of the calculus, treating the life table problem as just one example of a large class of familiar maximization problems which arise in economics and physics. In this spirit, Schaffer (1974) identified the life history problem as one that is in a form which can be approached through dynamic programming; Taylor et al. (1974) in fact applied dynamic programming techniques in their discussion; and Leon (1976), in a thorough mathematical review of the problem, discussed application of dynamic programming and optimal control theory.

In the course of the discussion I will attempt to elucidate the common features in the more interesting misapplications and misunderstandings of the role of reproductive value in the optimization, and in passing will indicate the correct form for the discrete time equations. Finally, to support the assertion that reproductive value can serve usefully as a unifying concept in life history optimization, I will demonstrate an application to the problem of sex ratio and sex changes.

There is hardly anything new in this paper. Indeed, by squinting just right when reading three particular pages of "The genetical theory of natural selection," one can conclude that Fisher, in his own laconic way, had anticipated almost the entirety of this discussion over 50 years ago. There still remains merit in restating the matter in the hopes that this can be accomplished with such clarity that a wider understanding will result.

OPTIMAL LIFE HISTORIES

Optimal Net Maternity Function for a Cohort

... it is worth noting that if we regard the birth of a child as the loaning to him of a life, and the birthlof.hisioffspringias asubsequentirepayment of;theldebt, thelmethod by which m is calculated shows that it is equivalent to answering the question-At what rate of interest are the repayments the just equivalent of the loan? (Fisher 1930, p. 26).

Consider a cohort of newborn. Over the lifetime of the cohort, members of the cohort will die and offspring of the cohort will be born at rates determined by the



REPRODUCTIVE VALUE AND OPTIMAL LIFE HISTORIES 805

life history. Let us characterize the life history in terms of an age specific per capita mortality schedule, Wx), and an age specific per capita fecundity schedule, m(X), where x designates age, or equivalently time since the birth of the cohort. The fraction of the cohort surviving to time x is then ef - 0 )dy, which is called the survivorship, and designated '(x). The rate of births accruing to the cohort at time x is m(x) times the number surviving, or t()m(x). This product is called the net maternity function, +(x).

At the end of the lifetime of the cohort, none of the original individuals will be left, so the contribution of the original individuals to population growth must, at this time, be measured in terms of the births to which they have given rise. Because those births appeared at different times over the life of the cohort, not all are equivalent, and a simple sum of all such births would not be an adequate measure.

From Lotka's proof of convergence to stable age distribution, we know that all the descendants of individuals born to the cohort at time x will eventually form a subpopulation, with this characteristic age distribution, growing at a constant per capita rate, r, where r is the one real root of Lotka's equation:

1 = f erxp(x)dx. (1)

The set of descendants of individuals born at time y will likewise come to form a subpopulation with the same characteristic age distribution, growing at the same rate r. Since these two subpopulations were initiated at different times, the ultimate subpopulation descended of individuals born at time y will have had (x - y) more time to grow than the subpopulation founded at time x, and will, during this extra time, have been growing at rate r. If these subpopulations had been initiated by the same number of births, then at any time in the distant future the size of the one founded at time y will be e i (xi) times larger than the one founded at x. So, relative to a newborn at time zero, a birth at time x contributes e -Ix ultimate offspring.

Thus, the reproductive contribution of the cohort can be measured as the sum of all its offspring discounted at the rate r for the time of their birth. That is, the total reproductive contribution, discounted to the time of the birth of the cohort, will be given by the integral:

T e -a4x+(Zxdx. (2)

By the same logic, the optimal life history for the cohort must be the one which maximizes this integral (2). This seems an empty statement in light of equation (1) which implies a value of one in any case, but the matter is resolved by noting that equation (1) was used to determine the growth rate of the population of descendants, so the 4 in this equation refers to their life history, whereas the 4 in expression (2) refers to the life history of the cohort in question. Designate the life history of the descendants as 4, while that of the cohort remains 4. On the other hand, the r in expression (2) was the discount rate derived from the growth rate of




the population of descendants, so this is the rate associated with k; designate it r. Thus, expression (2) is not really identical to the integral of equation (1).

As a mathematical dodge, we might consider the seemingly artificial question of finding the life history for the cohort, which will maximize its total reproductive contribution, under the peculiar assumption that the life history of the population of descendants is fixed at 4, with which a fixed growth rate, ri, is associated. Formally, then, the objective would be to maximize the expression

J = fe-P()ddx. (3)

where ( is chosen from a range of life history options, and ri is a parameter. Designate the life history which maximizes J as 0*. This life history will be the

optimal life history for the cohort, subject to the assumption that the cohort's descendants have a life history which confers a growth rate ri. The cohort's life history, 0*, will have associated with it, via equation (1), a growth rate, r*. If r* turns out not to be equal to the value of r used as the parameter in equation (3), then the net maternity function 0* will be optimal only in some strange world where indeed the life history of future generations is fixed, regardless of the life history of the founding cohort, quite unlike the world we know. But if, by some coincidence, r* does equal r, then we would conclude that 0* is in fact the optimal life history for the population, and that r* is the maximum achievable growth rate under the available range of life history options.

When the chosen value for the parameter r does coincide in value with r*, then the maximal value of J, obtained with 0* , will indeed be one, (since the right hand of eq. [3] then becomes indistinguishable from the right hand of eq. [1]). That does not vitiate the point, however, that the optimal life history maximizes the quantity J, where the parameter ri is the growth rate associated with the optimal life history. Any other life history substituted into equation (3), using this value for rj, will yield a value of J that is less than one. Since the value of the growth rate associated with the optimal life history cannot be known in advance, this does not seem a very practical way to proceed, but it does establish a formal point which will prove important.

The elegant way to find the maximal population growth rate as well as identify the optimal life history would be to develop a class of solutions for the net maternity function, (*, that maximizes J, as a function of the parameter ir. Since the maximal value of J, so obtained, will depend on the value of the parameter r, designate it J*(P). Then the actual maximum achievable population growth rate, rmax, is the solution to J*()f= 1; and the actual optimal net maternity function, 0Opt, is found by locating the (* which maximizes J with the parameter r equal to rmax. With sufficiently simple constraint surfaces defining the possible set of life history options, this program could be carried out analytically. Otherwise resort must be made to iterative methods.

We may cast this in a more biological light by beginning with a value for r that corresponds to the growth rate in a hypothetical population. Then the J computed using this r in equation (3) will indicate the effective reproduction for a cohort with the net maternity function (, in the context of a population with a growth rate of r. If the cohort has the same life history as the rest of the population, the value of




J for that cohort will be one, which follows immediately from substituting the population's life history, 4, in equation (3), recapitulating Lotka's equation (1). Now consider a mutant cohort, with some other life history, 4, and evaluate its effective reproduction J, still using the population's growth rate for the value of r . If the value of J conferred by this mutant net maternity function is greater than one, then this cohort would leave a larger number of ultimate descendants than a cohort composed of the same number of nonmutant individuals, even if the descendants of the mutants exhibited the nonmutant life history (recall that r was defined originally in terms of the growth rate of the cohort's descendants). Now, if the descendants of the mutant cohort retain the mutant life history, it follows that the ultimate number of descendants of the first mutant cohort would be larger still, with the discrepancy compounding in each generation. So, with perfect heritabil- ity of the net maternity function, a mutant cohort whose life history, 0, yields a value of J greater than one, where r is the growth rate of the nonmutant population, will give rise to a mutant subpopulation that will grow at a faster rate than the rest of the population.

A similar argument shows that if the mutant cohort's J, again computed where r is the growth rate of the nonmutant population, is less than one, then the subpopulation of descendants of the cohort must ultimately grow at a smaller rate than the rest of the population. Thus, we may interpret the value of J, computed for a mutant life history, 4, where the value of r equals the growth rate of the rest of the population, as a relative measure of the ability of the mutant to spread in the population, with the value J = 1 being the neutral point.

If the mutants' J, relative to the rest of the population, is greater than one, the mutant will spread until its life history characterizes the entire population, and then the value of r must be revised upward to reflect the growth rate associated with the mutants' life history. Now any further mutants must have a J greater than one, when computed with the new r, if they are to spread in the population. In this bootstrap fashion, each new mutant can be evaluated in comparison to the rest of the current population. The criterion for spread and ultimate fixation of the mutant is that its J be greater than one when the value of ir used in computing J is the growth rate associated with the life history of the rest of the current population; and upon fixation of each mutant, the value of ir is revised upward to reflect the growth rate of the type which most recently achieved fixation.

The process halts when no further mutants are available which can confer a value of J greater than one, under the current ir. In this situation, the J associated with the current life history is simultaneously maximum and equal to one. Thus the condition of maximal J equal to one describes an evolutionarily stable state, for then no mutant among those possible can invade the current population.

This evolutionary analogue may be mimicked in an iterative approach to the problem, in the following manner. Start with any life history in the possible set, and compute its associated growth rate, from equation (1). Using this growth rate for the r in equation (3), find the life history, f*, which maximizes J. Compute r*, the growth rate associated with 4*, and use this value for r, and again find the life history which maximizes J, etc. The sequence of values for r* will increase monotonically at each iteration, converging to rmax, while the values for the current maximum of J will decline monotonically at each iteration, converging to




the value one. At convergence, the current life history, f*, is the optimal life history. A formal proof is presented in the appendix.

Properties of the Optimal Net Maternity Function at Each Age

We may ask, not only about the newly born, but about persons of any chosen age, what is the present value of their future offspring; and if present value is calculated at the rate determined as before, the question has the definite meaning-To what extent will persons of this age, on the average, contribute to the ancestry of future generations? (Fisher 1930, p. 27).

The integral to be maximized in equation (3) is taken over the entire life of the cohort. What must be the situation at some particular age, x? Think of the net maternity function +(Z) as a description indicating the position of the cohort along some axis at time x. The optimal life history problem, then, is one of finding the optimal time trajectory for this position.

Since newborn invariably have zero fecundity, the initial position of the trajectory is necessarily 4(O) = 0. Similarly, if co represents the end of the cohort's lifetime, the survivorship f, is zero, so the final position in the trajectory is necessarily 0(,0, = 0. Finally, since the present cannot affect the past, we are assured that the position of the cohort at time x cannot influence the position at a prior time, though it may have an influence on the future position.

The integral of equation (3) may be rewritten as a sum of two integrals, one ending at time z, and the other beginning at time z:

fJe ex'(x)dx = efZ e P(x)dx + Le e-9 (xfdx. (4)

Now, consider a cohort which has followed the optimal trajectory up to time z. Then at time z it will be described by 0*), and the component of J corresponding to the first integral on the right-hand side of (4) will already have been fixed (and maximized). There remains the portion of the trajectory from time z forward. Therefore, in order to maximize J, the optimal trajectory at time z must be such as to maximize the second integral on the right-hand side of equation (4), subject to the constraint that this portion of the trajectory start with 0*W.

In other words, the optimal life history has the property than for any age z, the tail sum

K(Z) = Le- (Zxdx (5)

is maximized, where the admissable set of life history options that may be chosen among in the maximization comprises all those trajectories which pass through

Rewrite (5), multiplying and dividing the right-hand side by t(z)/ez:

K(z) = U(Z), et?(x)m(x)dx) e (Z

-~z __ 3(Z) (6) e P




The expression in parentheses above has the same form as Fisher's reproductive value v(z), but we note the peculiarity that the exponent ri is decoupled from the life history in this formulation by denoting this quantity v(z).

Now, (z), the survivorship to age z is determined solely by events prior to time z; and the exponent r is a fixed parameter; and z, of course, is not affected by choices of 4. Therefore, for purposes of maximizing K(z) with respect to 4, we may treat the expression tI(z)/erz as fixed, so the maximization of K(z) is achieved by maximizing v(z).

Maximizing Reproductive Value at each Age

Consider the situation where the parameter r in the expression for K(,) is in fact rmax, and where the life history up to age x is the optimal life history, k0pt. Obviously, the optimal life history past age x will then be the life history, meeting the constraint that +(Z) be equal to kopt(x), which maximizes K(X,). Under these circumstances, the value for v(,) which maximizes K(,), is v(x), which is the reproductive value at age x in the optimal life history.

Our fundamental result which I will call the life history optimality principle, may be stated as follows. In the optimal life history, the reproductive value at any age x is the maximum of the expression

epx rX v (X, i? e e-PY(y)dy , e(x) x

where the exponent r is fixed at the r associated with the optimal life history, and where the admissable set of life histories from among which the optimum is chosen is restricted to those where the product of the survivorship and fecundity at age x is the same as that product at age x in the optimal life history.

There are many ways in which attempts at application of this principle can go awry. Most of them have already been tried. The principle is equivalent to saying that the optimal life history "maximizes reproductive value at all ages" only in the particular and restricted sense developed above.

The optimal life history maximizes a particular expression, given as V(x) above, for any age, subject to two important constraints. One constraint is on the value of a fixed parameter in this expression, and the second is on the range of choices for possible life history segments that may be substituted into the expression during the course of the search for the maximum. Notwithstanding these constraints, the maximum value of the expression so obtained is identical to the reproductive value at that age in the optimal life history. If the constraints are ignored during the search itself, the results are unpredictable.

Sequential Optimization

The trajectory explored in maximizing K(x) involves only segments of the life history from age x till the death of the cohort. This suggests an advantage to searching out optimal short segments of the life history, starting from the oldest segments and working backward toward the youngest. The optimal segment, say




from y to z, will then be part of the solution for the segment from x to z, where y is greater than x.

Recall, however, that the life history segments that are searched in maximizing K(x) are constrained to pass through the position *4,. If this constraint only affects possible values of f(x) no difficulty arises, for this is but one point along a function which is being integrated over a finite interval, and so we could identify the optimal segment even if we did not know in advance the value of 0*W.

If, on the other hand, the constraint restricts possible values at times later than x, then the optimization is contingent on the value for 4*), and if this value is not known, much of the computational advantage of the backwards sequential optimization strategy vanishes.

What does it mean for the value of +(x) not to restrict possible values at times later than x? Only that the mortality rates and fecundity rates, at a given age, do not depend on prior states (or at least that in the range of values that is of reasonable interest, the dependency has negligible influence). When this is a tenable assumption, much simplicity is gained by a backward sequential search, by short segments. If the assumption is not tenable, that strategy is not valid unless it is conditioned on final verification regarding the earliest value within each segment.

There is a requirement in any case for final verification regarding the exponent r. Once an iterative scheme (such as that outlined above in Optimal Net Maternity Function for a Cohort) has been implemented, with the sequential maximization of K(x) embedded in the routine for maximizing J, the final iteration, where the value of r in equation (6) is r*, will indeed locate an optimal net maternity function, and the maximal K(x) so computed at every age will be the reproductive value in the optimal life history.

Properties of the Optimal Vital Rates at each Age It is probably not without significance in this connexion that the death rate in Man takes a course generally inverse to the curve of reproductive value (Fisher 1930, p. 29).

The problem of finding a function, such as 4, that will maximize an integral involving that function, such as J from equation (3), is the standard problem in the calculus of variations. The form of solution obtained in the calculus of variations is a set of necessary conditions that must be satisfied by the function at any instant. In our life history problem, these conditions would apply to the mortality rates and fecundity rates at any particular age.

Let us pose the maximization of J in the language of optimal control theory, a very general version of the calculus of variations. Readers unfamiliar with this body of mathematics may be discouraged to discover that the terminology is not entirely consistent across the several applied disciplines which employ it. Some small comfort may be taken in the observation that, regardless of nomenclature, these are at root just elaborate ways of simultaneously setting derivatives equal to zero. Here we will adopt the terminology used by Clark (1976) whose book attracted some audience among ecologists. This terminology diverges somewhat




from that employed by Leon (1976) whose paper is the sole instance to date of application of optimal control theory to the life history problem. In any case, Leon's formulation of the problem. differs from that employed here in that he describes the life history attributes in a very general state space, and he defines the controls as the set of energy investments in each attribute. By contrast I shall confine my state description to the survivorship and a single measure of physiological state of a cohort, and I shall define fecundity as the control. This formulation bypasses the question of cryptic intermediary variables, such as energy, and it makes explicit the trade-off between reproduction and survival in relation to reproductive value which is the focus of our interest. The appearance of the term "reproductive value" in the heading of section (B) of Leon's paper (p. 327) is almost certainly a typographic error, for that section nowhere mentions reproductive value, dealing instead with "reproductive effort," which Leon defines as the fraction of energy investment in life history attributes classified as pertaining to reproduction.

To begin, designate m( is) as the control, the value of which we seek to optimize. The state of the cohort at any time will be defined in terms of two state variables, t(X), the survivorship schedule, and -y(), some measure of physiological condition of individuals at age x.

The two state equations which follow from this formulation are

dx

=- e rf(ifly) (7) and

dy'r =g(m, Y) (8)

where the mortality rate at age x and the rate of change of condition at age x are both explicit functions, f and g respectively, of the current fecundity and the current physiological condition.

The Hamiltonian of this system is

e`#X( x) (m(/),- LL(e,)CXl(U)) + X2(XA) (9)

where Xl(x) and X2(x) are adjoint variables. From the maximum principle, the optimal life table will be one which maximizes

the value YC(x) for every age x. Thus, this formulation makes explicit the trade-offs between reproduction, survival, and condition at any age, for the Hamiltonian is a weighted sum of three terms respectively representing those processes.

At a particular age, the values of r, x, and f(,i. are given, either as parameters or as functions of the past. The trade-offs are among m(x) and Wx), and between both of these and the term dyldx, which is the rate of change of physiological condition at that age, given the present condition and the choice of m(x). If size, for example, were an adequate representation of condition, then dyldx would be the physical




growth rate at age x. From equation (9) we see that the trade-off between fecundity and mortality is governed by the expression ePX1(x), in that the sum of ux and m(x) weighted by this expression is to be maximized. Similarly, this sum plus the rate of change of condition times X2(xZerxff(x) is to be maximized, so X2(x) governs the trade-off between the change in condition (i.e., the growth analogue) and the weighted sum of fecundity and mortality.

The two adjoint variables are defined by the adjoint equations

dX,(x) = Xl(x)(x)(- e- Xm( ) (10) dx

and

_ _ _ __ _ _ x dy~x) \ dX2(X) =______f(Z) Il(Z) - X2(x) ( 1)

dx d0Y(x) OY(1) dx

Without knowing something more about the dependency of the rate of change of condition on reproduction and condition (i.e., about the function g in eq. [8]), we cannot proceed further with analysis of equation (11), though some explicit models involving growth rate would certainly be worth pursuing in this regard.

Equation (10), on the other hand, is an adequately determined differential equation. Note that the derivative of reproductive value with respect to age is

dv (x) - (r + /.(X))V(x) - m(x)* (12) dx

Accordingly, it can immediately be verified by substitution that

Xl(x) = V(-rx (13)

is a solution to equation (10). Whereupon we see that the expression for the trade-off between fecundity and mortality is v'(x). Then, in the optimal life history, where ?r is the r associated with that life history, the trade-off is v(x), the reproductive value, regardless of the involvement of growth or condition.

In economic terms, reproductive value is the current shadow price of an individual in the cohort. Substituting equation (13) back into the adjoint equation, and rearranging to obtain an expression for d(f(x i(x))!dx, the rate of "depreciation" of the value embodied in the cohort at a given time in the life history which maximizes J, yields the equation:

@(fwvwZ) = (t(X'(X))? - ((X^m(X). (14) dx

The rate of depreciation then for the entire cohort equals ̂ r times the current value, in units of vi, minus the current rate of births to the cohort. For the case where ̂r = rmax the value of a newborn is one, so births would already be scaled in the units of value. Then, adding t(X~m(x) to both sides of equation (14), we find that the instantaneous rate of change of total value embodied in the subpopulation com- prised of the cohort plus its offspring at that moment, is depreciation plus births,




which is equal to r times the total value of that subpopulation. Adding over all the cohorts represented in an arbitrary population, this of course recapitulates Fisher's result that the summed reproductive value of a population invariably grows exactly at the per unit rate r regardless of age structure.

In conclusion, in the optimal life history the reproductive value at age x deter- mines the trade-off between fecundity and mortality. In this trade-off the dif- ference between the fecundity and the product of mortality and reproductive value is maximized.

For this reason, the partial derivative of the fecundity rate with respect to the achieved mortality at any age in the optimal life history is equal to the reproductive value at that age (Goodman 1979), notwithstanding possible trade-offs involving physiological condition which go beyond age class boundaries.

OPTIMAL NOTATION

In a discrete time representation of the life history, choices must be made as to when, with respect to the time-age interval, the censusing of reproduction and survival is to take place. One internally consistent convention is as follows:

qx, the mortality in age class x, is one minus the fraction of individuals, censused as alive and aged x - 1 to x at time t that are censused as alive at time t + 1;

mx, the fecundity of age class x, is the number of offspring to age class x, censused as alive at time t, per individual censused as alive and aged x - 1 to x at time t - 1.

Accordingly, the smallest age subscript is one. That is, the youngest censused age class, one, refers to individuals from the time of their first census (when they appeared as the numerator of an mX) until the time of their second census, one time-age interval later, when those alive are censused as entering age class 2.

Then x-1

e(X) - H (1 - qj) (15) 3=1

and

{1= 1 (16)

Under this convention, we can verify, either through matrix manipulations or by long hand algebra, that the discrete time equivalent of Lotka's equation is the characteristic equation

co

1 = i e-`fxmx (17) x=1

where t is the oldest age achieved.




Notice in particular that if we choose to subscript the youngest age class as zero rather than one, the values in the exponent will then be shifted by r times one so that a summation of the form of (17) but going from zero to co would be incorrect. Instead, with this alternative age convention we would need to write

1 = A, e-'`(x+l)xmx. (18) x=O

The incorrect version, starting the summation from zero but without the added one in the exponent, occurs frequently in the ecological literature. Probably the convention of equation (17) is more congenial than (18), since it looks more like the continuous time version and in any case most Fortran compilers will not accept zero subscripts.

Having chosen to stay with the convention that yields equation (17), we will discover, to our chagrin, that things are now slightly amiss in the formula for reproductive value. Reproductive value is normalized to express the equivalent value of one individual in age class x in units of individuals of the youngest age class. Thus, in the continuous time version we had v(0) = 1, as we must. In the discrete time version, when age class one is the youngest age class, in order to arrive at the equivalent v1 = 1 reproductive value must be defined as

r(x-1) Co

Vx = C L' e-Xxmx. (19)

Almost everyone in the ecological literature omits the minus one in the first exponent and some, through sufficient oversight, have even contrived to derive such a formula, but then the reproductive value so computed is off by a factor of e r at every age.

There is no combination of definitions that will get around the underlying difficulty, which is that of a time lag between the censusing of parents and of offspring, so regardless of convention either reproductive value or the characteristic equation will require a discrepancy of one power of er relative to the continuous time formula.

Trying to define the reproductive value in units of newborn before their first census only defers the problem by creating a time discrepancy between the notation for reproductive value and the notation for fecundity. The consequence is that reproductive value no longer expresses exactly the trade-off between mortality and reproduction, but instead the trade-off is reproductive value times a factor of er. As it is, in the discrete time version the trade-off between mortality and reproduction in age class x is given by the reproductive value in age class x + 1, since mortality in age class x, under this formulation does not affect the survivorship component of the net maternity function until age class x + 1 (Goodman 1979). Since the sum of mx and (1 - qx) vx+l is equal to vx, it still is true that in the discrete time model the optimal life history is the history which maximizes reproductive value at every age, subject to the same constraints as apply to the continuous time model.




REPRODUCTIVE VALUE AND THE EQUILIBRIUM SEX RATIO

If we consider the aggregate of an entire generation of such offspring it is clear that the total reproductive value of the males in this group is exactly equal to the total value of all the females, because each sex must supply half the ancestry of all future generations of the species (Fisher 1930, p. 142).

Male Reproductive Value

Fisher's original formulation for reproductive value, like all the classical for- mulations in demography, refers solely to the female segment of the population. That is, the mortality rate schedule is that of females, the fecundity schedule is in units of female births per female, and the reproductive value is normalized to one female newborn-equivalent at age zero. Our problem now is to arrive at a similar age-specific measure for males where the male reproductive value is also normalized relative to a newborn female, thus permitting comparisons between the sexes.

It is a simple matter to tally up the male's mortality rates separately. Let us designate the male mortality and survivorship schedules as /4x,) and <'), respectively. The male fecundity schedule is not so obvious, in that the directly observable male reproductive activity is fertilizations not births. We define s(x,) to be the age specific schedule of relative fertilization rates achieved by males; that is, s(x,) is the fertilization rate achieved by a male aged x relative to the rate at some reference age. Our problem then centers on the means of converting this observable measure to m',), the effective male fecundity expressed in units of female newborn per male aged x.

Control of the Primary Sex Ratio

The male fecundity schedule, m',), must be some multiple of the relative fertilization schedule:

m (x) as (x). (20)

The value that ae must have is discovered by investigating the mean per capita birth rate in the population. The mean per capita birth rate of females per male in the population is

b J c'x~m'x~dx

- of J c'x~s(xsdx (21)

a e )'Xt(xs (X)dx

f e -(x)dx




where c',) is the stable age distribution of males, calculated in the usual way from the population growth rate and the survivorship schedule of males.

Since my definitions have not altered the fundamental quantities involved in the female schedule, the per capita population growth rate, r, which of course is the same for males and females is still given by Lotka's equation, equation (1), with +(Z) = f(X) m(x).

The mean per capita birth rate of females per female in the population is as usual,

b = f c(x)m(x)dx

1 d (22) Ie - xtP(x~dx 0

Consider time t when the total number of female births in the population is B(t). If the population has been in stable age distribution, B(t) will have been growing exponentially at rate r, so the total number of females in the population must be

N(t) = fB(t-X)e(x)dx

= B(t) e-rxt(x)dx. (23)

Similarly, the total number of males in the population, at the same time, is

N'(t) = B (t) f0e -rXtdX

- B't) fIe Ift(x)dx (24)

where Bit) is the total number of male births in the population, and p is the ratio of females to males at birth.

Thus the ratio of females to males of all ages in the population at stable age distribution is

N(t) j e-"XtP(xdx (25) N'(t)

f~~e -rx~fdx

Now the mean per capita birth rate of females per male must be equal to the mean per capita birth rate of females per female times the ratio of females to males in the total population, since these rates refer to the same set of births and differ merely by normalizing by the number of males or the number of females in the population. Therefore,

Not)

= _______P . (26)

fe x-rxtfidx




Equating this with the b' of equation (21),

a = ------- P (27) f -rxj)(Xd e W-txs W dx

and accordingly, the male fecundity in terms of measurable quantities is

m(x) I Ps(x) (28)

J'e -f(y)s (,)dy

Now the male reproductive value schedule may be written as

v lx) = (29)

f(X) j e- 'Yt(s (Y)dy

For age zero, which is where the choice of sex ratio is being exercised, the male reproductive value is therefore,

V(0) = p. (30)

This remarkably simple result shows immediately that the worth of a newborn male, relative to that of a newborn female (recall that v(0) = 1) is equal to the ratio of females to males at birth. If the ratio of male to female newborns is greater than one, males are of greater value than females, so the optimal p would decline; whereas if the ratio is less than one males are of less value, so the optimalp would increase. The equilibrium state clearly would be p = 1, as is well known (e.g., Charnov 1975).

In a life history where the sex control is exercised at birth, the equilibrium situation may be readily characterized by solving for the conditions under which the reproductive value at birth is the same for both sexes. This result is generaliz- able in at least two useful ways. One is to note that this can immediately accom- modate unequal "cost" of the two sexes: Simply express the cost of producing a male in terms of the trade-off resulting for m(x), bL(x) and U4r; and the principle of equalizing reproductive value of the two sexes remains operational. The second extension is to complex life histories encompassing the option of sex change which, as has been shown by Leigh et al. (1976), gives rise to formally similar optimization problems.

Control of Sex Changes

Consider, for example, a life history where the sex ratio at birth is fixed at the value p, and where a fraction h of the females become males at age f8. The definition of male reproductive value in the preceding section indicates that, at equilibrium, the fraction switching will be such as to equalize the reproductive values of males and females at the age where the sex change occurs.




Development of a usable expression for h, on the basis of this formal principle, proceeds as follows. As before, t,(,) will designate the mortality rate of an individual while it is female, and p4 will designate the mortality while it is male. The definition of survivorships requires some care in specifying whether we are interested in the fraction of a cohort of a given sex surviving to a particular age (and as which sex), or whether we are interested in the schedule appropriate to computing the age distribution of individuals of a given sex in the population, regardless of how they came to be of that sex. I will call this latter quantity the population survrivorship.

The quantities t) and ,(') will be defined simply in terms of the integrals, .) = e-f (y)dY and & `) =ef'(u)dY as before, but now they correspond to the actual cohort and population survivorships only up to age /3. Beyond that age, the survivorships are more complicated expressions.

Thus, the population survivorship of females is defined as the probability of appearance of a female aged x, at time t + x, per female birth at time t. The females in a cohort all follow the same survivorship schedule, t(), up to age /3, at which time a fraction h are removed from the female population, owing to the sex switch. The females that remain continue to follow the original mortality pattern for females, so per female aged /3 a fraction ((,) /!(O) will survive to age x, but at age ,8 only (1 - h) -C(f will have survived from birth. Thus, the female population survivorship is

e(X) X </3 (31)

(1-h) f(x) x ? A. (32)

The male population survivorship is defined as the probability of appearance of a male aged x, at time t + x, per male birth at time t, where males, in this model, can appear either because of survival of a primary male, or because of survival of an individual, born female, that subsequently changed sex at age ,/. The males in a cohort all follow the same survivorship schedule, . throughout their lifetimes. However, at time t + /3, when the cohort is aged /3, the cohort will be joined by secondary males, of chronological age /3, appearing at the rate ht(,) per female birth at time t, and this is equal to the rate h(f,)p per male birth at time t. From age ,/ on, the secondary males follow the male mortality pattern, so per male appearing because of sex change of a female at age /3, a fraction ,('x) will survive to age x. Thus, the male population survivorship is

tPx x < 8 (33)

a + h ___ _(X

(34) + P/I Km ) X 3.

e(3)




The population growth rate can be computed from female fecundity and survivorship

1 = J'el- 'i(()mx^,)dx + (1 - h) Je - 1r(x)rm (dx. (35)

Now the derivation simply repeats the strategy employed in the example of the prior section. The mean per capita birth rate of female offspring per male in the total population is

= oi( J' e ['Z (,X)s (,Xdx + (i1 + P m$7)?) J'e-' &,l)S (x)dx] (6

f e t[) ' (X) dx + (1 + Phl?)) f e [(x.dX

The mean per capita birth rate of female offspring per female in the population is

b= 1 (37)

t,, dx + (1 - ph) J'Cr(xtdX

The ratio of females to males in the total population is

N f J e-x'Zx(l.dx + (I - h) e-C (dxdx (38) )32

eI-xtI, dx ? (I ? ph{/) -I1ex[I.dxl

T e pt (X)dx + + p I e (x)dx

Thus b' can also be computed as

b- b N N' P (39)

JWe -bit ..,x+ + ph f(3'' J e - r'Xte Xadx f ,,e)dx ? ? v I exe~

Equating the two expressions for b', and solving for ar

P (40)

f e-xe X)s(x)dx + (I + P )L1e `Xetsxdx

which means that an expression for in (,.) can be written in terms of s(,.). Finally, enough of the component quantities have been derived to write out

formulas for the respective reproductive values of the two sexes at age f8:

V )e m m,,) dx (41)




pe J e t (XS (x)dx V(s) (42)

<13) [fe <X)S(x)dx + +-rx (x)dxl

It is readily seen that the male reproductive value at age /8 decreases as h, the fraction of females aged /8 that change into males, increases. Therefore the situation where V'(,/) = V(,/) is a stable equilibrium. Writing out this equality, and solving for h, gives

h fe 1()m ( _~X ( )foerOe X) s(X)dx

TX e t*m (x dx -XPfe (Xx)S (X) dx

This is not an esthetically compact expression, but it does get the job done, giving a value for the fraction of females that change into males at age /3, at equilibrium, in terms of strictly observable quantities. It should now be apparent that this same general strategy may be applied to any complex life history involving change from one sex to the other or back at any age or ages. The basic principle is that for any change the equilibrium occurs when the reproductive values of the two sexes are equal at the age of that change.

CONCLUDING DISCUSSION

The schedule of reproductive value is an eigenfunction which provides a means of resealing population in units such that each unit contributes equally to population growth. Thus it is that if population size is expressed in terms of the aggregate reproductive value of its members, this measure will grow exactly at the per unit rate r, regardless of age distribution. By the same token, this resealing effects a means of comparison between individuals of different ages, indicating how much the death of an individual at one age will reduce the asymptotic growth rate, relative to the death of an individual of some other age. Since a birth is just an individual of age zero, we may use reproductive value to compare the significance of reproduction versus mortality at any age. Finally, an extension of the reproductive value formula to include the reproductive value of males provides a means for comparing the significance of an individual of one sex versus the other, most notably at ages where a complex life history might allow the option of changing from one to the other.

The optimal life history maximizes a time-discounted sum of births over the life of a cohort. The discount rate is the maximum achievable asymptotic population growth rate, which is treated as a fixed parameter and not as a function of the life history, in the search for the life history which maximizes the discounted sum. In the usual situation where this maximal growth rate is not known in advance, the problem is approached by finding a family of maxima as a function of the discount rate, and then identifying the maximum which equals one as the maximum asso-




ciated with the maximal growth rate. The life history which confers this particular maximum, which may be located iteratively, is the optimum.

At any age, the optimal life history must maximize the discounted sum of births over the remainder of life of the cohort. This remainder has a form identical to the expression for reproductive value, but again the maximization treats the discount rate as a fixed parameter, so the remainder coincides exactly with the reproductive value at that age only when it has been maximized for the discount rate corresponding to the maximal asymptotic population growth rate. Thus in the optimal life history, the reproductive value at every age represents the solution to a constrained maximization problem. This paper has shown that the problem may be examined variously as a maximization over segments of the life history, as a maximization over segments taken in reverse sequence, as a decomposition into instantaneous contributions at a given age, or as a maximization over alternative states (such as sexes) available to the individual at that age.

An optimal life table must in some sense be optimal at every age. This optimality is expressed in a constrained maximization giving the reproductive value at every age. This principle remains true regardless of such complications as trade- offs operating across age classes. These complications have important conse- quences as regards restrictions on the life history options that may be searched through during the course on the maximization, but they do not affect the nature of the maximum itself, which still is reproductive value.

The recent literature arguing that optimal life histories do not "maximize reproductive value" has misconstrued the principle through failure to com- prehend the rules of the search.

SUMMARY

Reproductive value is the central construct in life history optimization. In the optimal life history, the reproductive value at every age represents the solution to a constrained maximization problem. This holds true regardless of trade-offs across age classes. The recent literature arguing to the contrary has misunderstood the principle and has neglected the constraints in the maximization.

In reviewing the matter, this paper considers maximization over segments of the life history, maximization over segments taken in reverse sequence, maximization of terms of a decomposition into instantaneous contributions at a given age, and maximization of reproductive value over alternative states such as the sexes. To clarify the relation between the maximization procedure and the constraints, a simple iterative approach to achieving maximization and simultaneously satisfying the constraints is described. To arrive at the instantaneous decomposition, it is shown that reproductive value arises as an adjoint variable in an optimal control formulation, where effects across age classes are explicitly incorporated in a variable representing physiological state. In considering alternative states, this paper defines a male reproductive value and concludes that at equilibrium sex composition, the reproductive values of each sex will be equal at any age where a sex change is possible. Coincidentally, some common errors in the discrete time formulation of reproductive value and Lotka's equation are corrected.




ACKNOWLEDGMENTS

The author is indebted to W. M. Schaffer for helpful comments on an earlier draft. This research was supported under NSF Grant DEB-78-22785 and NOAA Contract 80-ABC-00207.

APPENDIX ITERATIVE SOLUTION FOR rmax

Let us subscript net maternity functions according to iteration number. Thus do designates the net maternity function on the jth iteration. An unsubscripted net maternity function refers to any net maternity function in the possible set.

Convergence of r

We obtain qj?1 from

Le-iJxqj+l(x)dx = max {f e ̀J7(x)dxj (Al)

where rj was obtained from

I e-xjxqj(x)dx = 1. (A2) 0

From (Al)

e e- ' jX~ j+,(x)dx e e- Jxqj(x)dx . (A3)

From (A2) 0x e -ajJ j+#~)dx ?_-I

We obtain rj+1 from

e-'lj+l~kj+(x)dx = 1. (A5)

From (A4)

I>e-l~jxqj+(x)dx J e 'j+i?~ j+1(x)dx. (A6)

Since

dii; ` fe -a k(X)dX f= x- e k()dX dr e 0 x.

< 0 (A7)

it follows from (A6) that

rj+l ri (A8)

proving that the sequence of values for r is nondecreasing.

Case 1. If an equality is obtained in (A3), then (A4) will also yield an equality as will (A6) and (A8). From equality in (A8) we may substitute rj+1 for rj in (Al); then in the case of equality, (Al) simultaneously maximizes J and yields a value of one, so this case corresponds to rj+1 = rj = rmax-

Case 2. If an inequality is obtained in (A3), then the inequality will be strict in (A4), (A6), and (A8); so in this case, from the strict inequality in (A8), rj < rmax.




All the net maternity functions from which values of r are computed are members of the possible set, so no value of r in the sequence can exceed rmax. Since rj,1 cannot exceed rmax, and since rj,1 is greater than rj if rj is less than rmax, the sequence of r's converges absolutely to rmax-

Convergence of J

We obtain ij?2 from

I e - 'j+ lxcj+?2 (x) dx = max | fe -'j+?1X(c )dx (A9)

From (A7) and (A8)

J e -I O +j+2(X)dX 10 e -i iv d2~?j+2(x)dx.

From (Al)

Ie-' ijxj+l(x)dx 10 e-' jxj+2(x)dx. (A1)

From (A10) and (All)

Ie-rjqj+x w (x)dx . L

erIX4)?2(X)dX. (A12)

Following the same logic as cases 1 and 2, above, the inequality will be strict, except when rj =rmaxandthenf 0 e- 'ji+,(x)dx = 1. Therefore, the sequence of values forJ converges absolutely on the value one.

LITERATURE CITED

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Optimal Life Histories, Optimal Notation, and the Value of Reproductive Value

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