The Co-evolution of Intergenerational Transfers and Longevity: An Optimal Life History Approach

Abstract

How would resources be allocated among fertility, survival, and growth in an optimal life history? The budget constraint assumed by past treatments limits the energy used by each individual at each instant to what it produces at that instant. We consider under what conditions energy transfers from adults, which relax the rigid constraint by permitting energetic dependency and faster growth for the offspring, would be advantageous. In a sense, such transfers permit borrowing and lending across the life history. Higher survival and greater efficiency in energy production at older ages than younger both favor the evolution of transfers. We show that if such transfers are advantageous, then increased survival up to the age of making the transfers must co-evolve with the transfers themselves.

1 Introduction

A growing literature seeks the optimal solution to the “general life history problem”, how to allocate resources among fertility, mortality and growth from birth to death. Most optimal life history studies of which we are aware assume that the individual can use only the energy that it produces (forages) in each period, and the life history is optimized subject to this strict budget constraint (Cichon [1997], Cichon and Kozlowski [2000], Vaupel et al [2004], Abrams and Ludwig [1995], Taylor et al. [1974], Goodman [1982], Schaffer [1983], Stearns [1992], Clark and Mangel [2000]; the important exceptions are Kaplan and Robson [2002] and Robson and Kaplan [2003]). But what if individuals were permitted to borrow and lend over their life cycles? Markets for loans do not, of course, exist in nature, but intergenerational transfers from adults to juveniles are common and serve a similar function. Transfers permit a stage of nutritional/energetic dependence early in life with rapid growth and development, followed by a corresponding adult stage of “repayment” in which transfers are made to the young. The strict period-by-period energy constraint is then replaced by a looser version which, in a steady state, requires that the survival-weighted and discounted sum of transfers received minus transfers made over the life cycle must be zero, similar to a life cycle budget constraint with borrowing and lending at an interest rate equal to the population growth rate.³

Many species, including all mammals, most birds, many insects, and some fish and reptiles, make various forms of intergenerational transfers (see Clutton-Brock [1991]). The duration and magnitude of such transfers are extraordinary in the case of human beings and some dolphins and whales, and the longevity of these species (see Carey and Gruenfelder [1997]) motivates our exploration of the possible correlation between intergenerational transfers and the optimal life history strategies. We will consider how the life history changes shape when intergenerational transfers are permitted and confer a selective advantage. Lee (2003) took the existence of transfers as given, and did not consider physiological tradeoffs. In this paper we will examine the conditions under which transfer behavior (parental care) evolves, and consider how mortality co-evolves, when tradeoffs are explicitly modeled through the energy budget constraint.

The analysis we present is formally applicable to cooperative breeders, that is, groups of individuals in which some members across the age spectrum potentially provide food and care to young that are not necessarily their own offspring.⁴ In such cases, we can imagine a lineage carrying a mutation forming a stable population in aggregate, and living in small cooperatively breeding groups. Each group can be viewed as a microcosm of the lineage, with random departures from its stable age distribution. To obtain analytic results, we need the stable age distribution to write the balancing constraint on transfers.⁵ Within each group, all members share the same genotype which might include a gene promoting longevity, transfer behavior, or punishment of freeriding, for example. Transfers take place within these groups. Humans are cooperative breeders, and it has been argued that their longevity, particularly in postreproductive years, is related to their transfer behavior.⁶ There is also evidence (Brown [1987]) that cooperatively breeding bird species live longer than others.

The evolution of altruistic behavior raises difficult questions addressed by a large literature. We acknowledge these difficulties, but here we simply assume that some genotype can solve these difficulties and support transfer behavior. Although humans and other species we have in mind do not reproduce clonally, we believe that our analysis captures the central forces at play.

We begin by considering what life history for a lineage-founding individual would produce the greatest number of living descendants at a specified future date, optimizing subject to the usual budget constraint that does not allow transfers (section 2). We show that the appropriate measure of fitness to be maximized for this individual is the Malthusian parameter. This sets the stage for considering the conditions under which intergenerational transfers would be selected. We investigate when such transfers increase fitness (section 3), and if they do, how low mortality coevolves with them (section 4). The last two sections contain extensions and conclusions.

2 A Model of Optimal Life History

We first consider the case in which transfers are not an evolutionary option. The analysis could be carried out for a life history of potentially unlimited length, but we will instead consider the more realistic case of an individual who is not fertile past age y.⁷ To avoid the complications of mating and sexual reproduction, we will consider a population of females reproducing asexually. To unify the terminology and notation, we call the age interval [a, a+1) age a+1, and assume that all decisions affecting age a+1 are made at time a. The probability that a person survives from a to age just below a + 1 is denoted p_a+1. Fertility at age (a + 1) takes place just before a + 1, conditional on survival, and is denoted m_a+1.

At age a, a typical individual expects to have energy or resources which, following Abram and Ludwig (1995), Cichon (1997) and Vaupel et al. (2004), she allocates to fertility (m_a), maintenance (p_a) and growth (z_a). We can think of growth as an increase in body size, but we could also think of it as other kinds of physical investment such as development of the brain, as in Kaplan and Robson (2002) and Robson and Kaplan (2003). Because the individual can potentially reproduce in all periods before y, there is a tradeoff between energies devoted to reproduction, growth and maintenance: Having more children early in life comes at the expense of her growth and survival probability, which in turn affects her later fertility.

2.1 The Maximization Problem

The disposable resource or energy of an individual aged a depends on her body size, denoted w_a. Specifically, her age-a budget (energy) constraint is written as

$$b_a{p}_a+c_a{m}_a+d_a{z}_a\le {\zeta }_a{w}_a,\hspace{0.17em}\forall a$$ (1)

where b_a, c_a, d_a are constant coefficients, which express the rate at which energy can be used to achieve various levels of survival, fertility or growth. ζ_a is a production coefficient linking body-size with the net production, or acquisition through foraging, of disposable energy.⁸ It is easy to see that one of the four coefficients (b_a, c_a, d_a, ζ_a) in (1) is redundant, and so at each age a, we normalize ζ_a to be 1. This simplifies the expressions in what follows, but note that whenever we need to combine units of energy from different ages, we will have to convert the units appropriately. When this happens we will alert the reader.

The body size of an individual grows according to the following rule: w_a+1 ≡ w_a + z_a. The initial body size w₁ is itself an important intergenerational transfer from the mother. In our analysis, w₁ is given, while the adult size is part of the optimization problem through allocation of energy to growth. Thus the ratio of birth size to adult size is endogenous in our analysis. Given our linear homogeneous budget constraint, scale is irrelevant, so only this ratio matters. In this paper our emphasis is on intergenerational transfers occurring after birth.⁹

We expect that natural selection will maximize reproductive fitness, measured as the representation of an individual’s genes at some future date τ. Since we are assuming clonal reproduction, this is equivalent to maximizing the number of living descendants at some date τ, which may be far beyond the individual’s finite lifespan.¹⁰ Consider an individual age a at time t. Let V_a,t(.) be her contribution to the number of descendants at time τ. Here t will measure the remaining length of time until τ, when fitness is assessed, so for individuals closer to τ, t will be smaller. Bellman’s (1957) principle of optimality can be used to maximize the expected number of future descendants at τ. According to this principle, energy is allocated at age a and period t so as to maximize the contribution to fitness assessed at τ, assuming that the energy in all future ages and periods is also allocated optimally.

2.2 The Solution

For any a ∈ {1, 2,…,y}, let the age-a strategy be θ_a ≡ (p_a, m_a, z_a) and its feasible set be Ω(w_a). For any t, the Bellman equations can be written as follows, for which the interpretation is given in Appendix A.¹¹

$$\begin{array}{l}{V}_{1,t}(w_1)=\underset{{\theta }_1\in \mathrm{\Omega }(w_1)}{max}[p_1{m}_1{V}_{1,t-1}(w_1)+p_1{V}_{2,t-1}(w_1+z_1)]\\ ⋮\\ V_{y-1,\hspace{0.17em}t}(w_{y-1})=\underset{{\theta }_{y-1}\in \mathrm{\Omega }(w_{y-1})}{max}[p_{y-1}{m}_{y-1}{V}_{1,t-1}(w_1)+p_{y-1}{V}_{y,t-1}(w_{y-1}+z_{y-1})]\\ V_{y,t}(w_y)=\underset{{\theta }_y\in \mathrm{\Omega }(w_y)}{max}[p_y{m}_y{V}_{1,t-1}(w_1)].\end{array}$$ (2)

We denote the optimum in (2) by θ_a* = (p_a*, m_a*, z_a*). Now, we try to write (2) in terms of V_1,t for different t’s. Let φ_a ≡ (p₁*… p_a*m_a*) be the net maternity function. Starting from the age-y equation, lagging each equation by one period, substituting it into the equation one line above, and iterating the process, we obtain

$$V_{1,t}={\phi }_1{V}_{1,t-1}+{\phi }_2{V}_{1,t-2}+\dots +{\phi }_y{V}_{1,t-y}.$$ (3)

Manipulating (3) can give us a steady-state optimal solution for θ_a as well as the corresponding V_a(w_a). Instead of proceeding in this direction, we shall conform with the literature and apply the results derived by McNamara (1991).

Writing V_a,t as V_t(a), McNamara (p.235) transformed the problem of life history in (2) into the following recursive process:

$$V_{t+1}(a)=\underset{{\theta }_a}{max}{T}_{{\theta }_a}{V}_t(a),$$

where the operator T connects the state space across different ages. In this age-specific stochastic process, the transformation operator constitutes a Leslie matrix. Then the well-known Perron-Frobenius theorem can be applied to show the existence of a steady state. In that steady state, the value function V_t(a) of all ages grows at a constant rate, which is the dominant eigen value of the Leslie matrix. In particular, we can write V_1,t as V_1,t = A(λ*)^t ∀t. From now on, we shall normalize A to be 1 to simplify the notation.¹²

Let θ ≡ (θ₁, …, θ_y). We rewrite the maximization problem in (2) as

$$V_{1,t}=\underset{\theta }{max}\left[p_1{m}_1{V}_{1,t-1}+p_1{p}_2{m}_2{V}_{1,t-2}+\dots +p_1{p}_2\dots p_y{m}_y{V}_{1,t-y}\right].$$ (4)

Let l_a* = p₁* … p_a* be the probability of survival from birth to age a. In view of the definition of φ_a, we can rewrite (4) as

$$1=\sum _{a=1}^y{l}_a^{*}{m}_a^{*}{({\lambda }^{*})}^{-a},$$ (5)

This is the Euler-Lotka equation. Thus, we have

Proposition 1

The solution to the value function in (2) has the form V_1,t = (λ*)^t, where λ* is the Malthusian parameter solved from (5).

In the analysis above, we derive what a selfish agent that maximizes its own clonal replication would do, and show that the objective to be optimized turns out to be the Malthusian parameter. Explicit characterization of the generic optimization problem in (2) helps us specify and analyze the intergenerational transfers. The setting in (2) also provides us with a decision framework for the comparative static analysis of the next section.

2.3 A Corner Solution Pattern

In reality, many species first grow and then become fertile once they have reached their adult size. They cease growth or grow very slowly once they start bearing offspring. Our main interest is in species that make intergenerational transfers, such as mammals or birds, and these exhibit determinate growth of the sort described. The following proposition, proved in Appendix B, shows that the determinate growth pattern emerges in our model:¹³

Proposition 2

Values of m_a and z_a cannot both be interior solutions at the same time.

Given the linear energy constraint in (1), proposition 2 is intuitive and also consistent with that found in Taylor et al. (1974), and Vaupel et al. (2004) and is convenient for our later analysis. Biological interepretations and reasons behind the choice of determinate and indeterminate growth can be found in Heino and Kaitala [1999]). Simple differentiation of (2) tells us that any growth in size at age a has the benefit of increasing the number of future offspring at various ages by a constant factor: p_a+1/c_a+1 at age a + 1, p_a+1p_a+2/c_a+2 at age a + 2, …. Furthermore, in a steady state the value of a new-born at time t is proportional to λ^t. Thus the steady-state tradeoff between increasing size and bearing offspring is a constant, which depends on the parametric value of c_a’s and d_a’s. Therefore, a corner solution of either m_a or z_a must arise.

Substituting (1) into (2), one sees that the objective function at any age is a concave function of p_a. Thus, by restricting the range of the parametric values of (b_a, c_a, d_a), an interior solution of p_a can often be obtained (see also footnote 7). In what follows, we shall concentrate on analyzing the case in which an organism first grows for r periods, and then stops growing and reproduces. In our notation, the organism would have m_a = 0 in the first a ≥ r periods, and would have z_a = 0 when a ≥ r + 1.¹⁴

3 Optimal Life History and Transfers

In some species, parents invest in their offspring after birth by making transfers of food, guarding against predators, warming or ventilating them, and so on. We shall focus on the most prevalent form of transfers by mammals, when adult individuals aged j ≥ r + 1 transfer something to offspring aged i ≤ r. We ask when such a transfer would raise the intrinsic growth rate λ, and therefore be selected.

Recall that equation (1) was normalized at each age a by dividing through by ζ_a Because of this, when we consider transfers between ages i and j, we must use a conversion factor η_ij (= ζ_i/ζ_j). Let the transfer given by an individual at age-j be T_j and the amount received by an age-i be R_i. The demography imposes a feasibility condition on these transfers in a steady state:

$${\eta }_{ij}{\lambda }^{j-i}g(R_i)=p_i\dots p_{j-1}{T}_j.$$ (6)

where g(.) characterizes the technology for receiving transfers and converting them into the equivalent of energy directly produced by the age-i child recipient,¹⁵ with g(0) = 0, and g′(.) > 0. If there is no cost of converting the transferred energy, then g(R) simply equals R. We introduce g(.) to reflect the likelihood that the transfer process becomes less efficient at very high rates. This nonlinearity makes it possible to consider interior optimal transfers.

Because fertility is zero in the first r periods of life, from (1) we have z_s = (w_s−b_sp_s)/d_s, s ≤ r, s ≠ i. For the age-i, z_i = (w_i+R_i−b_ip_i)/d_i because the age-i agent receives transfer R_i. And because there is no body growth in periods s ≥ r+1, we know from (1) that m_s = (w_s−b_sp_s)/c_s ∀s ≥ r+1, s ≠ j. For the age-j, m_j = (w_j − T_j − b_jp_j)/c_j, because the age-j agent gives a transfer T_j. Finally, since body size does not grow after age r + 1, we have w_s = w_r+1 ∀s ≥ r + 1.

With this background information and assuming steady state, equation (4) can be rewritten as

$${\lambda }^t=\underset{{\theta }_s}{max}\hspace{0.17em}{p}_1\dots p_r[p_{r+1}{m}_{r+1}{\lambda }^{t-r-1}+\dots +p_{r+1}\dots p_y{m}_y{\lambda }^{t-y}],$$ (7)

where all p_s and m_s are evaluated at their optimal values. Note that the value of R_i is implicit in this equation, and is here taken as given and fixed. Note also that the form in which (7) is written assumes that r is given, whereas it is in fact endogenous, and varies as p_s, m_s and z_s vary. For small variations in the neighborhood of the optimum, however, r will not change. To see this, imagine that we carried out many otpimizations of the form of (7), sequentially taking r equal to a for every possible discrete age group a. If we now choose the value of r associated with the greatest maximum value of λ, that will be the optimal r which occurs in (7). Note that this value of r is a function of the level of transfers R_i. However, due to the discreteness of the age groups, for small variations of R_i the value of the optimal r will not change. For this reason, we can differentiate (7) with respect to transfers R_i to determine the effect on λ of a marginal increase in transfers.

Starting from a scenario with no transfers (R_i = 0), we shall evaluate how the steady state selection criterion λ will be affected by the introduction of a marginal transfer. The case of optimal transfers will be discussed briefly later. We now differentiate (4), and use the steady state condition V_1,t = λ^t to obtain¹⁶

$$\begin{array}{l}-\left[\frac{(r+1)p_{r+1}{m}_{r+1}}{{\lambda }^{r+2}}+\dots +\frac{y{p}_{r+1}\dots p_y{m}_y}{{\lambda }^{y+1}}-\frac{p_{r+1}\dots p_j{\eta }_{ij}g(R_i)(j-i)}{c_j{p}_i\dots p_{j-1}{\lambda }^{i+1}}\right]{\lambda }^{y}d\lambda \\ +\left\{K_i\left[\frac{p_{r+1}{\lambda }^{y-r-1}}{c_{r+1}}+\dots +\frac{p_{r+1}\dots p_y{\lambda }^0}{c_y}\right]-p_{r+1}\dots p_j\frac{{\lambda }^{y-j}}{c_j}{G}_{ij}\right\}d{R}_i=0\end{array}$$ (8)

where

$$G_{ij}\equiv \frac{d{T}_j}{d{R}_i}\equiv \frac{{\eta }_{ij}{\lambda }^{j-i}{g}_i^{\prime }(R_i)}{p_i\dots p_{j-1}}$$

is the conversion factor between the transfer T_j and the effective value of the transfer received, and

$$K_i\equiv \frac{1}{d_i}\left(1+\frac{1}{d_{i+1}}\right)\dots \left(1+\frac{1}{d_r}\right),$$

which is the compound factor of accumulating size from age i to maturity (the end of age r). That is, an increment to growth at age i will result in larger body size and increased foraging productivity at age i + 1, which in turn raise body size at age i + 2, and so on. In (8) the coefficient of dλ is negative by the stability condition of λ. In the case when R_i = 0, (8) only needs some minor revision: G_ij has to be evaluated at R_i = 0 and g(R_i) = 0.

Substituting the formula for G_ij into (8), we have

Proposition 3

The sign of dλ/dR_i, which is the selection impact of a marginal transfer from age-j to age-i, is identical to the sign of

$$B_{ij}\equiv K_i\left[\frac{p_{r+1}{\lambda }^{y-r-1}}{c_{r+1}}+\dots +\frac{p_{r+1}\dots p_y{\lambda }^0}{c_y}\right]-\frac{(p_{r+1}\dots p_j){\eta }_{ij}{\lambda }^{y-i}{g}^{\prime }(R_i)}{c_j(p_i\dots p_{j-1})}.$$ (9)

On the right hand side of (9), the first term (K_i[.]) is the lifetime expected sum of fertility increase, from age r + 1 to age y, due to the increased body size. Transfers to young individuals lead to larger adult body sizes, which in turn generate more energy for growth and other purposes. The envelope theorem tells us that the net marginal benefit of a change in R_i can be evaluated by the net increase in reproduction. Because m_s is weighted by λ^s, we obtain the first term as shown. The second term captures the lost fertility at age j due to the out-transfer. Term B_ij must be positive for the transfer to be selected.

Examination reveals that B_ij > 0 is more likely to be met under the following conditions: 1) When there are more age-j adults relative to age-i offspring to share the costs of the transfer (either larger p_i … p_j−1, or lower fertility, or both). Higher background mortality (larger coefficients b_a) would work against the evolution of transfers. It also follows from (9) that transfers are less likely to evolve in the context of rapid population growth (larger λ), for example for an opportunistic species or under favorable climatic conditions, and more likely to evolve when carrying capacity is saturated. 2) When the adults are relatively more efficient than the child at generating energy per unit body size (smaller η_ij ≡ ζ_i/ζ_j). Perhaps this is more likely for carnivores than herbivores, since catching prey requires more skill, speed, strength, and weaponry (teeth, claws). 3) When there is a lower cost to augmenting body size between age i and age j (smaller d_s, s = i, …, r in K_i), which makes the investment from adults more rewarding. 4) When survival from age r + 1 to j is high, so that low adult mortality is a predisposing factor for the evolution of transfers, at least up to the age of transferring.¹⁷

Transfers might be concentrated on younger offspring or older offspring. In general, the compounded returns to early transfers (larger K_i for smaller i) favor transfers to the young, and the more so when the young convert energy more efficiently into body size (smaller d_a for small a’s). Transfers to older juveniles might still evolve, if their survival were sufficiently high. Transfers to infants will be more likely if infants are relatively helpless, and unable to forage effectively (low ζ_i and hence low η_ij). This would be more likely true for carnivores but less so for herbivores. If older juvenile productivity relative to body weight increased, then transfers to that age would be less likely to evolve. Finally, a context of more rapid population growth favors transfers to older juveniles (as revealed by the λ^y−i term).

4 Coevolution of Transfers and Longevity

Natural selection should move the life history toward the optimal θ_s ≡ (p_s, m_s, z_s) to maximize the intrinsic growth rate λ. We now switch back to treating transfers, R_i, as given, and consider how the optimal levels of p_k depend on the level of transfers, for variations that are small enough such that the optimal age of sexual maturity r, corresponding to the given level of transfers R_i, does not change. We will develop one result (Proposition 4) that holds in the neighborhood of the optimal level of transfers, R_i*, and another result (Proposition 5) that holds when transfers are below this optimal level.

>From the corner-solution pattern presented in section 2.3, it follows that the problem of finding the optimal life history reduces to searching for the optimal p_a’s that maximize λ in equation (7). Given that the transfer in question is from age j to age i, it is natural to consider separately the first order conditions for p_k when k ≤ r and when k ≥ r + 1. We shall discuss these cases separately below.

The immature age range corresponds to k ≤ r. Differentiating the right hand side of (7) with respect to p_k and using the envelope theorem, we see that its first order condition is proportional to the following expression:

$$\begin{array}{l}{\mathrm{\Delta }}_{p_k}\equiv [p_{r+1}{m}_{r+1}{\lambda }^{y-r}+\dots +p_{r+1}\dots p_y{m}_y\lambda ]\\ -p_k{b}_k{K}_k\left[\frac{p_{r+1}{\lambda }^{y-r}}{c_{r+1}}+\dots \frac{p_{r+1}\dots p_y\lambda }{c_y}\right]\\ +\frac{(p_{r+1}\dots p_j){\lambda }^{y-i+1}{\eta }_{ij}g(R_i)}{c_j(p_i\dots p_{j-1})}\cdot I(k)=0,\hspace{0.17em}k\le \hspace{0.17em}r\end{array}$$ (10)

where I(k) = 1 if r ≥ k ≥ i, and I(k) = 0 otherwise. The term associated with I(k) is from the differentiation of (6) (dT_j/dR_i), which is nonzero only if k is in the range between i and j. The K_k factor enters (10) because it is the relevant compound growth factor up to age-k.

The sexually mature range corresponds to k ≥ r + 1. Differentiating (7) and using the envelope theorem yields the following first order condition:

$$\begin{array}{l}{\mathrm{\Delta }}_{p_k}\equiv (m_k{\lambda }^{y-k+1}+p_{k+1}{m}_{k+1}{\lambda }^{y-k}+\cdots p_{k+1}\cdots p_y{m}_y\lambda )\\ -\frac{p_k{b}_k{\lambda }^{y-k+1}}{c_k}+\frac{(p_k\cdots p_j){\lambda }^{y-i+1}{\eta }_{ij}g(R_i)}{c_j{p}_k(p_i\cdots p_{j-1})}\cdot I(k)=0,\hspace{0.17em}k\ge r+1\end{array}$$ (11)

where I(k) = 1 only if r + 1 ≤ k ≤ j − 1, and I(k) = 0 otherwise. In what follows, we shall ask the following comparative static question: how does the optimal p_k change when R_i increases toward its optimum R_i*?

We note that in the ongoing process of evolution, the cumulation of marginal changes should eventually maximize a species’ fitness and hence exhaust the selection advantage of increasing transfers by choosing the optimal R_i* such that

$$d\lambda /d{R}_i=0.$$ (12)

Suppose the optimal transfer from age j to age i, denoted R_i*, has been attained. Totally differentiating (10) we have

$$\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial R_i}d{R}_i+\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial \lambda }d\lambda +\frac{\partial {\mathrm{\Delta }}_{p_k}}{{\partial p}_k}{dp}_k=0.$$ (13)

If we evaluate the derivative around the optimal R_i*, we know from (12) that dλ = 0. The coefficient of dp_k is negative by the second order condition. Thus, we know that as R_i increases towards the optimum R_i*, whether p_k moves in the same or opposite direction hinges on the sign of ∂Δ_pk/∂R_i.

Partially differentiating (10) with respect to R_i, we get

$$\begin{array}{l}\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial R_i}=K_i\left[\frac{p_{r+1}{\lambda }^{y-r}}{c_{r+1}}+\cdots +\frac{p_{r+1}\cdots p_y\lambda }{c_y}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\le r\\ -\frac{(p_{r+1}\cdots p_j){\lambda }^{y-i+1}{\eta }_{ij}g\prime (R_i)}{c_j(p_i\cdots p_j)}+\frac{(p_{r+1}\cdots p_j){\lambda }^{y-i+1}{\eta }_{ij}g\prime (R_i)}{c_j(p_i\cdots p_j)}\cdot I(k).\end{array}$$

For r ≥ k ≥ i, I(k) = 1, the last two terms of the above expression cancel, and hence the we see that ∂Δ_pk/∂R_i is indeed positive. This means that if R_i increases towards its optimum R_i*, then p_k is also increasing for any r ≥ k ≥ i.

Following similar steps (see Appendix C) we see from (11) that for k ≥ r + 1, p_k moves in the same direction as R_i around the optimum R_i* if and only if ∂Δ_pk/∂R_i is positive for r + 1 ≤ k ≤ j − 1. Summarizing the above discussion, we have

Proposition 4

Consider a transfer R_i from age j to age i. As R_i increases towards the optimum within the neighborhood of the optimal R_i* that maximizes the fitness index, survival from age i to age j must increase.

What about the evolution of survival before age i? We summarize the result in the following proposition, and the proof is given in Appendix D.

Proposition 5

If an increased transfer from age j to age i improves fitness, the survival probability up to age i must also increase.¹⁸

Why is it that Proposition 4 applies only to survival improvements between the age of receiving and the age of giving the transfer, while in Proposition 5 survival also improves at ages from birth to i? Improved survival from age i to j always imparts an efficiency gain when transfers are increased, so it is selected. Increased survival from birth to age i does nothing to conserve the investment in transfers, and in this sense does not impart any efficiency gains when transfers increase. However, it raises the number of births surviving to age i, and thus raises λ, other things equal. In the neighborhood of the optimal transfer, the effect of this increase in survivors to age i is exactly offset by a reduction in transfers per offspring age i, so λ is unaffected, and the survival improvement to i is not selected.

Once an adult is both past the age of providing transfers and no longer fertile, her continuing survival makes no contribution, positive or negative, to reproductive fitness. Mathematically, with respect to a transfer from age j to age i, we can say nothing about the comparative statics with respect to p_s for s ≥ j.

5 Extension and Discussion

5.1 General Optimal Transfers

So far we have discussed the impact of a transfer from one age j to another age i, but of course transfers may be provided by adults of various ages, and received by children of various ages. The feasibility constraint in (6) need not hold for each (i, j) pair, but rather resources must be balanced over the lifecycle. Specifically, let g(R_i) indicate the energy cost of all transfers received by an individual at age i. The life cycle feasibility constraint is:

$$\sum _i{\zeta }_i{g}_i(R_i){\lambda }^{-i}=\sum _j{\zeta }_j{p}_i\cdots p_{j-1}{T}_j{\lambda }^{-j},$$

similar to that in Lee (2003).

In general it will be optimal for adults of many ages to make transfers, and then the marginal benefit of transferring from each age must be equalized. Likewise, the marginal benefit at each age of receiving must be equalized. As long as we have interior solutions, we should have a system of equations to solve for such optimal transfers. Details will not be provided here, but one should note that the co-evolution result of transfers and longevity we derived in the previous section would not be affected by such complications.

5.2 Selection and Population Density

For a given set of the parameters b_a, c_a, d_a and ζ_a, for all a, there will be some optimal growth rate associated with the optimal life history, and only by chance will it be zero. If the growth rate is positive, then nothing in our model prevents population density from increasing without limit. It is beyond the scope of this paper to consider the dynamic trajectory as density changes. However, we will sketch the way density could be introduced into the model and provided that density is at an unchanging equilibrium level, no change in the analysis would be required.

Density is measured as the total body mass of the population per environmental resource. The main effect of greater density would be to make foraging more difficult and thereby to reduce the energy yield for a given body weight, that is to reduce the coefficients ζ_a. The conversion of energy into body weight, fertility, or survival as expressed by the other parameters would not be affected to a first approximation, although a more elaborate analysis might permit density to affect mortality (through contagion) and fertility (through limited breeding sites) directly, in addition to the indirect effect through energy production that is now included. So long as the relationship is monotonic, its precise functional form need not concern us. We can simply multiply ζ_a at every age by some factor that makes the corresponding optimal growth rate λ* equal unity.

When transfer behavior evolves, the founder of the mutant line actually experiences a lower NRR than otherwise, because she foregoes some adult fertility and survival in order to divert energy to caring for her existing offspring, although she received no such care in her youth. In this way she gets the lineage started, and subsequent members of the lineage realize a higher NRR as a result of her initial sacrifice and the improved life history it made possible. Of course, the inclusive reproductive fitness of the first individual is also raised thereby, even though her own NRR is reduced.

5.3 The Linear Technology

We assumed that the budget constraint is linear homogeneous, reflecting a linear tradeoff technology. We could, instead, give equation (1) a nonlinear form, for example by allowing the cost of fertility at age a to be a nonlinear function c_a(.) of m_a. In this case, second order derivatives would appear in the comparative statics formulas. It would be possible to derive comparative static results if we are willing to assume the sign of these second derivatives,

6 Conclusion

The optimal life history approach seems well suited for exploring the positive selection of life history characteristics. In this paper, we are able to connect formally the optimization problem for an individual life history and the aggregate criterion of the growth rate. This enables us to carry out a comparative static analysis of the effects of these parameters, in contrast to the previous literature which has explored optimal life histories through numerical solutions.

Previous applications of the optimal life history approach have assumed that the individual’s energy budget must balance at every age. Intergenerational transfers replace this instantaneous budget constraint with one that holds over the life cycle. Here we model intergenerational transfers, and ask under what conditions they would improve reproductive fitness and be selected. With transfers, a period of juvenile dependency with more rapid growth and development can be funded by contributions from adults. We consider what features of an initial life history without transfers would make it more likely that intergenerational transfers would confer a selective advantage. Factors favoring the selection of transfers include the ratio of adults to juveniles, greater efficiency of the old relative to the young in producing energy per unit of body size, and the efficiency of juveniles in converting energy into body size. We also discussed the factors favoring transfers to younger versus older offspring, and favoring transfers from older versus younger adults. Because lower mortality favors the selection of transfers from old to young, and because lower mortality coevolves with increased transfers, we find the longevity and transfers should increase in a mutually reenforcing way, as argued in Carey and Judge (2001) and Lee (2003).

When transfer behavior evolves, the founder of the mutant line actually experiences a lower NRR than otherwise, because she foregoes some adult fertility and survival in order to divert energy to caring for her existing offspring, although she received no such care in her youth. In this way she gets the lineage started, and subsequent members of the lineage realize a higher NRR as a result of her initial sacrifice and the improved life history it made possible. Of course, the inclusive reproductive fitness of the first individual is also raised thereby, even though her own NRR is reduced.

A central finding is that on the one hand, lower mortality makes the evolution of transfers more likely, and on the other hand, if increased transfers do evolve then longevity should coevolve. With transfers from adults to juveniles, costly resources are diverted from immediate reproduction to care for existing offspring, and concurrent life history investments in reducing mortality serve to protect these investments.

Mathematical Appendix

Part A

The interpretation of equation (2) is as follows: p₁m₁ in the first term on the right hand side of (2) characterizes the event that an age-1 individual survives (with probability p₁) and bears m₁ offspring. Since each of these offspring is valued V_1,t−1 in period t − 1 (because the offspring is one period closer to τ), V_1,t−1 should be multiplied by p₁m₁ to obtain the expected value. The V_2,t−1 in the second term of (2) is the value function of this individual at age-2. With probability p₁ the individual will survive to face this state, and so V_2,t−1 should be multiplied by probability p₁. The age-2 body size should be w₂ = w₁ + z₁ instead of w₁. The interpretations of other expressions are similar, so we move forward to the last equation. For an individual aged y in period t, p_y and m_y are chosen to maximize the expected value of the last birth. At age y, there is no gain from further growth. This generates the age-y expected value p_ym_yV_1,t−1. Since y is the last fertile age by assumption, there is no second term for the last equation.

Part B

Let g(α) ≡ max_x f (x, α). The envelope theorem (see Simon [1976]) says that when x has an interior solution, dg/dα = ∂g/∂α around the neighborhood of the maximum, for the indirect effect through x is absorbed by the first order condition of x. Applying the envelope theorem to (2), we see that

$$V_{a,t}^{\prime }=p_a^{*}\{V_{1,t-1}/c_a+p_{a+1}^{*}[V_{1,t-2}/c_{a+1}+p_{a+2}^{*}(V_{1,t-3}/c_{a+2}+\cdots ]\}.$$

For the age-a problem, concerning the trade-off between m_a and z_a, we have the following first order condition (in terms of economics, MRS equals price ratio) for an interior solution:

$$\frac{p_a^{*}{V}_{1,t-1}}{p_a^{*}{p}_{a+1}^{*}\{V_{1,t-2}/c_{a+1}+p_{a+2}^{*}[V_{1,t-3}/c_{a+2}+p_{a+3}^{*}(V_{1,t-4}/c_{a+3}+\cdots )]\}}=\frac{c_a}{d_a}.$$

In the steady state, V_1,t is a constant of power λ for all t, as shown in the text, and so the above expression can be further rewritten as

$$\frac{p_a^{*}{\lambda }^{t-1}}{p_a^{*}{p}_{a+1}^{*}\{{\lambda }^{t-2}/c_{a+1}+p_{a+2}^{*}[{\lambda }^{t-3}/c_{a+2}+p_{a+3}^{*}({\lambda }^{t-4}/c_{a+3}+\cdots )]\}}=\frac{c_a}{d_a}.$$

Canceling p_a* in the numerator and the denominator of [A2], we see that both sides of [A2] are not dependent on any age-a choice variable. Thus, expression [A2] could hold only by accident in a steady state, which in turn implies that m_a and z_a cannot be interior solutions at the same time.

Part C

Partially differentiating Δ_pk in (11) with respect to R_i, we have

$$\begin{array}{l}\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial R_i}=K_i\left[\frac{{\lambda }^{y-k+1}}{c_k}+\frac{p_{k+1}{\lambda }^{y-k}}{c_{k+1}}+\cdots +\frac{p_{k+1}\cdots p_y\lambda }{c_y}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\ge \hspace{0.17em}r+1\\ +\frac{p_k\cdots p_j{\lambda }^{y-i+1}{\eta }_{ij}g\prime (R_i)}{c_j{p}_k(p_i\cdots p_{j-1})}\cdot I(k)-\frac{p_{k+1}\cdots p_j{\lambda }^{y-i+1}{\eta }_{ij}g\prime (R_i)}{c_j(p_i\cdots p_{j-1})}.\end{array}$$

For r+1 ≤ k ≤ j −1, I(k) = 1, the last two terms in the above expression cancel, and hence ∂Δ _pk/∂R_i s positive. As such, we know that p_k and R_i also move in the same direction for r + 1 ≤ k ≤ j − 1 as R_i moves toward the optimum, R_i*

Part D

For k ≤ i − 1, the first order condition for p_k is similar to that in (10), except that I(k) = 0 ∀k ≤ i −1:

$$\begin{array}{l}{\mathrm{\Delta }}_{p_k}\equiv [p_{r+1}{m}_{r+1}{\lambda }^{y-r}+\cdots +p_{r+1}\cdots p_y{m}_y\lambda ]\\ -p_k{b}_k{K}_k\left[\frac{p_{r+1}{\lambda }^{y-r}}{c_{r+1}}+\cdots \frac{p_{r+1}\cdots p_y\lambda }{c_y}\right],k\le i-1\end{array}$$

Partially differentiating the above expression with respect to R_i yields

$$\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial R_i}=\lambda B_{ij}$$

where B_ij is given in (9). We know that dλ/dR_i = 0, and hence B_ij = 0 around the optimum R_i*. Thus, for k ≤ i − 1, dp_k/dR_i is close to zero around the optimum transfer. But we are able to say more about the change of p_k in the process when R_i increases toward R_i*

Equation (13) says that for any dR_i, the sign of dp_k/dR_i is the same as that of [∂Δ_pk/∂R_i] + [∂Δ_pk/∂λ] · [dλ/dR_i]. Partially differentiating ∂Δ_pk with respect to λ, using (10) to simplify the result, and substituting in the dλ/dR_i formula, we have

$$\begin{array}{l}\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial R_i}+\frac{\partial {\mathrm{\Delta }}_{p_k}}{\partial \lambda }\cdot \frac{d\lambda }{d{R}_i}=\lambda B_{ij}E\\ +\lambda B_{ij}\left[1-\frac{\frac{p_{r+2}{m}_{r+2}}{{\lambda }^{r+2}}+\cdots +\frac{(y-r-1)p_{r+2}\cdots p_y{m}_y}{{\lambda }^y}-F}{\frac{(r+1)m_{r+1}}{{\lambda }^{r+1}}+\cdots +\frac{y{p}_{r+2}\cdots p_y{m}_y}{{\lambda }^y}-F}\right],\end{array}$$ (A3)

where

$$E\equiv p_k{b}_k{K}_k\left[\frac{p_{r+2}}{c_{r+2}{\lambda }^{r+2}}+\cdots \frac{p_{r+2}\cdots p_y}{c_y{\lambda }^y}\right]>0,$$

and

$$F\equiv \frac{p_{r+2}\cdots p_{j}g(R_i)(j-i)}{c_j{p}_i\cdots p_{j-1}{\lambda }^i}.$$

It is easy to see that terms in the square brackets of (A3) are positive. We showed in section 3 that whenever the increase of R_i improves fitness, B_ij must be positive. Thus, as R_i moves toward the optimum R_i* to improve fitness, B_ij > 0 must hold in the process, which means that dp_k/dR_i > 0 ∀k ≤ i − 1. Thus, we have proved proposition 5 in the text.