Age-specific optimal diets and optimal foraging tactics: A life-historic approach

THEORETICAL

POPULATION

BIOLOGY

36,

281-295 (1989)

Age-Specific Optimal Diets and Optimal Foraging Tactics: A Life- Historic Approach STEINAR ENCEN Department of Mathematics and Statistics, University of Trondheim, N-7055 Dragvoll, Nowa)

AND NILS CHR. STENSETH Division University

qf Oslo,

of Zoology, Department P. 0. Box 1050, Blindern,

qf Biology, N-0316

Oslo, Norway

Received September 30, 1988

By linking optimal foraging theory and optimal life history theory, we demonstrate that optimal diets, in general, may depend on the individual’s age even when everything else remains the same. Older individuals (i.e., individuals with lower reproductive values) are predicted to have diets composed of highly nutritious food types that are possibly dangerous to pursue. ‘i 1989 Academic Press, Inc.

INTRODUCTION

In optimal foraging models energy is often the currency to be optimized (see, e.g., Schoener 1971). This is a reasonable assumption since fitness often is found to be positively related to the rate of energy acquisition (e.g., Schoener 1971, Pyke 1984). For reviews and recent treatments of optimal foraging models, see Schoener (1971) Pyke et al. (1977), Pyke (1984) Stephens et al. (1986), Stephens and Krebs (1986), and Engen and Stenseth (1989). In optimal life history studies fitness is assumed optimized under the constraints provided by environmental conditions such as food availability, predator exposure, etc. The more food the forager obtains, the higher in general will survival and reproduction be. For reviews of optimal life history studies, see, e.g., Stearns (1976, 1977) and Charlesworth (1980). The obtained amount of energy and other nutrients available for increasing 281 cm40-5809/89 $3.00 CopyrIght ‘< 1989 by Academic Press. Inc All rights of reproductmn in any form reserved.

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fitness is in theory maximized by adopting the optimal foraging tactic in the particular environment. Hence, optimal foraging theory and life history theory are closely related. Nevertheless, the amount of food resources in the environment is only rarely incorporated explicitly into optimal life history models (but see, e.g., Stenseth et al. 1985). In this paper we attempt to tie optimal foraging theory more closely together with optimal life history theory. Similar attempts have been published by Mange1 and Clark (1986) and by McNamara and Houston (1986). By tying these two areas of ecological theory together, we are able to derive some new predictions in optimal foraging theory. In particular, we demonstrate that the optimal diet will, in general, depend on the age of the individual. By so doing, we extend optimal foraging theory beyond the point of studying only “typical” or average individuals in the population (compare Partridge and Green 1985). Our theory makes it possible to utilize individual features such as age and condition when predicting optimal diets of particular individuals. In life history studies the concept of reproductive value (Fisher 1930, Williams 1966a, b) is of central importance: Schaffer (1974) and Taylor et al. (1974) demonstrated independently that optimizing fitness is equivalent to optimizing the age-specific reproductive value simultaneously for all ages. This is a convenient result since it provides a key for deriving several specific predictions from optimal life history theory. (But see, e.g., Stenseth and Ugland (1985) and Engen and S&her (1988) for some qualifications of the general validity of this statement.) Two TIME SCALES It is convenient to distinguish between two time scales (see similar arguments presented by Stenseth and Maynard Smith (1984) and Stenseth (1985)): 1. Behavioural time scale applicable for analyzing behavioural optimization such as foraging. 2. Ecological (or life history) time scale applicable for analyzing optimal reproductive and survival strategies. The former is fast relative to the latter time scale. Commonly we may refer to minutes in the former and to years in the latter case. We return to this below. A biological feature may be (and commonly is) studied with reference to only one of these time scales. For instance, behavioural changes are often appropriately assumed to take place within an infinitesimally short time interval with respect to the ecological time scale. Certainly some simplifica-

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tion like this is necessary if any progress is to be made on the integration of foraging theory and life history theory. Such an assumption is in reality consistent with assumptions commonly made in behavioural ecology (for reviews, see, e.g., Krebs and Davies 1984): Behavioural ecologists generally study behavioural changes of an individual in a particular environment as if no dynamic changes occur in the surroundings of the considered individual.

OPTIMAL

FORAGING

Throughout this paper we adopt the framework and terminology of Engen and Stenseth (1984). See this paper for details. Here we only give a synoptic account of the theory. Let V denote the set of options with which the consumer may be faced. The elements of %‘, the options (c), are considered to be random experiments. Each c E +Z will require some time and give a certain contribution to the consumer’s diet with respect to both food types and nutritional components. Assume that the diet may be appropriately described by some vector of dimension n. The expected net contribution obtained from option c to the ith nutritional component is denoted G,(c), while T(c) is the expected time required to realize c. When the consumer is faced with certain options, it may always choose to do nothing and go on searching. This particular option, which is always available, is denoted cO. If “doing nothing” can be deciced instantaneously and without any cost of energy, T(c,) = G,(c,) = ... = G,(c,) = 0. (But see Hughes (1979) who assumed the existence of a “recognition time”.) When the consumer is searching for food, it has a loss cli per time unit of the ith nutritional component. If the expected length of the searching period is l/1, the expected loss of the ith nutritional component of CY,/A. The quantity 1, then becomes a measure of the food availability in the environment. One particular searching sequence ends (by definition) when the consumer is faced with one or more options; that is, the consumer is, by definition, said to stop searching when it is faced with a choice among the elements of some subset of 9, say s c %? (which, by definition, always includes co). Let Y denote the set of all such subsets SE Y that the consumer may encounter. We assume that the element SE Y which terminates the searching period is generated by some model. A feeding cycle is defined as one searching period plus the time required to realize the selected option (for a further discussion of the “feeding cycle” concept, see Engen and Stenseth (1984)). A tactic is a rule describing how to choose a CES when faced with s. This rule may in general be to choose

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the various elements of s with certain probabilities. The expected contribution to the ith nutritional component of the diet resulting from applying tactic r when faced with s is denoted Gi[r, s]; similarly T[z, s] denotes the expected time needed for acquiring the expected amount Gj. With this notation, the mean value of the ith nutritional component obtained per time unit in the long run may be written

- u,li + E(G,t-t,~1) Hi(z)=

l/A+E(T[r,s])

’

where the symbol E is the expectation with respect to the probability model generating the elements s E Y, a, is the loss rate while searching for food, and 1 is a measure of overall food abundance (see Engen and Stenseth (1984) for details). Engen and Stenseth (1984) found the tactic which maximizes one particular component, say H,(r) (which commonly is assumed to be the net amount of energy obtained per time; see, e.g., Schoener 1971): The optimal tactic is in that case simply to choose the c E s which gives the largest value of G,(c) -k, T,(c), where k, is the maximum obtainable value of H,(r). The theorem proven by Engen and Stenseth (1984) may also be used to maximize linear functions, like C ai. Hi(r), where the a;s are known constants. Using this, Engen and Stenseth (unpublished) showed that the set of obtainable diets (what they called the diet set), H(r) = (H,(t), H,(r), .... H,(r)), always is convex. Finally, they showed how a general fitness function W(H(r)) may be maximized. By so doing they found the particular tactic, r*, that maximizes the individual’s fitness. The corresponding optimal diet is H*. Below we present an approach for integrating optimal foraging theory and optimal life history theory, an integration which may be extended by using the above mentioned result. However, in this paper we do not specifically require this extension.

OPTIMAL

LIFE HISTORIES

This section provides a theoretical basis for the subsequent (and main) section, Its only purpose is to provide the framework to be used together with the results summarized in the previous section in order to derive the age-specific optimal foraging tactic of an individual. Consider an individual in a population with stable age distribution and a fixed specific growth rate, r. Let x denote the time (or age) of an individual so that x=0 represents the time of birth of an individual. The

AGE-SPECIFIC

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DIETS

time of death for such an individual is a stochastic variable X. Adopting standard statistical notation, we have F(x) = P(X< x) and l(x) = P(X> x) = 1 - F(x); hence, 1(x) is the survival schedule for the considered population. The probability density function of X is f(x) = dF(x)/dx, and the instantaneous death rate d(x) is defined by d(x) =f(x)/( 1 -F(x)) = -I’(x)/Z(x). (See for example Kalbfleisch and Prentice (1980) and Charlesworth (1980) for good technical treatments of this kind of model.) The fitness of this individual is measured by 30

l(x) m(x)ep’”

u() = s0

dx,

(2)

where m(x) is the rate of offspring production and r is the specific growth rate (Charlesworth 1980); in an evolutationarily stable population with stable age distribution, u. = 1 (e.g., Charlesworth 1980). On integrating by parts, or by a direct argument (Fisher 1930) Eq. (2) may be written as

v. = E

I

X

m(x)emrr

dx,

0

where the expectation, E, is taken with respect to the distribution of X. In optimal life history theory, the functions m(x) and I(x), or equivalently m(x) and d(x) (as defined above), are generally assumed to be molded by natural selection so as to maximize fitness u. (Charlesworth 1980, Maynard Smith 1982). The behaviour of an individual may affect its m(x) and d(x). Hence, an individual should behave so as to maximize u. (so that v,*, the maximimal vO, equals one). In particular, an individual should adopt a foraging tactic maximizing uo. An individual that is alive at an age t should therefore optimally adopt a foraging tactic which maximizes X

v,= E

which can, through integration

m(x) ercrm “‘dx

1X> t

1

by parts, be shown to be expressed as

e rr oc Z(x) m(x)eerx dx. s, O’=r(t)

(See Slagsvold et al. (1986) for an analogous argument.)

(5)

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Let u: denote the maximum

obtainable

value of v,. Then, it follows that

-$$m(r)-(d(r)+r)4:.

(6)

It further follows that a0 is maximized by maximizing m(t) - d(t). OF at each moment, t, in time. (This is equivalent to a result derived by, e.g., Crow and Kimura (1970, pp. 2&22). See also Stenseth and Ugland (1985).) For specified l,- and m,-schedules, a given r results. This r can be found by solving Eq. (2) or its derivatives, with u0 = 1. We can, however, for simplicity, assume a particular r-value together with a set of I,- and m,-schedules. These life-history functions (i.e., IXmX) must, however, be corrected by a multiplicative factor, say a, to ascertain that u0 = 1 and to make sure that Eq. (2) is satisfied. If the life-history functions corrected by the u-factor are used, we will in fact find the assumed r-value by solving Eq. (2). In the calculations reported below, we have used this easy way as we want to choose particular r-values. In order to tie together optimal foraging and life history theory, it now remains to be analyzed how the foraging tactic of an individual of age x affects m(x) and l(x). This we do in the next section.

AGE SPECIFIC FORAGING TACTICS: INTEGRATING FORAGING AND OPTIMAL LIFE HISTORY THEORY The General Model

We consider an individual of age x that adopts the foraging tactic ~~ to obtain a diet as described by the diet vector H(r,). We assume that the net rate of offspring production depends on the age of the reproducing individual as well as its diet. Hence, m =m(x, H(r,)). More specific assumptions will be made in the subsequent subsection. We now need to incorporate the effect of the foraging tactic on the death rate. It is reasonable to assume that the probability of death within a small time interval depends on the kind of activity performed by the animal in that time interval. In general, the probability of dying depends on the amount and kind of nutrients obtained by foraging as well as the age of the individual, In this paper we disregard such factors. To incorporate such factors will only complicate the analysis without contributing any new insight. Indeed, incorporating such factors may easily distract from the main point of the paper, which is that optimal foraging tactics (and diets) will depend on age even when no such additional factor is incorporated

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(see below). It should be realized, though, that such factors can fairly easily be introduced. To do this will, of course, be important when specific data sets are to be interpreted on the basis of the model. Let 6 be a time interval which on the behavioural time scale is large enough to include many feeding cycles (where one feeding cycle is, as described above, defined as one searching period plus the time required to realize the selected option). According to the assumptions made in the section “Two Time Scales,” 6 may be considered infinitesimally short on the ecological time scale; in practice it will, on this latter time scale, represent a point in time. Let v, denote the death rate at time t E (to, to + 6). Then, v, =

vO if searching at t v(c) if occupied with c at t.

(7)

The difference between vO and v(c) would then be the increased death rate due to chasing option c, i.e., a cost of choosing c. We assume that the survival-cost of choosing a particular option does not affect future reproduction and survival. For a given function vr, we have P(death in (to, t,+S)

1alive at to)= 1 -cI:~+“~~“‘.

(8)

Since 6 is small when measured on the ecological time scale, this probability is of order 6 and may be expressed as jz’” v, dt + o(6). We may now split 6 into feeding cycles, remembering that 6 is large when measured on the behavioural time scale and therefore contains, by assumption, a large number of feeding cycles (see Fig. 1). We have that P(death in (to, t,+S)

I alive at to)

$0 = E(cycle length) + 46)

= s vo/J.+ E(DCT>~1) ’ l/n + E( T[T, s]) + ‘(‘)’

(9)

where D(c) = v(c) . T(c) and C represents one feeding cycle (see Fig. 1). From Eq. (9) we see that a consumer adopting tactic T will in ecological time be exposed to the death rate

vo/Jb+ E(DCz, ~1) d(T)= 653.‘36’3-4

l/i + E(T[z, s])

(10)

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FIG. 1. The death rate of an individual depends on its behaviour. Here is depicted an intinitesimally small time interval as measured on the ecological time scale. As can be seen, eight full feeding cycles are shown. The depicted time sequence is a segment of the b-long time interval discussed in the main text. In the depicted time sequence the forager is exposed to three classes of options each with its own v-value. See text for further interpretation and discussion.

We further see that the expression for d(r) is mathematically equivalent to that of Hi(r) (see Eq. (l)), with Gi(c) replaced by D(c) = v(c) T(c) and the corresponding ai by - vO. Hence, d(r) may be considered just an additional component to the diet vector, H (unpublished results). The resulting vector with (n + 1) components will then characterize both the diet and the death rate resulting from adopting a particular foraging tactic. The expression to be maximized at age x,

(see Eq. (6)), is therefore just a function of the above generalized diet vector. The expression defined by Eq. (11) may therefore be maximized (with respect to z,). Since the age x is a parameter in this expression, it follows that the optimal foraging tactic, r,*, in general will change with age. The maximization procedure is in general mathematically rather complicated. The main points may, nevertheless, be illustrated by some simple examples. We present such simplified examples in the following main section. However, first we return to the two time scales discussed earlier in the paper. The Two Time Scales Revisited The quantities G and T refer to a feeding cycle (i.e., behavioural time). Similarly, ;1 refers to how frequently a choice situation (i.e., an s) occurs. In the demographic optimization carried out in this paper (by maximizing v,), these quantities must be transformed to the ecological (or life history) time scale. In practice, this means that G and T (referring to behavioural

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time) must be divided by a small quantity, say D ~ ‘, where CJis given by the ecological time unit divided by the behavioural time unit. Furthermore, J (referring to behavioural time) must be multiplied by the same quantity, (T-‘. Otherwise, the quantity given by Eq. (9) will be off by orders of magnitude. The quantities CI, refer to the behavioural time scale. These quantities need not be changed, since A already has been transferred to the behavioural time scale. This is so, because it is a,/J which enters the definition of H. The quantities v0 and v(c) refer to the ecological time scale. In the demographic optimization carried out in this paper, no changes of time units need therefore be done for these quantities.

RESULTS AND EXAMPLES

Some general results follow immediately from the model presented above: Suppose the fertility is zero for x > x0, where x0 is some age beyond which reproduction is physiologically impossible. Then v&= 0 and immediately before age x0 the forager should not at all take into account the risks associated with choosing option c (see Eq. (11)). Then only the effect of behaviour on the term m(x, H(T,)) is of importance. That is, the forager should concentrate only on obtaining the nutritionally “best” diet. As x decreases, v-z increases, and so does the term ~(z,~)v,* in Eq. (11). This means that a young forager should consider more carefully the risks associated with choosing various items: the nutritional value of the diet must then be weighted against how dangerous it is to obtain such a diet. Let H = (H, , H,, .... H,) be the diet vector. The fertility is, in its most general linear form, defined as (12)

where d(x) is a fecundity function being zero for young individuals, then increasing to a maximum and further decreasing towards zero at x,,. (Note that around the equilibrium H*, E q. (12) is, in fact, a Taylor approximation to the more general expression m = M(H(z,)). q(x).) The term in brackets, in Eq. (12), indicates that the fertility increases (for /Ii> 0) linearly with each of the nutritional components in the diet. The expression to be maximized is (omitting constant terms) given by (13)

290

ENGEN

Introducing

AND

STENSETH

b,,i = /Ii. cp(x)/o,*, we shall then maximize

1 b,,i. ffi(~)

- 4~)

= -(Cia,b,,i-v,)ll+E{C,GiCT,sl ~bx,i-DC~tsl} . l/A + E{ TCC sl >

(14)

Hence, the problem is reduced to a one-dimensional one. In order to simplify, we will in the following consider only one component in the diet vector; i.e., n = 1. As usual (e.g., Schoener 1971) we assume H, to be the net rate of energy acquisition. For our purpose, it is certainly sufficient to consider the case of n = 1 as an example. Many more data are furthermore needed before it makes sense to consider the more general case of n > 1. According to the result of Engen and Stenseth (1984) Eq. (14) (with II = 1) is maximized by choosing the option with the largest value of U(c,x)=b;G,(c)-v(c)

T(c)-k,T(c),

(15) with respect to r.K.

where k, is the maximum value of b,H,(z,)-d(z,) (Note that b, = b,,;.) In Fig. 2 we have depicted a situation with two options-one

being

g(c)= G(c)-b,.

D(c)

with

increasing

k x.0 FIG. 2. A forager is exposed to two food types. One no risk of pursuit (D = 0) whereas the other food type is (D>O). Assuming that q(x) = 1 for all values of x
kx of the food types is associated with associated with some risk of pursuit (where X, is the maximum age of age (x) as a result of v, decreasing 1) for further interpretation of this

AGE-SPECIFIC

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DIETS

A

1.5r

OPTIMAL s7 ‘(Cl

DIET : 1

333 333 333 12 2

#W3,~OJ

5.6 :(c,.c, ,c,l 55 =k,,c3,co) 5& =k, $2 ,c,l

: I

A I

0b

53s*=(C,.c01 ‘lCj,C0) 5, q(c, $0)

o

I

I

II

,

,

b2 , 1

I

2

3

4

5

6

Age,

: I I :

7

1

232323

,I

,

,I

II0

6

9

10

11 12

x

FIG. 3. Deriving optimal age specific foraging tactics. Three options, cl, cz, and c,, are available. Food fypes: G,(c,) = 0600003, G,(c,) =O.OOOOO45, and G,(c~) =O.O000055. T(c,) = 0.006601 for i = 1, 2, 3. The densities of the various S’S (see Table I) are 1, = 1, = I, = l,OOO,OOO and &=i,=&,=i,=300,000. a, = 1. Risks: v,=O.OOl, v(c,)=O.Ol, v(cJ= 1.0, and v(c3)= 1.5. Demography: r =0.4, &,= 1.0, and p, =0.29 (see below). The fecundity function, q(x), is defined as cp(x)=a, .e~(*-02)2!03, where a, = 1, a,=4, and a, = 3. The resulting fecundity function q(x) is depicted in A and the u,-function in B. The U-values (see Eq. (15)) for the various options and ages are depicted in C. The resulting optimal tactics when encountering the various s’s are depicted in D: the options that optimally should be chosen for the various s’s are given. Technical aspects of the numerical analysis: We have, in the recursive analysis, started at a maximum age for which o$ is approximately equal to zero. In all cases we have started at X, = 15 and then iterated down toward x=0. The iteration is, however, terminated when v, = 1 (see, e.g., Fisher 1930). This x’ is then used as the appropriate zero-point and all other parameters are transformed appropriately; that is, a2 is corrected so that a2 = a; - x’, where a; refers to the initial time axis, and pi =/I1e-” where /r; is the originally given /I,-value.

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ENGEN

AND TABLE

Definition

of the elements

s-element

STENSETH I in ,Y = (s,, s2, .. .. s,). Entering

c-elements

SI

co> (‘I

s2

co, (‘2

s3

co. (‘3

s‘l

co, c,,

c2

SS

(‘0. c,,

ci

S6

co, c2,

(‘3

s-i

(‘0, (‘I,

c2,

Note. The tactics mean that ~(3,) = c,,, various s’s are given by interpretation. See also

c3

(iI, i,, ... . i,), referred to in Fig. 3, j= 1, 2, ... . 7. The densities of the I, for s,. See text for definitions and Engen and Stenseth (1984).

dangerous to pursue, the other not being dangerous to pursue. Assuming that k, is kept constant experimentally, we see that the foraging animal should optimally choose to eat food items associated with low death rates when young, but the most nutritious items, possibly associated with higher death rates when old. This conclusion does of course assume a decreasing u,-function (Eq. (3)). It is, however, an empirical fact that u, is low at high ages for most, possibly all, organisms (e.g., Pianka 1981). In Fig. 3 we have depicted the results of a more detailed numerical analysis of our model (see also Table I). We consider three options, c,, c2, with increasing nutritional value (i.e., increasing G’s) but with c3. increasing death rates associated with the pursuit of these items (i.e., increasing v’s). As can be seen from the figure, the food types associated with high death rates enter the diet only at old age. The D-panel in the figure demonstrates indeed very clearly that the optimal foraging tactic, and consequently the optimal diet, is age specific.

DISCUSSION

Difference in diets may occur for a variety of reasons. Age and sex are often argued to be major causes of difference in feeding patterns (e.g., Werner and Gilliam 1984, Polis 1984, Partridge and Green 1985). Sex differences in diets are often assumed to reflect differences in parental or reproductive roles, whereas age differences in diets are often assumed to be a result of both dietary constraints and experience (e.g., Partridge and

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Green 1985). In this paper we have, however, explicitly demonstrated that age differences in diets, and implicitly sex differences in diets, easily may result even if the dietary constraints remain the same. In our case, the predicted dietary differences with age are a result of the fact that young and old individuals value the risky exposures associated with the various food items differently: If a young individual dies because it pursues an item associated with great dangers in obtaining it, but which is highly nutritious, this may have a large impact on that individual’s reproductive contribution to future generations. If, however, an older individual dies in a similar situation, this may have a much smaller impact on its reproductive contribution to future generations. To the best of our knowledge, this has earlier never been demonstrated on the basis of optimal foraging theory. The reason for this is, we believe, simple: until now, optimal foraging theory has only been concerned with an average-or “typical’‘-individual in a population. Furthermore, in most of the earlier theoretical discussions on optimal foraging, no distinction has been made between food item and option (Engen and Stenseth 1984). Distinguishing between these concepts, facilitates the formulation and analysis of much more complicated and realistic models (see Engen and Stenseth 1989). We have been unable to find good empirical examples interpretable within the theoretical framework presented in this paper, and demonstrating that the optimal diet will change as a result of aging even if everything else remains the same. There are, however, some examples indicating that our prediction saying that optimal diets are age specific in fact is valid (e.g., Clark and Gibbons 1969, Ballinger et al. 1977, Lituaitis et al. 1984, Stevens 1985). Unfortunately, none of these studies, nor any other we are aware of, presents the data allowing a detailed comparison between predictions derived from our model and observed patterns. Few, if any, of the earlier experimental studies on optimal foraging has been carried out with our, or a similar, prediction in mind. Lumping together animals of various ages in an optimal foraging experiment may, according to our predictions, result in higher variations in observed diets than would be the case if a group of equally aged animals were tested. Besides the predicted age-dependence of the optimal diets, this methodological cautionary note is a very important result of great relevance to experimental behavioural ecologists emerging from our study.

ACKNOWLEDGMENTS This

work

was supported

by a grant

from

(NAVF) as well as by our home institutions.

the Norwegian Discussions with

National Science Foundation participants in the Evolution-

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ary Ecology class at the University of Oslo were of great help. In particular we would like to thank Geir Sonerud for helping us search the literature for examples. Three anonymous referees provided very useful comments on an earlier version of this paper. Liss Fusdahl, Tove Noland, and Janne Juell Eide are all thanked for correcting an endless number of drafts to this paper: they made a mess into a tidy manuscript.

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CHARLESWORTH,B., 1980. “Evolution in Age-Structured Populations,” Cambridge Univ. Press, Cambridge. CLARK, D. B., AND GIBBONS, J. W. 1969. Dietary shift in the turtle Pseudemys scripta (Schoepff) from youth to maturity. Copeia 1969, 704706. CROW, J. F., AND KIMURA, M. 1970. “An Introduction to Population Genetics Theory,” Harper & Row, New York. ENGEN, S., AND STENSETH,N. C. 1984. A general version of optimal foraging theory: The effect of simultaneous encounters, Theor. Pop. Eiol. 26, 192-204. ENGEN, S., AND STENSETH,N. C. 1989. Optimal foraging theory: A set-theoretic approach (the Stene et al. book), in press. ENGEN, S., AND SBTHER, B.-E. 1988. Optimalization of constrained life histories: Some generalizations, J. Theor. Biol. 130, 229-237. FISHER,R. A., 1930. “The Genetical Theory of Natural Selection,” Clarendon Press, Oxford. HUGHES, R. N. 1979. Optimal diets under the energy maximization premise: The effects of recognition time and energy. Amer. Naf. 113, 209-222. KALBFLEISCH, J. Q., AND PRENTICE, R. L. 1980. “The Statistical Analysis of Failure Time Data,” Wiley, New York. KREBS, J. R., AND DAVIES, N. B., Eds. 1984, “Behavioural Ecology. An Evolutionary Approach,” Blackwell Scient. Pub]., Oxford. LITUAITIS, J. A., STEVENS,C. L., AND MANTZ, W. W. 1984. Age, sex and weight of bobcats in relation to winter diet, J. Wild/. Manage. 48, 632-635. MCNAMARA, J. H., AND HOUSTON, A. I. 1986. The common currency for behavioural decisions, Amer. Nat. 127, 358-378. MANGEL, M., AND CLARKE, C. W. 1986. Unified foraging theory, Ecology 67, 1127-1138. MAYNARD SMITH, J. 1982. “Evolution and the Theory of Games,” Cambridge Univ. Press, Cambridge. PARTRIDGE, L., AND GREEN, P. 1985. Intraspecific feeding specializations and population dynamics, in “Behavioural Ecology” (R. M. Sibly and R. H. Smith, Eds.), pp. 207-226. Blackwell Scientific, Oxford. PIANKA, E. R. 1981. “Evolutionary Ecology” (2nd ed.), Harper and Row, Publ., New York. POLIS, G. A. 1984. Age structure components of niche width and intraspecific resource partitioning: Can age groups function an ecological species. Amer. Nat. 123, 541-564. PYKE, G. H. 1984. Optimal foraging theory: A critical review, Amer. Rev. Ecol. Sysr. 15, 523-575. PYKE, G. H., PULLIAM, H. R., AND CHARNOV, E. L. 1977. Optimal foraging: a selective review of theory and tests, Q. Rev. Biol. 52, 137-156. SCHAFFER,W. M. 1974. Selection for optimal life histories: the effects of age structure, Ecology 55, 291-303. SCHOENER, T. W., 1971. Theory of feeding stragegies, Amer. Rev. Ecol. Sysf. 2. 369404.

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