Optimum body size: effects of food quality and temperature, when

J. evol.

Biol.

5: 677-690

(1992)

lOlO-061X/92/04677F14 0 1992 Birkhluser

$ 1.50 +0.20/O Verlag, Base1

Optimum body size: effects of food quality and temperature, when reproductive growth rate is restricted, with examples from aphids Pave1 Kindlmann’

and Anthony

F. G. Dixon*

‘Laboratory of Biomathematics, Czechoslovak Academy of Sciences, Branifovskh 31, 370 05 CeskP BudPjovice, Czechoslovakia *School of Biological Sciences, University of East Anglia, Norwich NR4 United Kingdom Key words:

Phenotypic

plasticity;

energy partitioning;

7TJ,

aphids.

Abstract Most models of optimum energy partitioning predict variability in adult size, although not always explicitly. Increase in size is usually attributed to an increase in the growth rate or decline in mortality. The model presented shows that this may not always be the case. Even when mortality is kept constant in organisms with overlapping generations, a constraint on the maximum reproductive growth rate may lead, when the rate of overall growth increases, to either an increase or a decline in the optimum adult body size. It is shown that adult size could be a consequence of the differential responses of life history traits to changes in temperature and food quality. This is clearly advantageous for short lived organisms, like aphids, each generation of which only experience a very small part of the great seasonal range in conditions. This hypothesis complements Iwasa’s (1991) explanation of the phenotypic plasticity observed in long lived organisms. The predictions are illustrated with empirical data from aphids. The model presented, which has been verified against a very large data set, indicates that for aphids the adult weight observed at a particular combination of temperature and food quality is that at which the population growth rate, r,,,, is maximized. We conclude that predictions about adult size from models based on the partitioning of energy are more likely to apply to organisms that scramble for resources, i.e., “r” selected species. The size of organisms that contest for resources is more likely to be determined by competitive status and avoidance of natural enemies. 677

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Introduction Marked phenotypic plasticity in size is common in plants (Harper, 1977). In contrast the size range in many species of animals tends to be quite small. However, certain animal groups show a great range of size within a species, e.g. aphids (Dixon and Wratten, 1971; Dixon and Dharma, 1980). Plasticity in size is accompanied by plasticity in other life-history traits and is associated with variations in environmental conditions: temperature, food quality. The result - intraspecific variability in size ~ is seen as the consequence of the optimization of life-history traits leading to the maximization of fitness, which is either defined by the Malthusian parameter r, in the Euler-Lotka equation, or lifetime fecundity. Many authors use the term “reaction norm”, introduced by Woltereck (1909), to describe the optimum relation between adult body size and age at maturity. Stearns and Koella (1986) and Gebhardt and Stearns (1988) accurately predict the form of the reaction norm for Drosophila mercatorum. Their assumptions include size dependent fecundity and constant juvenile and adult mortalities, which depend on growth rate and developmental time. They optimize r, with respect to the developmental time. Other hypotheses proposed to account for phenotypic plasticity assume an optimum partitioning of energy between growth and reproduction, which is dependent on environmental conditions. Most of these models were primarily intended as a description of annual (Paltridge and Denholm, 1974; Vincent and Pulliam, 1980; King and Roughgarden, 1982a, b; Schaffer et al., 1982; Kozlowski and Wiegert, 1987; Kozlowski and Ziolko, 1988) or perennial (Kozlowski and Uchmanski, 1987; Kozlowski and Wiegert, 1987; Pugliese, 1987; Iwasa and Cohen, 1989; Pugliese and Kozlowski, 1990) plant growth, but they may be also applicable to animals. Sometimes special assumptions are made, like non persistence of vegetative parts (Pugliese and Kozlowski, 1990), determinate (Kozlowski and Wiegert, 1987) or indeterminate growth (Kozlowski and Uchmanski, 1987), reduction in the vegetative body during adult life (Paltridge and Denholm, 1974; Oster and Wilson, 1978; Mirmirani and Oster, 1987), storage (Iwasa and Cohen, 1989; Pugliese and Kozlowski, 1990; Kozlowski, 1991) or limitations on the maximum rate of growth of reproductive tissues (Kozlowski and Ziolko, 1988). Often an explanation of plasticity in size is not the primary aim of these studies and only follows implicitly from a consideration of the reaction norm (Stearns and Koella, 1986; Gebhardt and Stearns, 1988), or of the optimal allocation of energy to growth and reproduction (Cohen, 1971; Kozlowski and Wiegert, 1986; Horn, 1988; etc.). Plasticity in size is related to that of life history traits in some studies. In the Ziolko and Kozlowski’s (1982) model, size depends on the mortality rate, the maximum life span, and the form of the growth curve with respect to body size. Stearns and Koella (1986) and Gebhardt and Stearns (1988) relate body size and age at maturity to the growth rate, which is assumed to reflect changes in food quality. They consider several submodels; in most of them increasing growth rate leads to an increase in adult size and in only one realistic case does size decrease when there is strong association between growth rate and juvenile mortality.

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size

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Kozlowski and Wiegert (1986) argue that small adult body size is a consequence of a short growing season or high mortality rate and Kozlowski and Wiegert (1987) that large adult body size is promoted by a high rate of somatic growth, high percentage increase in reproductive rate with body size increase and long life expectancy at maturity and life expectancy increasing with body size. High survival in winter is also claimed to lead to greater body size (Kozlowski and Uchmanski, 1987; Kozlowski, 1991). In summary, mortality and growth rate are mainly thought to determine the optimum adult body size. If mortality rate is kept constant, then increase in the growth rate never results in a decrease in adult body size in the above mentioned models. However, in many cases organisms thrive in a range of conditions and natural enemy inflicted mortality does not seem to play a significant role in determining their optimum strategy. Moreover, in rapidly changing environments the evolutionary response of short lived organisms is likely to be more to the immediate factors like food quality and temperature, than to infrequently acting density dependent mortality agents. In this paper it is revealed that when the mortality is kept constant, a constraint on the maximum reproductive growth rate in organisms with overlapping generations may, when the overall rate of growth increases, result in either an increase or a decline of the optimum adult body size. The above mentioned models do not make this prediction. This ambivalent response is associated with changes in temperature and food quality. We illustrate our predictions with empirical data on aphids.

The aphid case The black bean aphid, Aphis fabae, shows a very wide range of adult sizes, which may vary by more than one order of magnitude when it is reared at different temperatures and on foods of different qualities (Dixon and Dharma, 1980). This supports the common prediction of the previously mentioned models: associated with particular environmental conditions there is a particular size, a feature common to many species of insects. However, variation in size by more than one order of magnitude is not common. Moreover, aphids as a group have certain specific features that should be taken into account when trying to explain their plasticity in size. During larval life parthenogenetic aphids simultaneously commit energy for the development of their assimilatory organs (soma) and gonads (Kindlmann and Dixon, 1989). This results in the gonads growing exponentially and the soma logistically (Brough, Dixon and Kindlmann, 1989). At maturity somatic growth stops and all the energy assimilated is devoted to reproduction: production of offspring. This, together with parthenogenesis, enables aphids to achieve enormous growth rates (about .5 mg/mg/day). Kindlmann and Dixon’s (1989) model of this behaviour satisfactorily describes the course of somatic and gonadal growth, and reveals the adaptive significance of telescoping of generations, in which an imma-

Kindlmann

680

and Dixon

ture aphid can have her granddaughters developing in her gonads. Moreover, this model has provided an explanation of the relationship between population and individual growth rates (Kindlmann, Dixon and Gross, 1992) and the investment in gonads shown by aphids (Kindlmann, Dixon and Brough, 1992). During their life, wingless parthenogenetic aphids suffer a certain, usually not very high mortality, as they live on lush plants and the pressure from predators and parasites starts to appear only later in each population cycle, when alate forms develop rather than the wingless ones that we are considering. Therefore, we assume an age independent mortality coefficient, as all the other models do, but in contrast we do not assumethat this coefficient is dependent on either the growth rate or the developmental time, for which we have no evidence and, moreover, the empirical data suggest the contrary is true for aphids (Dixon, 1985; Howard, 1988). Below we use our model to predict norms of reaction of life-history traits and to determine the adaptive significance of phenotypic plasticity in adult size in aphids. Summary of the Kindlmann and Dixon (1989) model Notation: (Nl) (N2) (N3) (N4) (N5) (N6)

the size of soma at time t is s = s(t), the size of gonads at time t is g = g(t), the developmental time is D, initial size of soma (size of soma at birth) is s0= s(O), initial size of gonads (size of gonads at birth) is g, = g(O), adult size of soma (size of soma at the adult moult, fixed during adulthood) is sA = s(D), (N7) size of gonads at the instant of maturation is g, = g(D), (N8) assimilation rate per unit weight is a, (N9) mortality at age t is m = m(t), (NlO) fecundity at age t is energy gained + energy from gonads f=f(t> = energy content of one offspring ’ (Nl 1) the net energy production (the amount of energy used for growth) is Eprod. Assumptions: (Al) E,,,,(s) = US.~~. (A2) There exists a constraint R on the gonadal growth rate. (A3) The fecundity function is triangular, with the peak at age D, linearly declining and becoming zero at age 20, that is, the reproductive rate declineswith age. (A4) Mortality before age 20 is constant and equal to m. (A5) s0 is species-specific. (A6) The fitnesscriterion is rmdefined by the Euler-Lotka equation. After inclusion of (A3) -(A4) this becomes: 20 I=

e-(‘m

sD

++

.f(t)dt,

(1)

Optimum

body

681

size

The assumptions are in accord with our knowledge of aphid biology and would appear to be soundly based. More details of the biological reasons for our assumptions are given in: Kindlmann and Dixon (1989) Kindlmann, Dixon and Brough (1992) Kindlmann, Dixon and Gross ( 1992). From (Al), (A2), (A4) and (A6) it follows that the optimum strategy while immature is to maintain the gonadal growth rate at a maximum throughout development and devote the rest of energy to the growth of soma (Kindlmann and Dixon, 1989). Mathematically: s’ = us.67 - Rg,s(O) = so,

(24

g’ = &,g(O)

(2b)

= go,

where ’ = d/dr. Necessarily, s’ becomes zero at a finite time D, which is the best time to cease somatic growth and mature (Kindlmann and Dixon, 1989). At this instant as;’

= Rg,.

(3)

After that sA remains constant (N6) and all the energy gained from assimilation should be devoted to reproduction. Because of (A3) and (A4), the explicit form of the fecundity function, f(t), is f(t) = 2Rg,(2 - t/D), if fecundity is expressed in terms of energy contained in biomass. Initially, the reproductive investment will be 2Rg, at age D and 0 at age 20. This gives the average reproductive investment per unit time equal to as, 67 = Rg,, (eq. (3)). Initially, the reproductive output is larger than the energy gained (i.e., 2Rg, > RgA), which leads to the observed decline in gonadal size (c.f. Kundu, Dixon and Kindlmann, in preparation). Therefore, in accordance with (A3) and (A4), equation ( 1) becomes:

s 20

e

-r,r

D

(For follows

convenience, (A4).) After e -r,D

2Rg, ~ so +

(4) go

r,,, + m is simplified to rk ; this change does not affect what simplification, (4) gives

. (r;D

- 1 + eP’kD) r:D

so + go = 0. - 2RgoeRD

(5)

If so (A5), a (Al) and R (A2) are known, then D and subsequently g, follow from the solution of (2), as D is uniquely determined by s’(D) = 0. Therefore, the only free parameter in (5), which can influence rk, is go. Although it has not been possible to prove it analytically, all simulations for a broad range of parameters have shown that the dependence between rk and go is concave (its second derivative is negative) with exactly one maximum. The peak of this curve gives us the go value that maximizes rk and also r,, as this is only different from rl, by a constant m. As shown in Kindlmann, Dixon and Gross ( 1992), this value depends on a/R and not on a and R individually (Fig. 1.).

682

Fig. 1. The optimum

Kindlmann

gonadal

size at birth

as a function

of a/R

and Dixon

and sO

The model’s predictions There are three basic parameters in the model, when one considers one species under a range of environmental conditions: the assimilation rate, a, the constraint on the gonadal growth rate, R, and the somatic size at birth, sO. The value of s,, is assumed to be species-specific: an aphid should be as small as possible at birth conversant with maximizing its instantaneous population growth rate, r,,,, i.e., its soma at birth has to be of a certain minimal size in order to be able to feed (Kindlmann, Dixon and Gross, 1992). Therefore the value of s, is not dependent on environmental conditions and was kept constant in our simulations. The model’s predictions are depicted in Figs. 2-3. In each figure, A shows changes in a map in a - R space, where the lines indicate the isometric values of the corresponding dependent variable. In B the predictions are depicted as the dependent variable in three dimensional space, in relationship to the independent variables, a (assimilation rate) and R (constraint on the gonadal growth rate). The developmental time, D, adult weight, wA, and birth weight, wi, followed directly from simulations of the system (2). Some other widely used life history traits have also been computed: the relative growth rate during development, RGR = In(w, /wj)/D and the initial fecundity (the fecundity achieved in the first period of adulthood equivalent in length to the developmental time, D): Md = RgAD/(q, + 8,). Both of these life history traits are frequently used, especially for aphids. The predicted values of all the life history traits are very dependent on the a/R ratio, because - as shown in Kindlmann, Dixon and Gross (1992) - the optimum g, is dependent on this ratio and all the life history traits are a consequence of optimizing g, and thus maximizing r,,,.

Optimum

body

683

size

Fig. 2. The model’s prediction of the developmental time and relative growth rate (RGR), when assimilation rate, a, and the constraint on the embryonic growth rate, R, vary. The arrows indicate expected influence of food quality (F) and temperature (T).

the the

An increase in either a or R always results in an increase in RGR and in a decline in D (Fig. 2.). However, the values of wA, wi and Md depend on the a/R ratio (Fig. 3.): if both a and R increase and a/R also increases, then wA, w, and Md increase. When both a and R increase, but a/R declines, then wA, w, and Md will decline.

Influence of environmental

conditions on life history parameters

It has not been possible to relate changes in the assimilation rate and/or gonadal growth rate directly to changes in environmental conditions. The quantification of food quality is especially difficult. Therefore we cannot compare our quantitative model’s predictions directly with reality. However, the model’s qualitative predic-

684

Fig. 3. The model’s prediction of the adult weight, assimilation rate, a, and the constraint on the embryonic expected influence of food quality (F) and temperature

Kindlmann

birth weight and initial fecundity, growth rate, R, vary. The arrows (T).

and Dixon

when indicate

the the

Optimum

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685

tions can be compared with empirical data. For this the estimates of the effects of changes in temperature and food quality on a, R and on a/R are needed. The aphids Acyrthosiphon pisum, Aphis fabae and Cavariella aegopodii were reared at a range of temperatures and on different food qualities at a particular temperature. Their birth and adult weights (wi and w,), gonadal growth rate, R, developmental time, D, and RGR were measured. The results are summarized in Table 1. An increase in temperature and/or food quality results in an increase in R, RGR and a decline in D. Improving food quality and a decline in temperature both result in an increase in wA and wi. The same was observed by Dharma (1979) and Watson (1983) for Aphis fabae and Sitobion avenue. Moreover, Watson (1983) reports a decline in Md with increasing temperature. Improved food quality and/or a higher temperature not only result in a higher assimilation rate, a, but also positively affect the gonadal growth rate, R, even if indirectly. An improvement in food quality will affect directly the assimilation rate more than the gonadal growth rate (indirectly), and result in an increase in the a/R ratio. In contrast, an increase in temperature, which speeds up all processes, is likely to have a bigger influence on R than an improvement in food quality. This results in a decline in the a/R ratio. These considerations are illustrated in Figs. 2.-3, where F and T represent increases in food quality and temperature, respectively. Increase in temperature results in an increase in both a and R, but not a/R. Improving food quality increases both a and R, and a/R. An increase in either food quality or temperature, or both results in an increase in RGR and in a decrease in the developmental time, D. However, the influence of changes in environmental conditions on the adult weight, birth weight and initial fecundity differs. Improving food quality results in larger adult size, larger birth size and a larger initial fecundity, while an increase in temperature results in a decline in adult weight, birth weight and initial fecundity. All these predictions are well supported by empirical data from many species of aphids (Table 1.; Dharma, 1979; Watson, 1983).

Discussion

and conclusions

Using the model of Stearns and Koella ( 1986) and Gebhardt and Stearns ( 1988) and parameters observed in aphids gave a series of reaction norms (Fig. 4), which are very different from that observed in aphids. This is partly because offspring size in aphids also depends on environmental conditions, a feature not included in Stearns and Koella’s model. Maybe a more important discrepancy is the assumption that both adult and juvenile mortality depend on the growth rate and developmental time, for which there appears to be neither good empirical, or theoretical evidence, at least in aphids. Modification of Stearns and Koella’s model to take account of the above might give a more appropriate reaction norm. However, their model assumes certain dependences of fecundity on size, size on growth rate etc. and is more appropriate for those organisms, for which there is a poor basic understanding of their life

predictions: when \

Model’s

/

aegopodii*

pisumt

Carariella

Aphis fabae*

Acyrthosiphon

Species

increases

constant

Poor

Good

Food quality

.17 .32 .53

10 15 20

constant

increases

increases

.31 .53 .74 .79

10 15 20 25

increases

.50

.24 .40 .66 .85

R

20

10 15 20 25

Temperature ‘C

f & + *

.7 .2 .l .4

increases

decreases

24.3 k .3 22.8 & .2 21.0 f .2

29.5 23.5 25.3 15.5

119&275

119k 16 119k6 132 &2 89 + 8

W,

+ f + k

435 364 53 1 344

weight

16 11 9 17

increases

decreases

625 + 14 468 k 10 310+7

776 & 552k 586 k 398 f

1890 + 275

4713 4273 4026 2901

W.4

Table 1. The effect of temperature and food quality on the embryonic growth rate (R), birth weight (NJ,), adult relative growth rate (RGR) of 3 species of aphids (t-unpublished results, *-unpublished results of R. Kundu).

(w,,),

decreases

decreases

31 15 8

18 10 7 7

12

28 16 11 8

D

developmental

increases

increases

.I0 .20 .34

.18 .32 .45 .46

.23

.13 .22 .31 .43

RGR

time (D) and

x 5’ a z 2

Optimum

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687

size

Fig. 4. The reaction Koella’s model.

norms

for age and size at maturity

calculated

for Aphis fubae

using

Stearns

and

histories and a dependence on empirically derived relationships between life-history traits. For aphids, these relationships are described in terms of size related assimilation (assumption (Al)) and the optimum partitioning of energy between the various parts of the body in our model (eqs. (2)-(5)). All the other traits like fecundity, developmental time, size etc. follow from the basic assumptions. Thus, e.g., fecundity, is not only dependent on adult size, but is affected by birth size (which is also optimized), by assimilation rate as a function of somatic size and by the optimum allocation of energy during larval life, which results in gonads of an optimum size for the energy available to an adult. Therefore, in the case of aphids it may be more appropriate to start with basic assumptions about energetic constraints. The pattern of energy partitioning between growth and reproduction in aphids (Kindlmann and Dixon’s (1989) model) is different from that proposed in all the other models. Although similar, Kindlmann and Dixon’s (1989) model differs from those of Ziolko and Kozlowski (1983) Pugliese (1987), Kozlowski and Wiegert (1987) and Iwasa (1991), as it includes constraints on the gonadal growth rate and the minimum size of the soma. Ziolko and Kozlowski (1988) include a constraint on the gonadal growth rate, but their fitness criterion is lifetime fecundity, which is appropriate for organisms with non-overplapping generations, but not for aphids. Therefore these models are not applicable to aphids. Sibly and Calow’s (1986) approach presents the same problems as Stearns and Koella ( 1986): the assumption of mortality being dependent on the growth rate which, as already stated appears unrealistic for aphids. Most models predict variability in adult size although not always explicitly. Explicit predictions about phenotypic plasticity were made recently by Iwasa (1991). According to his model, relatively long lived organisms living in slowly changing environments should show great phenotypic plasticity, while those adapted to a quickly changing environment should use an almost fixed growth schedule. Short lived organisms like aphids, which each only experience a very small part of the total seasonal range of conditions, could in this context be considered

688

Kindlmann

and Dixon

Fig. 5. The relationship between the population growth rate, r,, and adult size, wA, for 2 sets of conditions of food quality and temperature (a = 2.16, R = .54 and a = 3.3 and R = .3) predicted by the model (2)-(5).

to belong to the first group and show great phenotypic plasticity, which aphids do. However, Twasa’s (1991) further prediction is that these organisms should prolong their development in a favorable environment but should shorten their development and start reproduction earlier in an unfavourable environment. This contradicts both the results of our model and the empirical data for aphids. The difference is due to the constraint on the reproductive growth rate in our model. Thus ours and Iwasa’s model are complementary explanations of phenotypic plasticity in different situations: when this constraint is important, and when it can be neglected. The model presented, which has been verified against a very large data set, indicates that for aphids the adult weight observed at a particular temperature and food combination is that at which T,,, is maximized (Fig. 5). That is, the marked phenotypic plasticity in adult size exhibited by individuals of clones in aphids is adaptive. Given the adaptive variability in adult size shown by aphids it is pertinent to ask: why does the adult size of most organisms vary so little? Energy partitioning models are only applicable to those groups of organisms that are constrained mainly by the availability of energy. A marked feature of the way of life of aphids are periods of intense intraspecific competition. Thus predictions of the variation in adult size from models based on the partitioning of energy are more likely to apply to organisms that scramble for resources, i.e., “r” selected species. The size of organisms that contest for resourcesis likely to be determined by competitive status

Optimum

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and avoidance of natural enemies. However, to resolve this problem, competition needs to be incorporated into the model, which is a task for the future. Variation in adult size is seen as a means of coping with a variable environment (Cockburn, 1991). However, in the species that are energy limited variation in adult size is not a means of coping but a consequence of maximizing r,,,.

Acknowledgements This work was supported by the British Council, NERC Grant GR3/8026 and Czechoslovak Academy of Sciences Grant 68002. The authors thank Mrs Shirley Wilkins for drawing the figures and monitoring the performance of the pea aphid at different temperatures and R. Kundu for the data on A. fabae and C. aegopodii.

References Brough,

C. N., A. F. G. Dixon, and P. Kindlmann, 1990. Patterns of growth and fat content of somatic and gonadal tissues of virginoparae of the vetch aphid, Megoura uiciue. Ent. Exp. Appl. 56: 269-275. Cockburn, A. 1991. An Introduction to Evolutionary Ecology. Blackwells, Oxford. Cohen, D. 1971. Maximizing final yield when growth is limited by time or by limiting resources. J. Theor. Biol. 33: 2999307. Dharma, T. R. 1979. Fecundity and size in the black bean aphid, Aphisfabae Stop. PhD Thesis, University of East Anglia, Norwich, UK. Dixon, A. F. G. 1985. Aphid Ecology. Blackie, Glasgow. Dixon, A. F. G. and T. R. Dharma, 1980. ‘Spreading of the risk’ in developmental mortality: size, fecundity and reproductive rate in the black bean aphid. Ent. Exp. Appl. 28: 301-312. Dixon, A. F. G. and S. D. Wratten. 1971. Laboratory studies on aggregation, size and fecundity in the black bean aphid, Aphis fabue Stop. Bull. Ent. Res. 61: 97- 111. Gebhardt, M. D. and Stearns, S. C., 1988. Reaction norms for developmental time and weight at eclosion in Drosophila mercatorum. J. Evol. Biol. 1: 335-354. Harper, J. L. 1977. Population Biology of Plants. Academic Press, London. Horn, C. 1988. Optimal reproductive allocation in female dusky salamanders: a quantitative test. Am. Nat. 131: 71-90. Howard, M. T. 1988. The factors determining the pest status of Metopolophium dirhodum (Wlk.), the rose grain aphid. PhD Thesis, University of East Anglia, Norwich, UK. Iwasa, Y. 1991. Pessimistic plant: Optimal growth schedule in stochastic environments. Theor. pop. Biol. 40: 246-268. Iwasa, Y. and D. Cohen. 1989. Optimal growth schedule of a perennial plant. Am. Nat. 113: 480-505. Kindlmann, P. and A. F. G. Dixon. 1989. Developmental constraints in the evolution of reproductive strategies: telescoping of generations in parthenogenetic aphids. Funct. Ecol. 3: 531-537. Kindlmann, P., A. F. G. Dixon, and C. N. Brough. 1992. Intraand interspecific relationships of reproductive investment to body weight in insects. Oikos, in press. Kindlmann, P., A. F. G. Dixon and L. J. Gross. 1992. The relationship between invidividual and population growth rates in multicellular organisms. J. Theor. Biol., in press. King, D. and J. Roughgarden. 1982a. Multiple switches between vegetative and reproductive growth in annual plants. Theor. Pop. Biol. 21: 1944204. King, D. and J. Roughgarden. 1982b. Graded allocation between vegetative and reproductive growth for annual plants in growing seasons of random length. Theor. Pop. Biol. 22: 1 - 16. Kozlowski, J. 1991. Optimal energy allocation models - an alternative to the concepts of reproductive effort and cost of reproduction. Acta Oecologica 12: 11-33.

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Kozlowski, J. and J. Uchmanski. 1987. Optimal individual growth and reproduction in perennial species with indeterminate growth. Evol. Ecol. 1: 214-230. Kozlowski, J and R. G. Wiegert, 1986. Optimal allocation of energy to growth and reproduction. Theor. Pop. Biol. 29: 16637. Kozlowski, J. and R. G. Wiegert. 1987. Optimal age and size at maturity in annuals and perennials with determinate growth. Evol. Ecol. 1: 231-244. Kozlowski, J. and M. Ziolko. 1988. Gradual transmission from vegetative to reproductive growth is optimal when the maximum rate of reproductive growth is limited. Theor. Pop. Biol. 34: 1188129. Mirmirani, M. and G. Oster. 1978. Competition, kin selection, and evolutionary stable strategy. Theor. Pop. Biol. 13: 304-339. Oster, G. F. and E. 0. Wilson. 1978. Caste and Ecology in the Social Insects. Princeton Univ. Press, Princeton, New Jersey. Paltridge, G. W. and J. V. Denholm. 1974. Plant yield and the switch from vegetative to reproductive growth. J. Theor. Biol. 44: 23-34. Pughese A. 1987. Optimal resource allocation and optimal size in perennial herbs. J. Theor. Biol. 126: 33-49. Pugliese, A. and J. Kozlowski. 1990. Optimal patterns of growth and reproduction for perennial plants with persisting or non persisting vegetative parts. Evol. Ecol. 4: 75-89. Schaffer, W. M., R. S. Inouye and T. S. Whitham. 1982. Energy allocation by an annual plant when the effects of seasonality on growth and reproduction are decoupled. Am. Nat. 120: 7877815. Sibly, R. M. and P. Calow. 1986. Physiological Ecology of Animals, Blackwells, Oxford. Stearns, S. C. and J. C. Koella. 1986. The evolution of phenotypic plasticity in life-history traits: predictions of reaction norms for age and size at maturity. Evolution 40: 8933913. Vincent, T. L. and H. R. Pulliam. 1980. Evolution of life history strategies for an asexual annual plant model. Theor. Pop. Biol. 17: 2155231. Watson S. J. 1983. Effects of weather on the numbers of cereal aphids. PhD Thesis, University of East Anglia, Norwich, UK. Woltereck, R. 1909. Weitere experimentalle Untersuchungen iiber Artveranderung, speziell iiber das Wesen quantitativer Artenunterschiede bei Daphniden. Verhandlungen der Deutschen zoologisthen Gesellschaft 1909: 1 lo- 172. Ziolko, M. and J. Koziowski. 1983. Evolution of body size: an optimization model. Math. Biosci. 64: 1277143. Received 12 December 1991; accepted 12 January 1992. Corresponding Editor: Y. Iwasa