Evolution of body size: an optimization model

Evolution of Body Size: An Optimization Model MARIUSZ ZIoLKO Institute of Automatic

Control, Systems

Engineering

Technical University of Mining and Metallurgy,

and Telecommunication,

Mickiewicza

30, 30-058 Cracow, Poland

AND

JAN KOZLOWSKI Department of Animal Ecology, Institute of Environmental Biology, Jagiellonian iJniversi(y, Karasia 6, 30 -060 Cracow, Poland and Department

of Zooloa,

University of Georgia, Athens,

Georgia 30602

Received 22 June 1982; revised 13 December 1982

ABSTRACT The growth of individuals is often described by bioenergetic equations which partition assimilated energy into maintenance, growth, reproduction, and so on. Such energy flows vary with body size. The bioenergetic equations in this paper, under the assumptions that organisms

will adopt

behavior

that

channels

a maximum

amount

of energy

into

the

production of surviving progeny, produce a model optimizing growth and reproduction. Using the Weierstrass theorem, we show that a solution of the optimization problem exists. The problem is further solved analytically using the Pontryagin maximum principle. The main conclusion of the paper is that in a given environment an optimal body size exists, one which maximizes the energy channeled to the production of progeny. This body size depends on the mortality rate, the maximum life span, and the derivative of the growth equation with respect to body size. The biological results predicted from the model are compared with ecological data for zooplankton and vertebrate species, which support the conclusions

1.

obtained.

INTRODUCTION

The evolution of body size and the existence of connections between body size and various life history parameters continue to be lively topics in modern ecology [ 11. Nevertheless, the most important questions have not been solved. On the contrary, different approaches frequently lead to contradictory results [2, 31. One source of these ambiguities is neglect of physiological constraints in the evolution of life history parameters; these parameters are assumed to evolve under the influence of demographic forces only [ 1, 31. In this paper, physiological constraints on the bioenergetics of individuals are a starting point. Using a simplified demographic assumption, a model is MATHEMATICAL

BIOSCIENCES

64: 121-143

QElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017

127

(1983)

00255564/83/03127+

17$03.00

128

MARIUSZ

ZIkKO

AND JAN KOZLOWSKI

developed which can predict an optimal allocation of energy between growth and reproduction. The idea of maximization of the energy allocated to reproduction has been developed by Cohen [4], Mirmirani and Oster [5], Oster and Wilson [6], and Vincent and Pulliam [7]. All these authors assumed individual fitness to be proportional to the biomass of reproductive growth during life span, and they looked for optimal division of the energy used for vegetative growth and reproduction. All of them except [4] used the mathematical techniques of dynamic optimization in analyzing their models. Unfortunately, these models only considered a semelparous life history, i.e., reproduction occurring at the end of life. They also assumed that no mortality occurs before reproduction. Additionally, everyone except [5] assumed that the rate of production is a linear function of body size. The model presented here deals with the iteroparous mode of reproduction, not with semelparity. It also includes constant mortality during the growing season. The production rate is the difference between the rates of assimilation and of respiration. Thus, our model can be viewed as proceeding from the most general model for semelparity given by Vincent and Pulliam [71. All the papers cited above lead to the conclusion that the optimal strategy is to divide the life cycle into two separate periods: (1) when all surplus energy is allocated to growth, and (2) when all is allocated to reproduction. This kind of optimal solution will be referred to as the “bang-bang strategy,” according to the control theory terminology. The models cited also give conditions of optimal switching from vegetative growth to reproductive growth. In this paper we ask: (1) Is the bang-bang strategy still optimal if the model is nonlinear, mortality occurs during the growing season, and reproduction is iteroparous and continuous? If the answer is yes, we ask: (2) Is it possible to find the condition for optimal switching from vegetative growth to reproduction? If this answer is positive, a further question is: (3) Does the form of the switching condition suggest that the model is modifiable for more realistic situations than those considered here? Predictions of the model are compared with field data. A more complete discussion of the biological consequences of this model is presented in [8]. Here we are concerned with the mathematical basis of the model. 2.

THE MODEL

Let us assume that the rate of energy assimilation of an individual of size w is an allometric function of w, so it can be expressed by the equation dA x = aw”,

(1)

EVOLUTION

OF BODY

SIZE

129

and the rate of energy spent for maintenance

costs by the equation

The expression dP dt=aWa

-pWh

gives the quantity of energy which can be used for body growth or reproduction. If all surplus energy is used for growth, (3) is a representation of Winberg’s equation of growth [9]. For the sake of simplicity w is measured in energy units. Assume that a function y exists, depending on the age of an organism, which takes values from [0, 11. This function describes the fraction of surplus energy used for reproduction. Then the growth rate of the organism can be expressed by the equation.

$=(l-y)(awa+vb). Assume that the individual fitness is proportional to the total energy used for reproduction during the life span. If size of the individual young or egg is constant and optimal (for other reasons), then maximization of this energy is equivalent to maximization of the total number of progeny produced during an average lifetime. Taking the above arguments into consideration, it can be assumed that a function y* is sought, which maximizes the functional

where Q is the quantity of energy used for reproduction during the whole life span, T is the maximum life span, and 1(,) is the probability of surviving to . age t (it 1s assumed that ICo = 0 for t > T, i.e., the additional bang-bang mortality occurs at T). In further considerations it is assumed that the instantaneous mortality rate is independent of age for 0 < t < T, so that

(but see Section 9 for a suggested relaxation of this constraint). The maximum life span T does not have to be treated as a physiological property. Very frequently periodic climatic changes or transient resources limit the maximum life span.

130

MARIUSZ ZIbtKO

AND JAN KOZLOWSKI

The maximization function (5) can be used as a fitness measure only if one of alternative assumptions is met. The first is that there is a diapause period and all offspring start to grow at the same time of the next year regardless of birthdate. The alternative assumption is that the population number is constant. This assumption may seem to be very restrictive but in fact almost all populations are stable over a long time period.

3.

FORMULATION

OF THE OPTIMIZATION

The system under consideration

PROBLEM

is described by a differential

equation

(7)

with the initial condition

w(o,= w, > 0. The parameters

in Equation

(7) are constant,

(8) and they satisfy inequalities

O
(9)

These assumptions make the problem clearer, but they need not be so strong (see Section 9 for a generalization). The optimization problem consists in finding a function

(10) which maximizes the functional

(11) where the positive parameters

4.

STUDIES

4, T are constant.

OF SYSTEM DYNAMICS

It is biologically reasonable to consider the function few,for only positive values of its independent variable. Using the inequalities (9), which are also biologically justified, it is possible to study elementary properties of the function&,, and to sketch its form.

131

EVOLUTION OF BODY SIZE COROLLARY

I

(a>

f~w,,=m~f(w),

(b)

f(,) = 0,

wm= wg =

cc>

(2 1‘, (s1=.

wherec=&.

a

wm< wg

(4 (e)

f(w) ’

0

for

O
(f)

f (w)< O

for

w>w,.

(g)

f;w, = 2

for

O
(h)

f(w)< 0

for

w>w,.

f(0)= 0.

> 0

The differential equation (7) has a unique solution because the right side of the equation is continuous and satisfies the Lips&k condition. Taking into account the assumption (9), it is easy to verify the next corollary. COROLLARY

2

(4

w(,,>O

(b)

wet, < wg for t 2 0

if

wO
(c)

W(f)= wg for t > 0

if

wO=wg.

(d)

w(,)> wg for ta0

if

w,> wg.

5.

EXISTENCE PROBLEM

for t>O.

OF A SOLUTION

OF THE OPTIMIZATION

The problem of finding the function y(:, which maximizes the functional (11) is a problem of nonlinear optimization. Using the Weierstrass theorem it is possible to prove the existence of such a solution. Notice that every solution of the differential equation (7) can be presented in the form

(12) where z is a solution

of the equation dz

x =f(z),

Z(O)= W(O).

(13)

132

MARIUSZ ZIbtKO

AND JAN KOZLOWSKI

In order to state whether the Weierstrass theorem can be used, we can examine topological properties of mapping, by which solutions of the differential equation (7) are assigned to the function y. Let us define U, as

The mapping

(15)

is continuous, and most important of all, weakly continuous, i.e., continuous with the weak topology in the set U, which results from the compactness of the integral operator. Notice that the set of values of Hcu,, of mapping H is a subset of nonnegative functions in C,, Tj, and moreover

for abitrary y E U,. The mapping =:

c,o,3 Hw~,-+c,o, T)

T) 1

( =d,,, = %(,))

(17)

[where z is a solution of the differential equation (13)] is well defined and continuous, which is a consequence of the fact that the function z is continuous. The solution of the differential equation (7) can therefore be written in the form

The mapping

=H: L:o,.+L -+c(O,T)

(19)

is weakly continuous as a product of two mappings: a continuous one Z and a weakly continuous one H. Taking into account the form of the functional (ll), continuity of the function f defined by (7), and weak continuity of the mapping ZH, it is possible to prove weak continuity of the performance index Q. The set U, is

EVOLUTION

133

OF BODY SIZE

weakly compact as a convex, bounded, and closed subset of L&,. Now, using the Weierstrass theorem, it is possible to conclude that the weakly continuous and limited functional

attains its minimum 6.

TRIVIAL

and maximum

SOLUTIONS

on a weakly compact set.

OF THE OPTIMIZATION

PROBLEM

An initial condition w, for the differential equation (7) is fixed and positive in view of its biological sense. The maximum value of the functional (11) and the function y * which is a solution of the optimization problem are dependent on the initial condition ~1~.Using the corollaries given in Section 3, it is possible to find immediately a solution y* for the cases when w, > wm: (1) Assuming w, > wg it can be found from corollaries 1f and 2d that the maximum value of the functional Q is equal to zero and yc”;,= 0 for 1 E [0, T]. Q is equal to zero and (2) If w(J= wg, the value of the functional independent of the function y. It is easily verified that wn is an equilibrium point of the differential equation (7) and a multiplicative and bounded input unaction has noinfluence over a trajectory of the system. (3) If wm < w,, -C wn, then the maximum value of the performance index is obtained for y(:, = 1 with t E [0, T]. Such an input function keeps the value of the function fCw) at the highest possible level. Any other function y causes a decrease of the function f(,+,, and so a change for the worse in the performance index. Thus it is very easy to find a solution of the optimization inequahties w, < w, < cc hold for the initial conditions. 7.

UNIQUENESS

OF A NONTRIVIAL

problem

if the

SOLUTION

When the initial value of the differential equation (7) satisfies the inequalities O< w,< w,, it is possible to find an optimal solution and to test its uniqueness using the Pontryagin maximum principle [lo], which gives the necessary condition for optimal control. For the problem considered the Hamiltonian has the form H= and the adjoint equation

4

-= dt

[YC~-4’-P)+Plf(w)

(21)

[IO] is

-f;w)[Y(e-q'-P)+Pl~

(22)

134

MARIUSZ

ZI&KO

AND JAN KOZLOWSKI

with a final condition (23)

P(T) = 0. An optimal solution which maximizes the Hamiltonian

I$,=

1

if

e-4’>~Cr),

i 0

if

eeq’
has the form

(24)

The optimal solution is a bang-bang function, so it is very useful to find an analytical solution of the adjoint equation for two cases, when y* has the constant value 1 or 0. For y* = 0 the adjoint equation has the form + = -f;,,P, dt

(25)

where flW, is the derivative of the function f with respect to the variable The equation (25) has the solution

w.

fWd P(r) = PO-

f(w)

9

(26)

where PO = P(fo))

For y* = 1 the adjoint equation

wo = W(to).

has the form (27)

and its solution is

where wp is a fixed value of the variable w. Equation (25) with the condition (23) has the solution (26) equal to zero, i.e. p(,) = 0. In this case y* = 0 and e-qr z P(~). This is contradictory to the Pontryagin principle (24) of maximization of the Hamiltonian. On the other hand, Equation (28) with the condition (23) has a form

EVOLUTION

135

OF BODY SIZE

where wP is a value at a switching point, if one exists, or wP is equal to w, if there is no discontinuity. The function pet) is positive because the derivative of the function f is positive for 0 < wP < wm. Thus, there exists a time when the inequality e-+ > pcr) is satisfied. According to (24) this means that for a sufficiently large time the optimal solution is y * = 1. Continuing the considerations for decreasing time (i.e., from time T to the initial time, denoted by 0), we must search for a hypothetical switching point Tp that fulfills the condition

(30)

The switching, if it occurs, will do so when the condition (31) is satisfied. Up to the time of switching, variable is described by the formula

pct)=-e

after the initial time 0, the adjoint

fw _@ p5

(32)

f(w)

which is the solution of the differential states that the inequality 4<

equation

f;w,,

(25). The condition

(31)

(33)

holds. Approaching the initial time 0, variable w decreases monotonically, so Taking into account this fact, the the function r;W, increases monotonically. inequality (33), and the equation

4 _=_ dt

4p

which is fulfilled by the function e-q’, one can see that there are no further switching points, because the curve given by Equation (32) does not cross the curve e-+. Because of the uniqueness of the solution of the adjoint equation (22) and the affine dependence of the Hamiltonian (21) on the function y, we conclude that there exists only one solution which maximizes the Hamiltonian. This solution has one switching point at most and maximizes the functional (11) subject to the constrains (7), (8), (9), (10).

136 8.

MARIUSZ

NUMERICAL

ZIbtKO

AND JAN KOZLOWSKI

EXAMPLE

The equation (31) shows that the switching point (adult body size) is dependent on the mortality rate 4, the maximum life span T, and the size of the newborn individual, w. (assuming that the parameters u, b, (Y, /I are constant). In order to illustrate these dependences, consider a numerical example (Figure 1), in which a = 0.67, h = 0.75, cr = 2.0, and /? = 1.0. For many organisms, the individual metabolic rate varies as the weight to the power 0.75. On the other hand, the maximum assimilation rate is probably proportional to the surface area of the intestinal tract, which should vary as w 2/3. The numerical values of the parameters y and j3 are not very important, because they depend on the units used. However, the condition cr > p must be satisfied. Growth curves when the mortality rate q = 0.01 are shown by solid lines in Figure 2(a). Different curves correspond to different initial values w,. The dashed line indicates the switching curve obtained from the condition (31) assuming a different initial body size w,. Each growing individual stops its growth after crossing this line and puts its surplus energy into reproduction.

fCu>

30 T

c

Organism growth rate fCHj versusits weight w. The curve represents given in the numerical example.

FIG. 1,

equation

the growth

EVOLUTION

OF BODY

SIZE

137

Obviously the individual with smaller initial body size will attain the switching curve later, i.e. at a lower point. Thus the positive correlation between newborn size w, and adult size wp follows. Figure 2(b) shows switching curves for different maximum life spans T. An influence of this parameter on body size is evident. Figure 2(d) depicts switching curves for the same maximum life span but different mortality rates 4. The mortality rate increases in geometrical progression from 5 X lop4 (upper line) to 0.256 (the lowest line). The switching curves indicate that the optimal body size falls when the mortality rate increases. When the mortality rate is very high, its further increase has a rather small effect on optimal body size. Figure 2(c) shows optimal growth curves for organisms which have various initial sizes. Very big individuals appropriate all energy for maintenance costs, and they do not procreate. The amount of assimilated energy is too small for maintenance, so their weights decrease with time. Individuals which have smaller initial size (central part of the figure) use all their surplus energy for reproduction, and their sizes are constant. In the lowest part of Figure 2(c) there are curves which are given in more detail in Figure 2(a).

9.

DISCUSSION

Although most life history studies have been based on maximization of the Malthusian parameter, the idea of maximizing the energy allocated to reproduction, adopted in this paper, was developed successfully during the 1970s [4-71. Thus, another look at life history evolution, is strongly physiologically oriented and leads to verifiable predictions about the optimal age at maturity and optimal mature body size, all topics important to modem evolutionary theory [ 111. Conditions of optimal switching from growth to reproduction have been found analytically in both this paper and all the cited papers. These conditions are simple, and they contain well-defined, measurable parameters. Thus, the approach presented here is a useful alternative to maximization of the Malthusian parameter. Possible application of the model to make quantitative predictions is limited mainly by two assumptions: (i) that the growth rate is described by (3) with the conditions (9) on the parameters, and (ii) that the mortality rate is constant during life span. Limitation (i) makes the model familiar in that Winberg’s growth equation is frequently used, especially in limnology and fish biology. In fact, the growth equation in the model can be much more general. Apart from Section 4 (where system dynamics is discussed), we assume only that the function describing the growth rate is continuous and differentiable with respect to w, and the derivative decreasing monotonically. This last condition seems to be biologically justified because the specific

of initial

Dependence

of switching

of switching

Dependence

broad

range

C

A LIFE

WORTRLITY RATE IJA’AR llRXIHUfi LIFE SPAN 1 =I00

llRXIflUM

size; both

and

nontrivial

curves are given.

8.

(d)

for a see Section

growth solutions

life span. (c) Optimal

rate. For more explanations

trivial

curves on mortality

body

curves on maximum

of optimal solutions. The growth equation dw/dt = ~wO.~‘- w’,‘~ growth curves (solid lines) and switching curve (dashed line). (b)

RATE qzO.01 LIFE SPRN T =I00

flORTALITY flRXlllUtl

FIG. 2. Examples was used. (a) Optimal

RATE qzO.01 LIFE SPRN 1 _-LOO

llORTRLITY MlXIflUf!

EVOLUTION OF BODY SIZE

139

growth rate (per unit weight) is always a decreasing function of body size. Suggestions regarding possible moderation of limitation (ii) are given later. Three basic questions were asked in the introduction. The first one was about the general character of the solution. The results clearly indicate that an optimal division of energy for growth and reproduction is still in the bang-bang form, even when the model of body growth is nonlinear, continuous mortality occurs, and cases other than semelparity are considered. Nevertheless, many cases are reported for which growth does not cease with reproduction. Such cases may be divided into two groups: first, when the switch from growth to reproduction is complete but reversible, so that there is more than one period of growth and more than one period of reproduction during a life span, and second, when surplus energy is channeled to both growth and reproduction simultaneously. The term “simultaneous growth and reproduction” is often applied even to the first group because long time units, say years, are commonly used in demographic studies. When smaller time units are used, i.e., months, days, hours, or even minutes (depending on body size), a division of life into phases of growth and phases of reproduction is probably the rule in nature. Most trees, fishes, and amphibians belong to the first group. Their life span is usually longer than 1 year, and, therefore, there are unproductive periods with heavy mortality in a seasonal environment. The model presented in this paper deals with annual habit (if applied to a seasonal environment) and cannot predict growth patterns of longerliving organisms. Nevertheless, Kozlowski and Wiegert [8] have shown, using an approach similar to that of this paper, that a generalized switching condition (31) can be used not only to optimize age at maturity but also to predict switching back from reproduction to growth, and again to reproduction. Growth in the years following first reproduction is likely in long-living organisms in a seasonal environment. Values of parameters (not the model structure) define whether determinate or indeterminate growth is optimal. It is more difficult to explain the cases belonging to the second group (simultaneous allocation of energy to growth and reproduction), or examples when many periods of growth and reproduction occur in one season. Random variation of parameters may be the cause of these phenomena [4, 61. Nevertheless, this subject needs further study. Before environmental variability is invoked as an explanation, the inherent short term differences in productivity and mortality patterns must be eliminated as a cause of recurrence of growth after the first reproduction. Our model gives the simple condition (31) for optimal switching. An individual should switch from growth to reproduction whenever the derivative of growth rate with respect to body size [!(,,,,I, multiplied by the factor 1 - P [where P is the conditional probability that an individual alive at T,, (i.e., at maturation) is still alive at T], is equal to the mortality rate 4. Thus, individual bioenergetics and mortality are reduced to the same units (time-‘)

140

MARIUSZZI6tKOANDJANKOZtOWSKI

fL> I ?

.‘q.0.80 q'0.50 qe0.40 q-0.30 q-0.20 4'0.10

I

I 0 FIG.

BOGY SIZE w 3. Graph for the estimation

of the optimal

250 body size when life span is unlimited.

The optimal body size is obtained when/(,) curve falls to present mortality rate level. flN,, is the derivative of the growth equation with respect to body size. Parameters of the growth equation are the same as for Figures 1 and 2.

and can be compared directly. When the maximum life span T and/or q are so large that the probability of survival to age T is small enough to be replaced by zero, (31) takes the form

The graphical representation of (36) for the same parameters as in the previous sections is given in Figure 3. Note that, unlike our model, the model given by [7] leads to a trivial solution when T becomes very large. Thus, our model can be extended to reasonable biological situations when the maximum age is not precisely defined. The switching condition (31) does not contain time, but only the difference between T and switching time Tp. Therefore, we suspect that (31) is still valid even if neither growth rate nor mortality is constant during life, although some limitations for these functions are required. This problem is explored in [8].

EVOLUTION OF BODY SIZE

141

Both our model and the models in [4], [5], [6], and [7] predict switching from growth to reproduction before the maximum of the production rate is reached. Thus, production rate maximization cannot be adopted as a criterion of body size optimization, as was assumed in [ 121. Further qualitative predictions of the model are as follows. The optimal body size has a tendency to increase when (i) the life span T increases, (ii) the mortality rate ~7decreases, (iii) the individual growth rate increases, (iv) the initial body size wOincreases. Additionally, (v) the total clutch volume should be an increasing function of body size. Prediction (i) can be reformulated for a population with several generations during one season as follows: body size has a tendency to decline from spring through autumn. These predictions have been tested against data from the field in detail [8]. A brief discussion of this analysis follows here. The physiological assumptions of the model seem to be quite realistic for many zooplankton species. Thus independent mortality is an agent selecting for small body size among zooplankters. Hence, the size-efficiency hypothesis [ 13, 151 should be modified to include the influence of mortality neutral to body size. This hypothesis states that the gain in fecundity observed in large zooplankters selects for large body size, and selective fish predation (more intensive for large specimens) is a counterbalance selecting for small body size. There are, indeed, many examples where body sizes of zooplankton are inversely correlated with the intensity of predation by fish. When fish are scarce, predation by invertebrates prevails, which is selective on small individuals. Yet, fishes are likely to be more efficient predators than the smaller invertebrates. Then, according to our results their predation, even if sizeindependent, causes a decrease in body size. This effect should not be neglected, although selective preying upon large zooplankters is an additional pressure toward small body size. Lynch [ 141 in his simulation studies also has found that size-independent mortality is not neutral to body size. Unfortunately, he did not pay much attention to this important finding. Another difficulty also arises when zooplankton life history data are used to test the predictions of our model. In a rapidly changing aquatic environment the forces (i)-(iv) act simultaneously. Furthermore, switching from growth to reproduction is not always in a bang-bang form in the case of zooplankton, so we must assume that the size at first reproduction is a measure of adult body size. Nevertheless, we have found published support for prediction (i) (e.g. [16], [17], [18]; see also [17] for more examples). Additional food caused an increase of body size in a Duphnia pulex population [16], which is in agreement with prediction (iii). If we assume that fish act as nonselective predators (this is an oversimplification, but e.g. [19] indicates that size selectivity by fish has been overestimated), there is rich literature supporting prediction (ii) [ 131.

142

MARIUSZ

ZIhKO

AND JAN KOZLOWSKI

The size of newborn organisms depends on factors not considered here, and the ratio of newborn size to adult size ratio varies greatly. Nevertheless, a general correlation exists between these two variables both for zooplankton [13, 171 and vertebrates [20]. One of the possible reasons is that adult size depends in some degree on initial size, as has been shown in the numerical example. Applying the model to vertebrate species is more complex because additional physiological constraints appear. The growth rate of the nervous system is sometimes reported as a limiting factor in the whole body growth rate [2 11. Homothermy also obviously influences body size. Therefore, only a very general correspondence of the model predictions to real data can be expected. Fortunately, numerous allometric functions of different life history traits versus body size are given in [20] both for poikilotherms and for homeotherms. They support the predictions of the model; body size is positively correlated with maximum life span as well as an average life span (which measures mortality rate). Total clutch or litter volume increases with body size. Adult and newborn body sizes are also positively correlated. The correlation of average life span and body size must be treated cautiously as evidence of dependence of body size on mortality, because the opposite relationship (mortality rate being less for larger animals) gives the same results. There is probably a feedback between these two causal systems in nature. Such a feedback could explain the phylogenetic increase in body size (Cope’s rule). The problem presented in this paper is only partially solved. The model can be and should be a subject of further study and generalization. Biological predictions should be extended, too. Nevertheless it seems to us that in the approach opened by [4], [5], [6], and [7] and advanced in this paper, there is the chance to solve some paradoxical problems of life histories (see [8], 1221).

We wish to thank R. G. Wiegert for help in preparing this version of the manuscript. We especially thank the anonymous reviewer, who provided very valuable criticism. The paper was partially supported by a postdoctoral fellowship from the University of Georgia to the second author. REFERENCES 1 2

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7:177-209 (1976). M. Lynch, Predation, enrichment and evolution of cladoceran life histories: a theoretical approach, in Evolution and Ecology of Zooplankton Communities (W. Ch. Kerfoot,

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18

J. P. Milssen,

19 20 21 22

When

and

pond

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to reproduce:

a dilemma

for

in

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Optimal

allocation

of energy to growth

and reproduc-