Evolut ionary Ecology, 1987, 1,214-230

Optimal individual growth and reproduction in perennial species with indeterminate growth JAN KOZ:LOWSKI Institute of Environmental Biology. Jagiellonian University, Oleandry 2a, 30-063 Krak6w. Poland

JANUSZ UCHMAIqSKI Department of Paleobiology, Polish Acadenly of Sciences, Zwirki i Wigurv 93, 02-089 Warszawa, Poland

Summary

A model predicting optimal timing of growth and reproduction in perennial species with indeterminate growth living in a seasonal environment, is presented. According to the model, the optimal fraction of growing season devoted to growth decreases with increasing individual age and size, which leads to S-shaped growth curves. Winter mortality seems to be a crucial factor affecting the timing of growth and reproduc- tion, under the same function describing the dependence of growth rate and reproductive rate on body size. When winter mortality is heavy, it is often optimal to start reproducing in the first year, and to devote a large proportion of the subsequent years to reproduction, thus leading to small adult body sizes.

The model has been applied to two species of mollusc and one species of fish. The model predictions fit well to the field data for these three species.

Keywords: Life history; maturation age; body size; growth curve; mathematical model; Conus pennaceus; Chlamys islandica; Salvelinus alpinus.

Introduction

Kozlowski and Wiegert (1987) presented the model of optimization of age and size at maturity for annuals or perennials with determinate growth (i.e. growth ceasing at maturity). Their model starts from the assumption that surplus energy (not used for maintenance) should be optimally allocated to growth and reproduction in order to maximize fitness. They found that optimal age and size at maturity depend on three factors: the individual growth rate, the specific rate with which reproductive rate increases with adult body size, and life expectancy at maturity.

Perennial species with indeterminate growth (i.e. not ceasing at maturity) must optimize not only the age at maturity but also the division of all the years following maturation between growth and reproduction. The problem of such an optimization process is addressed in this paper. To make an analytical solution possible, it is necessary to assume that population numbers are either constant or fluctuate around the mean with the average period of fluctuation shorter than the generation time. Such an assumption, certainly realistic for many species (Connell and Sousa, 1983; Schoener, 1985), equates reproductive value to the expected number of offspring. Contrary to the model presented by Kozgowski and Wiegert (1987), it is necessary to assume size- independent adult mortality in order to solve the optimization problem analytically. It is also assumed that the switch from growth to reproduction is complete although reversible (for discussion of this point see Kozlowski and Wiegert, 1987), and it can occur at most once a year.

0269--7653/87 $03.00+.12 © 1987 Chapman and Hall Ltd.

Growth and reproduction in perennials 215

The model for a maximum life span of two years is presented in the next section, and the conditions of the optimal switching from growth to reproduction are derived in Appendix I using relatively simple mathematics. The optimization condition for any length of life span is derived in Appendix II, and only the final result is presented in the next section.

Numerical examples of optimal solutions are given, and the cost of adopting a suboptimal strategy instead of the optimal one is discussed. Some field examples supporting the model are provided. Finally, the conflict between growth and reproduction, and the effect of adult mortality (both natural and caused by exploitation) on the optimal age at maturity and on the shape of the growth curve are considered.

The model

Let us assume that individuals of the species under consideration can live no longer than two years, and can reproduce either in the second year only or in both years. Let reproduction be continuous, i.e. energy allocated to reproduction be released immediately. Let us also assume that mortality rate is size-independent after reaching some threshold body size, but can differ between the growing season and winter. Reproductive rate can be measured either as the amount of energy allocated to reproduction or the number of offspring of the same size and quality.

Under such assumptions, the average lifetime amount of energy allocated to reproduction (or offspring production) can be represented by the sum of volumes of two solids depicted in Fig. la. The axes of the three-dimensional space in which the solids are placed are as follows: age (x), probability of surviving to a given age (l(x)) and the rate of reproduction m(x). The space between two solids represents the unproductive part of the year, when neither growth nor reproduction is possible ( 'winter ') , and also the part of the second growing season devoted to growth.

Because of the assumption that the switch from growth to reproduction is always complete, the growth curve has the form shown in Fig. lb: body size increases until an individual matures at age tl, then remains constant until the beginning of the second growing season, then growth is

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216 Kozlowski and Uchmahski

continued until the switch to reproduction at age t2. Thus function m(x) has the following form: (1) it equals zero for immature individuals; (2) from the onset of maturation occurring at age tl and corresponding size wl to the end of the first growing season, reproductive rate is equal to H(Wl); (3) then again re(x) = 0 throughout winter and in the part of the second growing season preceding the switch to reproduction occurring at age t2 and corresponding size w2; (4) from t2 to the end of the second growing season reproductive rate equals H(w:). Such a form of the function representing reproductive rate reflects the assumption that reproduction (if present at a moment) is size-dependent, not directly age-dependent. It is obvious that the second solid is higher than the first one if any growth in the second year takes place.

The sum of the volumes of the solids can be expressed as:

V(h,t2) = Vl(t,) + V2(tl, tz) (1)

The volume of the first solid (V~) depends on the age at maturity only (h), and the volume of the second solid on both the age at maturity and the time of the switch from growth to reproduction in the second year. This results because body size in the second year depends on the sum of growth periods in both years.

Let us find now such a pair tl, tz for which partial derivatives of V with respect to t~ and tz take the value zero, i.e.:

dVl + 31"2 = 0 (2a) dtl 3tl

and simultaneously

~V2_ at2 0 (2b)

If the pair h, t2 satisfying both Equations 2a and 2b can be found, the variable V reaches its maximum for such a pair. As shown in Appendix I, conditions 2a and 2b are equivalent to:

H'(w1) f(wl) El + Ht(w2) f(w2) pE2 = H(wI) (3a)

H'(w2) f(w2) E2 = H(w2) (3b)

where H(wa) and H(w2) are the rates of reproduction (measured as the rates of offspring production or the rates of energy allocation to reproduction) at switching in the first and in the second year, respectively, p stands for the probability of surviving from switching time in the first year to switching time in the second year, Ea and E2 for the expected number of future productive days (life expectancy minus the expected number of winter days) at switching times t~ and t2, respectively. Function f(w) represents somatic growth rate at body size w, when the entire surplus energy is allocated to growth.

After dividing both sides by H(w2), Equation 3b is identical with the condition for optimal age and size at maturity in annual species (Kozlowski and Wiegert, 1987). Equation 3a, dealing with optimal switching from growth to reproduction in the first year, cannot be solved if we attack the problem forwards. This is because body size in the second year and life expectancy at switching in the second year are not known in the first year, for which Equation 3a applies.

However, the problem can be solved backwards which is common practice in control theory (e.g., Intriligator, 1971). Instead of initial body size, final body size w2 is assumed. Life expectancy at the end of the second season as well as the left-hand side (called henceforth LS) of Equation 3b equals zero. This means that it is optimal for an individual to reproduce, since LS is smaller than the right-hand side (RS). The value of E2 increases for decreasing time whereas the

Growth and reproduction in perennials 217

values of H(w2) and f(w2) hold constant, because all surplus energy is allocated to reproduction, and no growth takes place. Thus it is likely (although not obligatory) that for some time in the second growing season LS will become even with RS, which gives the optimal switching time from growth to reproduction in the second year. Having this optimal time t2 we can solve the growth equation f (w) backwards to the beginning of the second growing season, which gives us the value of Wl, i.e. the final body size in the first growing season. Now we can repeat this procedure for the first year since w2 and E2 in Equation 3a are known. After t t is found, solving the growth equation backwards to age zero gives us the initial body size. If obtained initial body size differs from that required, the final body size w2 can be adjusted in the proper direction and all calculations can be repeated.

It may happen that either Equation 3a or 3b cannot be satisfied for any t~ or tz. In such a case it follows immediately from the Maximum Value Theorem that it is optimal to devote the entire first year to growth if the LS of Equation 3a is larger than the RS of Equation 3a, even at the end of the first season. Similarly, it is optimal to devote the entire second year to reproduction if LS of Equation 3b is smaller than RS of Equation 3b, even at the beginning of the second season.

The procedure described can easily be generalized for a life span of any length m (see the Appendix II). The condition of optimal switching in year j reads:

t r l

Z H ' ( w , ) f ( w i ) E, pii = n (w/ ) (4)

where pj/stands for the probability of surviving from switching to reproduction in the year j to switching to reproduction in the year i (when switching does not occur in a given year, the end of the season for LS > RS, or the beginning of the season for LS < RS is formally treated as switching time). It can easily be checked that after substituting m = 2 we arrive at Equation 3a if j = 1, and at Equation 3b if j = 2 .

Many perennial species reproduce only once a year after reaching maturity. It is optimal to release reproductive energy at the ends of growing seasons as long as the season is considered uniform with respect to offspring future. In such a case the parameter pj~ stands for the probability of surviving from the end of the jth season to the end of the ith season, and

Pji = qi-j (5)

where q stands for yearly survivability. The dependence of growth and reproduction schedules on the different parameters in Equa-

tion 4 is discussed later.

Numerical examples

Let us consider a species living for a maximum of 10 years, with production rate f (w) = H(w) = 0.05w °Ts, reproducing at the end of each growing season following maturation. Figure 2a shows the optimal fraction of consecutive seasons devoted to energy allocation to reproduction, when yearly survivability equals 0.41, 0.65 and 0.72, respectively. Growth curves resulting from such a schedule of energy allocation are represented in Fig. 2c. If yearly survivability is low (0.41), maturation takes place in the first year and the fraction of the season in which energy is allocated to reproduction increases gradually, reaching 100% in the two last years. As a result, the growth curve is step-like from the beginning, and final body size is small (about 1000 units, assuming that newborn size equals one). Under moderate yearly survivability, reproduction starts in the fourth year, and growth does not take place in the last three years. This

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gives a final body size of about 4000 units. When annual survivability equals 0.72, maturation occurs as late as in the fifth year, and the period of mixed growth and reproduction is limited to three years. Final body size is six times larger than under yearly survivability 0.42, and the growth curve is smooth until the fifth year.

Optimal fractions of the consecutive years used for growth under the same three survivabilities, but under production rate expressed as 0.05w °68, are given in Fig. 2b. The effect of mortality is similar to the previously considered case, but lower productivity and lower rate of its increase with body size result in earlier maturation, larger fraction of growing seasons devoted to growth, and much smaller body sizes, reaching a mere 200 units for the lowest survivability.

The above examples clearly show that growth rate and reproductive rate (both dependent on production rate) as well as mortality rate can strongly influence the pattern of growth and reproduction. Under the same function describing the dependence of production rate on body size, optimal body size may vary many-fold for different levels of mortality. The same is true for the same mortality and various production rates.

Growth and reproduction in perennials 219

The cost of not being optimal

As discussed by Kozlowski and Wiegert (1987), there are three ways by which an optimal schedule of growth and reproduction is reached in nature. First, one species may be replaced by another one better adapted to local conditions. Second, natural selection may shift the schedule closer to optimum. Third, an individual may vary its life history within the limits of its develop- mental plasticity in order to maximize an expected lifetime reproductive output.

Let us consider the third way. As shown in the explanation of the model, the optimal trajectory of growth can be found if the problem is considered backwards (for decreasing time), whereas an individual must adjust its life history to present conditions in the forward manner. Thus, optimization cannot be very precise. The physiological mechanisms of recognizing the necessary parameters and adjusting the schedule of growth and reproduction are undoubtedly limited, thus additionally lowering the accuracy of the adjustment attainable. It is important, therefore, to discuss the cost of being suboptimal instead of being strictly optimal. This will be only done for a species with a life span of two years, in order to make possible a graphical presentation of the problem.

Let us assume that the species under consideration releases reproductive energy at the ends of growing seasons, initial body size equals 4, probability of surviving from the beginning to the end of growing seasons is 0.82, probability of surviving winter is 0.65, growth rate f(w) is 0.05w °'75, and the rate of energy allocation to reproduction H(w) can be expressed by the same function as f(w). Isolines in Fig. 3 show an average lifetime amount of energy allocated to successful (completed) reproduction for different switching times from vegetative to reproductive growth in both years. At the point with coordinates 80 and 26 (80% of the first season and 26% of the second season used for growth) lies the maximum, 42.6 units of energy allocated to reproduction on average. Deviation from these 80 and 26% by several units of time in any year, or even in both years, results in only a small decline in reproductive energy. Substantial losses in fitness occur for coordinates far removed from the optimal ones. Thus it is not necessary to be optimal, especially in a changing environment for which optimal switching times may vary slightly in different years. It is sufficient to have switching times close to the optimal ones. The same is true for life spans longer than two years.

The sufficiency of being close to the optimum leads to a substantial variability in body size. For the example given above, the range of body sizes in the second year may vary from about 40 to 80 for those pairs of switching times which give a measure of fitness less than 1% smaller in comparison with the maximum value.

Field examples

It is relatively easy to extract data on growth curves, relations between fecundity and body size and mortality rates (which are very often size-independent for adult animals) from existing literature. Sometimes it is even possible to find such a set of data for the same population. It seems, therefore, that fitting these data to the model presented in this paper is an easy task. Unfortunately, the function f(w), describing growth rate in the model, and H(w), describing the dependence of reproductive rate on body size in the model, cannot simply be calculated from existing data. This is because both f(w) and H(w) represent potential rates when the entire energy is allocated to growth (in the case off(w)), or to reproduction (in the case of H(w)), not into both these processes simultaneously. Average yearly growth rates reported for age classes in which both growth and reproduction take place are thus underestimated, and the bias increases with age if the fraction of the growing season devoted to growth decreases with age. Annual fecundity

220 Kozlowski and Uchmahski

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increases with body size as a rule for animals in which egg or newborn size remains constant. This phenomenon results perhaps from (i) a physiologically determined rate of energy allocation to reproduction, increasing with body size, and (ii) an increase with age of the fraction of the season devoted to allocating energy to reproduction. Function H(w) occurring in the model represents the first component only, i.e. size-dependent physiological limitation of reproductive rate. Hence, function H should increase with body size much more slowly than dependence of fecundity on body size.

To overcome difficulties resulting from the lack of data on the fraction of consecutive seasons used for growth, the following procedure is applied. It is assumed that growth rate f(w) is described by the function aw b, with constant parameters throughout life. Similarly, it is assumed that reproductive rate (per day) can be expressed by the function cw a, with c and d constant during life. Then the optimality condition (Equation 4) is applied for a given set of parameters a, b, c, d. This gives the growth curve and the curve describing the dependence of fecundity on age predicted from the model, which can be compared against real field data. In the next step parameters a, b, c, d are adjusted to obtain the best fit of predicted curves to real data. The examples of such a fit are given in Figs 5, 6 and 8. In all the examples given further on, reproduction does not occur in the first year, and growth is negligible in the last year. This makes it possible to reduce the number of estimated parameters to two (b and d). For any value of b the

Growth and reproduction in perennials 221

corresponding value of a can be calculated, this giving an increase in body size observed in the field in the first year. Similarly, for any value of d the corresponding value of c can be calculated, which gives the prediction of fecundity in the last year (under the assumption that growth does not occur) exactly as reported from the field. It must be remembered that the growth curve between the first year and the second one, as well as fecundity in the last year must be omitted when we evaluate the goodness of the fit.

We are conscious of the weakness of our procedure - - the assumption derived from the model that the fraction of the growing season used for growth decreases with age is again implied in the verification procedure. Thus, the examples considered below show that our model is not contradictory to field data, not that it fits these data. To evaluate the model properly, it must be found whether the fraction of the season used for energy allocation to reproduction decreases with age according to the model's predictions. It was possible to estimate roughly the division of the growing season between growth and reproduction in the case of Iceland scallop (Fig. 7), and here the fraction of the growing seasons devoted to reproduction does increase with age.

The case of gastropod Conus pennaceus in Hawaii

Perron (1983) examined growth, fecundity and mortality in a natural population of the tropical gastropod Conus pennaceus in Hawaii. Shell lengths given by Perron for animals of different age are shown by circles in Fig. 4a, and fecundity by circles in Fig. 4b. Perron (1983) reports

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222 Kozlowski and Uchmahski

instantaneous mortality rate in the area studied equal to 0.536 per year, which is equivalent to the annual survivability 0.585. Using the procedure described above, we obtained the best fit of model 's predictions for growth rate f(w)= 0.038w °'38 mm per 1% of the growing season), and reproductive rate H(w) = 0.60w 1"24 eggs per 1% of the growing season. The growth curve and fecundity predicted from the model for different ages are given by the unbroken lines in Fig. 4a and b, respectively. According to the model, the entire first year should be used for growth, then the fraction of the season devoted to growth should decrease gradually from 82% in the second year to 21% in the sixth year, and to less than one percent for animals older than 15 years. The broken line in Fig. 4a shows the potential growth curve if reproduction were absent, and the broken line in Fig. 4b shows the potential fecundity in consecutive years if growth were absent.

As seen in Fig. 4, the agreement between the model 's predictions and field data is very good. Hence, we cannot reject the hypothesis that physiological abilities would permit individuals of Conus pennaceus to have a concave upward growth curve if it were not optimal to allocate some surplus energy to reproduction early in life because of heavy mortality.

The case of the Iceland Scallop Chlamys islandica in Norway

Vahl ( t98 ia ) gives the growth curve of Iceland scallop at Troms0 (circles in Fig. 5a). He also gives the amount of energy allocated yearly to reproduction by animals of different ages (circles in Fig. 5b). Using instantaneous mortality rate reported by Vahl (1981b), equal to 0.17 per year

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Growth and reproduction in perennials 223

(which gives annual survivability 0.84), it is possible to apply the same procedure as for Conus pennaceus. The best fit of the model's predictions to field data was obtained for f(w) = 0.024w °'42, and H(w)= 0.82w °84, where body size is expressed as shell height, and reproductive rate as calories per individual per 1% of the length of the growing season. The lines of best fit are shown in Fig. 5a and b, respectively (unbroken lines).

The growth curve and the amount of reproductive energy predicted from the model fit field data well, although worse than in the case of Conus pennaceus, especially for age classes 7-9. Total production reported by Vahl (1981a) increases monotonically from the beginning of life to age 5, then again for animals older than 8 years, whereas for age classes 6 to 8 there are large irregularities (at age 4 production equals 2783 cal, in the fifth year 5323 cal, which drops to 3934 in the sixth year, and even to 3376 in the seventh). Perhaps this is the reason for discrepancy between the model's predictions and Vahl's data in the age interval 7-9 years. It is assumed in the verification procedure that power functions f(w) and H(w) have the same parameters for the entire life span.

According to the model, maturation should take place in the fifth year. The fraction of the growing season devoted to reproduction should increase from 32% in the sixth year to 73% in the 12th, and 90% in the 15th. Vahl (1981a) reports that in the third year 5% of animals were mature, in the fourth 33%, in the fifth 70%, and 100% in the sixth year. Moreover, for age classes 3-5, mature animals were significantly larger than immature ones.

Vahl (1981a) reports the amount of energy allocated to growth and reproduction by animals of different age. If we assume that production rate when energy is being allocated to growth is the same as when energy is being allocated to reproduction, the fraction of energy allocated to growth throughout a year is a good estimator of the fraction of the growing season devoted to growth. Such estimates for scallops of different ages are given by open circles in Fig. 6, whereas the unbroken line denotes the predictions from the model. The agreement between the model's predictions and the estimated values is satisfactory, and it would be even better if reproductive

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224 Kozlowski and Uchmahski

growth were more costly than the somatic growth (see filled circles in Fig. 6 for the case when 1.0 calorie allocated to reproduction is equivalent to 1.5 calories allocated to growth).

The case of the arctic charr Salvelinus alpinus in northern Labrador

Dempson and Green (1985) report body length, the equation describing the dependence of fecundity on body length, and mortality for arctic charr in the Fraser River. Body lengths of females of different age are shown by circles in Fig. 7a, and fecundity expected from Dempson and Green's equation by circles in Fig. 7b. According to Dempson and Green (1985) annual mortality estimated from a catch curve for charr aged 8 to 14 years was 49%, whereas the estimation based on tag recovery was slightly lower. The authors conclude that annual mortality for this stock appears to be in the range 44--49%.

When an annual mortality of 49% was applied in the model, no reasonable fit of the model's predictions to reported data could be achieved. When annual mortality of 44% was assumed, the predictions were much better, but still not satisfactory. This is because for such a high mortality it

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Growth and reproduction in perennials 225

is optimal to mature in the second year (starting with very small clutches), not in the seventh, as in the case of charr from Labrador. To obtain satisfactory fit of the model's predictions to real data (unbroken lines in Fig. 7), it was necessary to assume additionally that at least 60% of the growing season must be devoted to reproduction if it is to be completed. This reasonable assumption, supported by very high fecundity at first reproduction, will be discussed in the next section.

Discussion

The model presented in this paper makes sense only if the conflict between growth and reproduction really does exist. The evidence of such a conflict is given by Bertness (1981): three species of hermit crab in Panama reproduce more intensively if there is a shortage of shells large enough to make growth possible. Reduction of growth was recorded in some tree species in the years of mass fruiting (Law, 1979a).

Perhaps the best example of the conflict between growth and reproduction is provided by Law (1979b). The grass Poa annua can live two years and reproduce in both years or in the second year only. There is a negative correlation between the number of inflorescences per plant in the first year, and plant size in the second year. The number of inflorescences in the second year is size-dependent and, therefore, the negative correlation between the number of inflorescences in the first and second year does also exist. The concept of trade-off between current reproductive effort and residual reproductive value is traditionally applied to explain this kind of correlation. It seems, however, that considering the relation between size and reproductive rate makes explan- ation more straightforward.

The model presented in this paper makes it possible to predict age and size at maturity, and the optimization of growth and reproduction in mature life, leading to sigmoidal growth curves as a rule. Both the shortage of resources and high mortality rate are among the factors promoting low age and small size throughout life. The occurrence of dwarf forms under poor trophic conditions is well known among, for example, fish or plants. The dependence of body size on mortality rate is less obvious, but numerous examples can be found in nature. Let us cite a few here.

Law et al. (1977) give an interesting example, supporting qualitatively the model presented in this paper. Individuals of Poa annua reproduced moderately in their first year in transient habitats in which winter survivability is low. This resulted in low reproductive rate in the second year. On the contrary, individuals living at permanent sites had a low reproductive rate in the first year, and a huge one in the second year. Winter survivability was very high in the permanent environment.

Reznick and Endler (1982) investigated several local populations of guppy Poecilia reticulata in Trinidad. They divided these populations into three groups, differing in the pressure of predatory fish. In the first group there was strong predation pressure on adult guppies, in the second one on juveniles, and in the third one mortality of both adults and juveniles was low. Exactly as predicted by the model, size at maturity was the lowest in the first group, and also more energy was allocated to reproduction by guppies from this group.

The relation between body size and mortality was also described for reptiles and amphibians. Tilley (1973) reports large altitudinal differences in size at maturity in the salamander Desmog- nathus ochropaeus. At low altitude (900 m), 50% of females of body length 34 mm were mature, whereas at high altitude (2000 m), 50% of females were mature at body length 41 mm. The density of predators, especially of predatory salamanders, was much lower at high altitude. Tilley (1973) connects this difference in size at maturity mainly with the inequality of age at maturity, although small differences in growth rate seem also to occur.

226 Kozlowski and Uchmanski

Inguanid lizards Sceloporus jarrovi from Arizona mature in the first year at the altitude of 1675 m, and in the second year at the altitude 2542 m where predation pressure is lower (Ballinger, 1979). Deferring maturation results in much higher size at maturity in the highland population.

Two modes of reproduction were considered in this paper, i.e. continuous reproduction and producing a single clutch at the end of the growing season. Other constraints can be incorporated into the model, as shown for the case of arctic charr. When reproduction occurs as a large clutch, a certain period is necessary for its completion. Hence the fraction of the growing season devoted to reproduction cannot be shorter than the lengthof this period. Such a fraction can, however, be longer because energy can be channeled to storage tissues which are later used up for producing reproductive tissues. This is the case for the Montane lizard Sceloporus scalaris in Arizona. A full clutch is completed during about 140 days in this species, and yearling females devote perhaps such a period to reproduction (Ballinger and Congdon, 1981). Older females contain much more fat before the egg production, and this store is almost completely exhausted during viteliogenesis. This suggests that energy is allocated to reproduction for longer by older females, if fat reserves are considered a component of reproductive energy.

In the case of arctic charr, it was possible to obtain a satisfactory fit of the model's predictions to field data only under the assumption that at least 60% of the growing season must be devoted to reproduction. It is interesting that raising this percentage above 70% leads to a qualitatively new result: it is optimal to divide the life span into three parts, i.e. the phase of immature life, the period in which it is optimal to reproduce every two years, and the period with reproduction appearing every year. Indeed, Sparholt (1985) reports that arctic charr in Greenland do not reproduce every year.

The model presented in this paper allows for quantitative predictions of ages at maturity and individual growth curves under various production rates and mortality intensities. Such predic- tions seem to be realistic for some species, as shown by the comparison against field data. It would undoubtedly be possible to find some species not growing according to the model's predictions. This is because additional constraints, not considered in the model, may be operat- ing. Sometimes they can be incorporated into the model, as in the case of arctic charr. The scope of the model's applicability is limited to species with size-independent adult mortality, and to populations at equilibrium. More complicated models are required for populations which do not meet these two assumptions.

Appendix I

Let us assume that the species under consideration can live no longer than two years. Its fitness is equivalent to the sum of the volumes of the two solids represented in Fig. 1, and denoted by V. The variable V reaches its maximum for such a pair tt, t 2 for which partial derivatives of V with respect to t t and t2 take the value zero, i.e:

dV1 ~V2 0 dtl + --~-1 = (la)

and simultaneously

0v2 Ot2 - 0 (lb)

The volume of each solid is a product of its height and the area of the base. ThUs:

VI = H(wt) Si (2a)

Growth and reproduction in perennials 227

and

I"2 = H(w2) $2 (2b) Keeping in mind that wl is a function of tl, we have:

dS 1 dV I _ d 1) f(wl ) S1 + H(wI ) dtl (3) dtl

where function

dw f ( w ) = -d7

represents growth rate at size w, and

dS~ dtl - l(tl)

because St is the area under the survival curve, i.e. it is equal to the integral of l(x). Furthermore, let us introduce the notation:

H'(w) - dH(w) dw

Thus Equation 3 takes the form:

dV~ dt~ = H ' (w t ) f(wl) SI - l(q) n ( w i ) (4)

The partial derivative of the volume of the second solid with respect to age at maturity tt can be expressed as:

H(w ~S____z ~V2 _ ~ 2) S2 = H(w2 ) 3tl (5) ~t~

But:

8S2_ 0 (6) ati

because of the assumption that mortality rate is size-independent. Furthermore:

~ n ( w 2 ) - all(w2) (7) 8tl 8t2

if we assume that growth rate depends on size, not directly on age. Under this assumption, body size attained at switching time in the second year will not change if growth period is extended by At in the first year, and shortened by the same At in the second year.

Taking into account Equations 6 and 7, Equation 5 reads:

~--~12- H'(w2) f(w2) S2 (8)

The derivative of the volume of the second solid with respect to switching time in the second year reads as follows:

~ 2 dS2 (9) 8V2 _ d 2) f(w2 ) S2 + H(w2) dt2 ~t2

228 K o z l o w s k i and Uchmahsk i

or

aV2_ n,(w2) f(w2 ) $2 -/(/2) H(w2) (10) &2

The total volume V, being a measure of fitness, reaches its maximum when the sum of derivative Equations 4 and 8, and simultaneously derivative Equation 10 take the value zero. This gives the pair of optimality conditions for switching from growth to reproduction in two years:

H ' ( w l ) f(wl) $1 = n ' ( w 2 ) f (w2) 52 = n ( w l ) l(tl) ( l la)

H'(w2) f (w2) $2 = H(w2) l(tz) ( l lb)

After dividing Equation l l a by l (q ) , and Equation l lb by l(t2), we obtain:

H'(w~) f ( w l ) E1 + H ' (w2) f(w2) p Ez = H ( w l ) (12a)

H'(w2) f (w2) E2 = H ( w z ) (12b)

which are equivalent to Equations 3a and 3b, respectively, in the paper.

Appendix II

Let us assume that the species under consideration can live no longer than m years. We are looking for the optimal switching time from growth to reproduction in the kth season. We are maximizing a volume V(q . . . . . tin), being a sum of volumes:

V(q , . . . , tin) = Vl( t l ) + . • • + Vk(tx . . . . , tk) + . • • + Vm(q . . . . . tm) (1)

each one representing an amount of energy allocated (on average) to reproduction in the ith season. The volume Vi depends on switching times in the year i and all the preceding years (q, . . . . ti), but does not depend on future switching times (ti . . . . . . . tm).

For the maximum value V ( q , . . . , tin), all the partial derivatives of V with respect to t t . . . . . tm must equal zero. Let us choose the derivative with respect to tk:

a v = v" av , (2)

Otk i~k ~tk

But we have:

Vi = H(wi) S i (3)

where

wi = w(tl + . . . + ti) (4)

(which means that body size in the ith year depends on the length of the sum of the growth periods in the years 1 , . . . , i) and

Si = Si(ti) (5)

(which reflects the assumption that mortality is size-independent). Thus:

~ V i _ dH(wi ) d w i S. + H(wi ) dSi ~tk dwi d t , ' ~ (6)

Growth and reproduction in perennials

But:

d w i _ dwi dtk dti

which results from Equation 4. Furthermore:

d S ~ = { O for / :/: k

dt k l(tk) for i = k

Thus:

where

and

229

H'(wi) - dH(wi) dwi

dwi f ( w i ) - dti

Substituting Equation 7 to Equation 2 we obtain:

~V m - - = E n ' (wi ) f (wi ) Si - l(tk) H(w~) (8) btg i=k

Derivative Equation 8 equals zero if:

m

H'(wi) f(wi) S i = l(tk) H(Wk) (9) i=k

After dividing both sides of Equation 9 by l(tk) we obtain:

m Si E H'(w,) f(wi) ~ = H(w~) (10)

i=k

The ratio Si/l(tk) represents the expected (at switching in the kth year) number of days devoted to energy allocation to reproduction in the ith year. This ratio can be expressed as:

Si l(tk) -- Ei Pki (11)

where Ei stands for expected (at switching in the ith year) time of energy allocation to reproduc- tion in the ith year, and Pki for the probability of surviving from switching in the kth year to switching in the ith year. After substituting Equation 11 to Equation 10 we have:

r r l

~. H'(wi)f(wi) E, Phi = H(wk) (12) i=k

which is equivalent to Equation 4 in the paper.

f o r i . k

at k l(tk) for i = k (7)

230 Kozlowski and Uchmahski

Acknowledgements

We are grateful to Adam Lomnicki, Nils Ch. Stenseth and two anonymous reviewers for helpful comments, and to Danuta Padley for correcting the manuscript. This paper was supported by grant CPBP 04.03 of the Polish Academy of Sciences to the senior author.

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