© 2004 by The Society for Integrative and Comparative Biology
Can Optimal Resource Allocation Models Explain Why Ectotherms Grow Larger in Cold?1
owski2,1
ski1
ko1
1 Institute of Environmental Sciences, Jagiellonian University, Gronostajowa 7, 30-387 Kraków, Poland
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Basically all organisms can be classified as determinate growers if their growth stops or almost stops at maturation, or indeterminate growers if growth is still intense after maturation. Adult size for determinate growers is relatively well defined, whereas in indeterminate growers usually two measures are used: size at maturation and asymptotic size. The latter term is in fact not a direct measure but a parameter of a specific growth equation, most often Bertalanffy's growth curve. At a given food level, the growth rate in determinate growers depends under given food level on physiological constraints as well as on investments in repair and other mechanisms that improve future survival. The growth rate in indeterminate growers consists of two phases: juvenile and adult. The mechanisms determining the juvenile growth rate are similar to those in determinate growers, whereas allocation to reproduction (dependent on external mortality rate) seems to be the main factor limiting adult growth. Optimal resource allocation models can explain the temperature-size rule (stating that usually ectotherms grow slower in cold but attain larger size) if the exponents of functions describing the size-dependence of the resource acquisition and metabolic rates change with temperature or mortality increases with temperature. Emerging data support both assumptions. The results obtained with the aid of optimization models represent just a rule and not a law: it is possible to find the ranges of production parameters and mortality rates for which the temperature-size rule does not hold.
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INTRODUCTION |
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SYNOPSISINTRODUCTIONIF BERTALANFFY'S MODEL IS...IF BERTALANFFY'S MODEL IS...THE EFFECT OF TEMPERATURE...DISCUSSIONReferences |
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There is a very clear pattern in nature: most ectotherms grow slower in cold but reach a larger size than at high temperature. This effect of temperature differs from the effect of limited resources, growth retardation accompanied by smaller adult size (for review see Atkinson, 1994
; Atkinson and Sibly, 1997; Angilletta and Dunham, 2003; Angilletta et al., 2004). Atkinson (1996) coined the term "temperature-size rule" for this effect of temperature. Larger size in cold is attained by a prolonged growth period overcompensating slow growth. The proximate reason for larger size is known for many species: animals reared in cold have larger cells, as in Drosophila (e.g., Robertson, 1959; Partridge et al., 1994; McCabe et al., 1997), Caenorhabditis elegans (Van Voorhies, 1996) and the rotifer Synchaeta pectinata (Stelzer, 2002; the author does not mention cell sizes, but their change with changing body size seems unavoidable because rotifers have a set number of cells). Van der Have and de Jong (1996) provide a proximate biophysical model of this phenomenon based on the assumption that the temperature coefficients for cell growth and cell division differ. It seems, however, that cold must also be an ultimate factor for larger cells, because it is difficult to imagine that mechanisms compensating the increase of body size associated with larger cells would not evolve if larger body size in cold decreased fitness. Smaller cell numbers in cold would provide such a mechanism. Moreover, it is known that sometimes increase of size is caused by simultaneous increase of both cell size and cell number (James et al., 1995). Numerous examples of geographical clines also mimic thermal reaction norms (e.g., James et al., 1995; Ashton, 2002; Ashton, 2004). Huey et al. (2000) revealed that size of Drosophila subobscura introduced to the New World diverged genetically after two decades along a latitudinal cline at the speed of 0.22 standard deviations per generation. Thus the prevalence of temperature-size rule and different proximate mechanism leading to size increase suggest that larger adult size in cold is adaptive.
If larger size in cold is adaptive (see Partridge and Coyne, 1997 for a discussion of this), the temperature-size rule requires explanation. Bergmann's classic explanation, the adaptiveness of decreased heat dissipation through the reduction of surface-to-volume ratio in cold, cannot be applied to ectotherms. Alternatively, Berrigan and Charnov's (1994) explanation is based on several assumptions: (i) growth follows Bertalanffy's (1957) curve even when energy is not drained for reproduction, (ii) fecundity increases with size, and (iii) juvenile and adult mortality rates are constant. For the temperature-size rule to appear, it is necessary to assume that the asymptotic size and the growth coefficient in Bertalanffy's equation are negatively correlated. Berrigan and Charnov (1994) invoke field data supporting such a correlation, but they call the appearance of a tradeoff between these two parameters a major puzzle. Using a bioenergetic equivalent to Bertalanffy's model, Perrin (1995) claims to have a solution to the puzzle: the negative correlation between the parameters can be obtained if metabolism is more sensitive to temperature than resource acquisition. Perrin presents a numerical example in which Bertalanffy's growth coefficient is larger at high temperature and asymptotic size is larger at low temperature. However, a careful examination of Perrin's example reveals that growth rate measured as body mass change in time is in fact higher in cold. Perrin (1995, p. 138) assumes such values consciously, as he states: "However, the relevant parameter here is not the growth rate (which measured the size increase per unit time), but the growth coefficient k (which relates to the time needed to reach the asymptote)." Under higher temperature the low real growth rate in Perrin's example translates to a higher Bertalanffy's growth coefficient because of strong suppressing by a low asymptote. Thus Perrin's solution can be restated as follows: Animals grow larger in cold because we assume that they can grow to larger sizes (we assume a higher asymptote). This is not an explanation of the empirical temperature-size rule, under which larger size in cold is obtained under a lower real growth rate. A rigorous evaluation of the Bertalanffy-Perrin model by Angilletta and Dunham (2003) also shows its very limited explanatory power. We may conclude, therefore, that the temperature-size rule still remains unexplained.
In the next section we discuss why Bertalanffy's growth equation is not a proper framework to study life-history optimization, and especially the temperature-size rule. Then optimal resource allocation models are discussed, and finally the models are applied to explaining the temperature-size rule. In the Discussion we give some data supporting the assumptions necessary to obtain the optimality of the pattern resembling the temperature-size rule.
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IF BERTALANFFY'S MODEL IS CORRECT, WHERE DO CHILDREN COME FROM? |
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SYNOPSISINTRODUCTIONIF BERTALANFFY'S MODEL IS...IF BERTALANFFY'S MODEL IS...THE EFFECT OF TEMPERATURE...DISCUSSIONReferences |
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Von Bertalanffy (1957)
assumed that the rate of change in body weight is the difference between the rates of anabolism (tissue production) and catabolism (tissue dissipation), with both the anabolism and catabolism rates being allometric functions of body weight w
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To make the function (1) integrable, he assumed the exponents 
and 
as follows:
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According to Bertalanffy, the diminishing (with size) difference between anabolism and catabolism slows down growth with age; growth finally stops when catabolism balances anabolism (Fig. 1A). For body mass, after integration of equation (2) we obtain:
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where 
= (
/ß)3, k = ß/3 and 
= 1 – 
where w0 represents initial body mass. The equivalent Bertalanffy's growth curve for body length is most often given in the form
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where lt represents body length at age t, l
asymptotic length, k the growth coefficient and t0 the hypothetical (negative) age at which length equals zero. The equivalent form of equation (4) is
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where c = 1 – l0/l.
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There is a logical problem with Bertalanffy's model: how can natural selection favor organisms that keep growing beyond the size at which the difference between their anabolism and catabolism rates is at maximum and continue to grow to the point where the difference equals zero (Day and Taylor, 1997
; Czarnoski and Kozowski, 1998)? Production of offspring tissue, like the production of the animal's own tissue, requires a surplus of anabolic over catabolic processes. If the ultimate purpose of organisms is to maximize their reproductive success, why should they reach the point where reproduction is lower than the maximum possible or is even not possible at all? To keep Bertalanffy's model logically consistent, it is necessary to restrict it to tissues not involved in reproduction. In such a case the model becomes strictly descriptive and explains nothing. The logical problem with Bertalanffy's model is even more visible if Winberg's (1956) energetic equivalent is used, as applied by Perrin (1995):
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where A and R represent the resource acquisition rate and metabolic rate. Growth must stop before A and R are balanced, because the difference is converted either to offspring production in a form of direct and metabolic costs of tissue synthesis or to metabolic costs of mating.
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IF BERTALANFFY'S MODEL IS NOT CORRECT, WHY DO SO MANY SPECIES HAVE BERTALANFFY-LIKE GROWTH TRAJECTORIES? |
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SYNOPSISINTRODUCTIONIF BERTALANFFY'S MODEL IS...IF BERTALANFFY'S MODEL IS...THE EFFECT OF TEMPERATURE...DISCUSSIONReferences |
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General modeling framework
It has been known since the work of Taylor et al. (1974)
that reproductive value at birth,
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should be maximized by natural selection. It depends on vital statistics: the probability of surviving to age x, l(x), and fecundity at age x (female offspring only taken into account), f(x), defined for the entire life span. The value of future offspring production must be discounted to the present by the expression e–rx, where r represents the real population growth rate. To find the optimal life history we must take into account the tradeoffs between l(x) and f(x) at different ages. In optimal resource allocation models a change of one parameter changes the other one through alterations of an organism's state: for example, increased reproduction impairs body size and limits future reproduction.
The logic behind allocation models is simple: organisms are limited by their available resources and must allocate them optimally to competing functions in order to maximize fitness. Although different resources can be considered, it seems reasonable to focus on energy. Energy may be limited either by resource scarcity or by the ability of an organism to process food, whatever comes first (Weiner, 1992). Resource scarcity may mean either a low absolute amount of food or the need to avoid the risk of foraging in a place rich in food but dangerous (Werner and Anholt, 1993). The energy stream coming into an animal may be used to meet maintenance costs, may be retained as tissue growth or storage, or may be channeled to reproduction. This complex problem of energy management is usually simplified in allocation models: (i) no distinction between external and internal limitations on energy input is made; (ii) it is assumed that maintenance costs are paid first; and (iii) it is assumed that surplus energy, equivalent under some assumptions to production rate P(w), is a function of body size w as a result of assumptions (i) and (ii). Allocation models are built to find proportions of surplus energy that should go to growth, reproduction, storage, repair, etc., at each age. Repair is often not considered explicitly. In fact, allocation to repair would be measured by the physiological ecologist mostly as heat production and part of the energy content of excreta, because repair means tissue replacement and maintenance of the immune system. Explicit inclusion of allocation to repair, not done in this paper, can lead to interesting results, especially concerning the relationship between growth rate and maximum life span (Cicho, 1997; Cicho and Kozowski, 2000; see also section on switching curves).
The optimal proportions of surplus energy going to different sinks can change during life. Although the number of energy sinks considered is arbitrary, two of them cannot be removed: reproduction and growth. This is because reproductive allocation throughout life defines fitness, and growth changes body size, on which the production rate depends. Such a simple model is analyzed in this paper.
The fitness measure is another critical point in allocation models (Mylius and Diekmann, 1995). We must remember that reproductive value at birth, and not the Malthusian parameter, is a proper measure of fitness. The parameter r in (7) is often improperly interpreted as the solution of the Euler-Lotka equation (after the right-hand side of equation [7] is set to one) obtained under given l(x)'s and f(x)'s; such a solution is called the Malthusian parameter or intrinsic growth rate. In fact r in (7) represents the real population growth rate, which can differ from the Malthusian parameter if density-dependence is not taken explicitly into account as is usually the case in life history optimization models (Houston and McNamara, 1992; Kawecki and Stearns, 1993; Kozowski, 1993). Because many populations are close to the stable population size needed in the long run for the evolutionary process, real r equals zero, although the Malthusian parameter is positive; the excess of individuals migrates or is eliminated another way. If density-dependence operates early in life whereas we are interested in optimal traits late in life such as age at maturity or energy allocation in mature individuals, maximization of reproductive value at birth simplifies to maximization of the net reproduction rate R0.
A large part of life-history optimization models is based on maximization of the Malthusian parameter. Except for very specific situations (see Kozowski, 1999) this must lead to erroneous results. Population size cannot increase in a period long enough for evolution of any trait. R0 is by no means a universal measure of fitness, but it covers a much broader range of ecological situations than r.
Switching curves
Models considering optimal allocation of energy to growth and reproduction usually produce switching curves that divide a plane defined by age and size axes into two parts: it is optimal to grow below the switching curve, and to reproduce above it (Fig. 2A). The switching curve in an aseasonal environment is formed by a straight line parallel to the age axis if there is no aging (Fig. 2B, upper row, left column; e.g., Kozowski and Wiegert, 1987; Perrin and Sibly, 1993) or it is a monotonically decreasing curve for animals with mortality increasing with age (Fig. 2B, upper row, middle column). In the latter case, the switching curve goes down with age because life expectancy decreases toward the end of life. Thus determinate growth is optimal in an aseasonal environment because once an animal has crossed the switching curve, it has no chance to be below it in the future. The picture is changed by repair (Cicho, 1997). Resources used up for repair postpone the aging and in this way move the switching curve up (Fig. 2B, upper row, right column). But repair is not for free: it drains energy from growth, slowing it down. Thus the growth curve crosses the switching curve later. Slower growth may last longer and produce an animal larger than in the case without repair as illustrated in the figure. It is also possible for growth to be so retarded that the growth curve crosses the switching curve at a lower size than in the case without repair. Although the allocation to repair always delays reproduction, the effect of repair on adult size depends on repair efficiency (Cicho, 1997). Under high efficiency, aging is postponed substantially at low cost (growth not slowed much), which gives increased adult size.
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The type of growth depends on the form of the switching curve. In the cases described up to now an instantaneous and irreversible switch from growth to reproduction occurred, leading to a growth pattern often called exponential, although in fact it should be described by a power equation with age raised to a power in the range 3–4. If size is expressed in energy units, the growth rate is equivalent to the production rate because all production is allocated to growth before maturation.
There is a possible exception to the rule that optimal switching from growth to reproduction is instantaneous and irreversible in an aseasonal environment. It appears when the switching curve is humped and the growth curve would cross it to the left of the peak under instantaneous switch, as shown in Figure 7A (Perrin and Sibly, 1993; Perrin et al., 1993). In such a case it is optimal to have a period of pure growth, then a period of mixed growth and reproduction, with the fraction of energy allocated to reproduction increasing steadily during growth, then finally a period of pure reproduction. Reproduction starts before the growth curve reaches the switching curve, then growth follows a so-called singular arc, to cross the switching curve just at its peak (Fig. 7A). The switching curve should be interpreted in such a case as a line dividing the age-size plane into a part where only reproduction occurs and a part where growth occurs, possibly accompanied by reproduction. The necessary (but not sufficient) condition for the relation between the growth curve and the switching curve to lead to optimality of the singular arc is for both the production rate and mortality rate to increase with size (Perrin and Sibly, 1993; Perrin et al., 1993; for the sufficient condition consult these papers), as is the case, for example, in cladocerans endangered by fishes catching larger individuals more efficiently. The resulting growth curves have a phase of rapid growth, which slows down after switching to mixed allocation. The shape of the curves may resemble Bertalanffy's curve, but the reason for such a shape is completely different than in his model.
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Seasonality complicates the shape of the switching curve (Fig. 2B, lower row). From the point of view of resource allocation, seasonality means that a year is divided into a favorable part when growth and reproduction are possible (called summer hereafter), and an unfavorable part when the energy balance equals zero or is even negative (winter). If the energy balance is negative in winter, storage should be explicitly considered. To simplify a model, it may be assumed that summer is shortened by a period necessary to gain resources to balance winter expenditures.
Annual organisms have monotonically decreasing switching curves, reaching zero at the end of summer (Fig. 2B, lower row, left column; e.g., Vincent and Pulliam, 1980; Kozowski and Wiegert, 1986). This is because it is optimal to use all surplus energy for reproduction (producing diapausing propagules) at the end of the season beyond which no life is expected.
Seasonality makes switching curves nonmonotonic if life extends to longer than one season: the curve decreases during summers and increases during winters (Kozowski and Uchmaski, 1987; Kozowski, 1996a; Kozowski and Teriokhin, 1999). The switching curve never falls to zero at the end of the summer (except at the end of life; a sharp end of life is defined in models only to make them mathematically tractable) because for iteroparous organisms there is always some chance to survive and reproduce in the future (Fig. 2B, lower row, middle and right columns). The elevation of the switching curve at the end of the summer is positively related to this chance, or more precisely to the residual reproductive value. The teeth on the curve mean that an organism that has crossed the switching curve and reproduced is likely to be back below the switching curve at the beginning of the next season, which makes growth optimal again. Such switching curves lead to either indeterminate growth that approaches an upper limit asymptotically (Fig. 2B, lower row, middle column), or to slowing-down growth that stops completely after several years (Fig. 2B, lower row, right column). Which of these solutions is optimal depends on the mortality schedule. If the maximum life span is unlimited and mortality is age-independent, the teeth on the switching curve are of the same shape and size during the entire life, and asymptotic growth is optimal. If the life span is limited and/or mortality increases with age, the teeth go down, becoming smaller toward the end of life. This leads to limited final size, which is attained after some years of mixed growth and reproduction. In practice, it may be often difficult to distinguish between asymptotic growth, with very little growth late in life, and growth ending after several years of mixed growth and reproduction. Both forms of growth are called indeterminate here.
Optimal allocation of energy in an aseasonal environment
According to the optimal resource allocation models, the size at which it is optimal to switch from growth to reproduction in an aseasonal environment depends on the production rate P(w) and mortality rate m(w), and more exactly on their dependence on body size w measured in energy units (Kozowski, 1992; Perrin and Sibly, 1993). If the production rate is the difference between the resource acquisition and metabolic rates according to the equation
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optimal size exists if b1 
b2. If b1 < b2, the production rate curve is humped. For size-independent mortality it is always optimal to stop growing at a point well before the maximum production rate is reached, unless the mortality rate is negligible. In the case of mortality rate decreasing with size, not considered in this paper, it may be optimal to grow even beyond the size maximizing the production rate, in order to escape high mortality (Kozowski, 1996b). The optimal size at maturity (at switching from growth to reproduction) is the one satisfying the following condition:
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(Perrin and Sibly, 1993; Kozowski, 1996b). Equation (9) is a necessary but not sufficient condition for optimal size. Lifetime reproductive output will be maximized at the optimal size wopt satisfying condition (9) only if the function P(w)/m(w) is convex (concave downward) in the vicinity of wopt (Kozowski, 1996b; Sebens, 2002). For size-independent mortality m, condition (9) simplifies to the form
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This condition together with production equation (8) is exploited in the next subsection to show the effect of temperature on optimal body size.
Optimal allocation of energy in a seasonal environment
The optimality of indeterminate growth not ceasing at maturation in a seasonal environment was shown by Kozowski and Uchmaski (1987) and Kozowski (1996a) for the very specific case of constant mortality with reproduction occurring just at the end of the season. Only under such assumptions was it possible to use basic calculus to solve the dynamic problem. Kozowski and Teriokhin (1999) studied the problem more rigorously using the Pontryagin Maximum Principle (Pontryagin et al., 1962). Their model makes it possible to study optimal allocation of resources under mortality that changes with both the age of the organism and the season.
According to Kozowski and Teriokhin's (1999) model, first reproduction appears at different ages and sizes, depending on season length, mortality and the size-dependence of the production rate. Furthermore, year after year following maturation a larger and larger fraction of resources is allocated to reproduction, which means that growth slows down. As a result, the growth curve closely resembles Bertalanffy's growth curve, albeit for reasons completely different from those suggested by von Bertalanffy (1957). The difference between these two mechanisms is shown in Figure 1. As presented by Kozowski (1996a), the resulting Bertalanffy's-like growth curves show so-called Beverton-Holt (1959) patterns: the mortality rate is positively correlated with Bertalanffy's growth constant, length at maturity is positively correlated with asymptotic length, and the growth constant is negatively correlated with asymptotic length. Such patterns have been described for fishes (Beverton and Holt, 1959; Beverton, 1992) reptiles (Shine and Charnov, 1992; Shine and Iverson, 1995) and zebra mussels (Czarnoski et al., 2003).
The important feature of indeterminate growth is that part of it is realized after maturation. The proportions of growth before and after maturation depend on mortality. If mortality is low, most growth is fulfilled before maturation; if mortality is heavy, most growth is fulfilled after maturation. Thus the shapes of the growth curves vary, as shown in Figure 3 for natural populations of zebra mussels. Because mortality also affects maximum size, a negative correlation between Bertalanffy's growth coefficient and asymptotic size results, as shown both in modeling studies (Kozowski, 1996a) and real populations (Czarnoski et al., 2003).
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The pattern of growth resulting from optimal resource allocation may lead to several misinterpretations. Before maturation, the growth rate is defined only by the production rate. After maturation, the growth rate is defined by both the production rate and the optimized fraction of surplus resources allocated to growth. Bertalanffy's growth curve fitted simultaneously to the prereproductive part of life and the period following first reproduction has a purely phenomenological meaning. Thus Bertalanffy's growth coefficient k in equation (3) or (4) also is purely phenomenological, and does not refer to real growth potential. This means that all bioenergetic considerations of the dependence of this parameter on temperature are completely irrelevant. Specifically, the negative correlation between the growth coefficient and asymptotic size, as well as the intercrossing of growth curves, cannot be used to support Berrigan and Charnov's (1994)
explanation of the temperature-size rule, because they assume that growth follows Bertalanffy's equation irrespective of resource allocation. When the temperature-size rule is discussed, the effect of temperature on the growth rate (being a derivative of the growth curve) in the region of pure growth must be considered.
Why can the growth curve have an inflexion point in an aseasonal environment?
Although some determinate growers, for example many holometabolous insects, have "exponential" (power in fact) growth (e.g., Sevenster, 1995), others have growth curves with an inflexion point. There are several possible reasons of slowing down growth. First of all, growth is a complex phenomenon, and it is difficult to imagine that it can stop immediately and that biochemical production machinery running at full speed can be instantly switched to reproduction. Growing tissues usually have different features than mature tissues, especially different water content. Tissue maturation may be responsible for gradual slowing of growth (Ricklefs, 1979; Konarzewski, 1988; Ricklefs, 2003), which may lead to logistic, Richard's, or Bertalanffy's growth curves. Furthermore, developing reproductive structures may not be able to absorb the entire production previously used for growth. Under such a constraint, it may be optimal to start reproduction earlier and to continue growth together with reproduction (Kozowski and Zióko, 1988); the resulting growth curves also have inflexion points. When bioenergetic limits for growth are studied, only periods with growth not limited by tissue maturation or reproductive allocation should be taken into account. Singular arc, described earlier, can also cause growth to continue after maturation.
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THE EFFECT OF TEMPERATURE ON GROWTH RATE AND ADULT SIZE ACCORDING TO OPTIMAL RESOURCE ALLOCATION MODELING |
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The effect of temperature in an aseasonal environment under instantaneous switch from growth to reproduction
It was shown earlier that it is optimal in an aseasonal environment to switch completely and irreversibly from growth to reproduction when condition (9) is satisfied. The condition contains the derivative of the quotient for the production/mortality rates with respect to body size. If mortality is size-independent, condition (9) simplifies to condition (10), saying that at optimal adult size the derivative of the production rate with respect to body size expressed in energy units must be equal to the mortality rate. If an optimal resource allocation model can explain the temperature-size rule, temperature must work through modification of the mortality and/or production components of the optimal condition (10).
There exist data indicating that the body size exponents for the resource acquisition and metabolic rates change with temperature (see Discussion). Let us now assume that constants a1 and a2 of production equation (8) increase with temperature, but exponents b1 and b2 decrease. An example is shown in Figure 4. The parameters change according to the rules shown in the insert of Figure 4. As a result, the curves describing the production rate as a function of body mass (expressed in energy units) cross at the size of roughly 250 units: below this size the production rate increases with temperature, but above it decreases with temperature (Fig. 4). Reaction norms of optimal adult sizes, dependent on both the production rate and mortality rate, are shown in the figure. For different mortality rates we get different shapes of the reaction norms of adult sizes: for high mortality m = 0.0009, adult size slightly increases with temperature; for m = 0.0008, adult size is almost invariant (not shown), then for lower mortality rates, between 0.0007 and 0.0005, adult size decreases more and more with temperature. If the mortality rate is further decreased, the sizes for low temperature are shifted to the right of the production rate intercrosses. Although the rule of the lower size in higher temperature is still fulfilled, in fact the production rate (and growth rate as a result) is higher for low than for high temperature, which is against the rule that larger sizes are attained under lower growth rates in cold. Thus there exists a broad but limited range of mortality rates for which the temperature-size rule is fulfilled. Such a range of mortality rates can always be found if production rate increases with size more rapidly at low than at high temperatures. Obviously such a pattern of production curves calls for biological explanations (see Discussion). However, production efficiency decreases with temperature in the model; in ectotherms this is the exception rather than the rule (Angilletta and Dunham, 2003).
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Let us assume now that the constant a1 of the resource acquisition rate increases and the constant a2 of the metabolic rate decreases with temperature, but exponents b1 and b2 behave the opposite way, according to the rules shown in the insert in Figure 5. (Although a2 decreases with temperature, the metabolic rate increases with temperature because of accompanying changes in b2.) The curves describing the production rate as a function of body mass intercross, but unlike the previous example along a broader range of sizes (Fig. 5). For high mortality m = 0.0009, adult size increases with temperature; for m = 0.0007 and m = 0.0006, adult size increases and then decreases with temperature; and for m = 0.0005, adult size monotonically decreases with temperature. Thus the temperature-size rule is fulfilled not only for a limited range of mortality rates, but also, at some mortalities, for a limited range of temperatures. Interestingly, production efficiency increases with temperature in this example.
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Bertalanffy (1960)
assumed that the parameters and ß in (1), but not the exponents, change with temperature. For bioenergetic model (8) it is equivalent to changing constants a1 and a2 with temperature (insert in Fig. 6). Under such an assumption the production curves do not cross if Q10 of the assimilation and metabolic rates are the same (Fig. 6). The temperature-size rule can be caused in such a case by the temperature-dependence of the mortality rate, as suggested by Atkinson (1994) and Atkinson and Sibly (1997). The insert in Figure 6 shows three curves depicting the increase of mortality with temperature. The reaction norm of adult size to temperature depends on the rate at which mortality rises with temperature. If this increase is slow, that is, with Q10 lower than for the bioenergetic parameters, optimal sizes increase with temperature; if the increase is moderate, that is, with Q10 the same as for the bioenergetic parameters, optimal sizes are temperature-invariant; but when mortality increases with temperature rapidly, that is, with Q10 higher than for the bioenergetic parameters, optimal sizes become larger in cold (Fig. 6). Q10 for the assimilation and metabolic rate constants applied in this numerical example, roughly 1.3, are realistic but low (Angilletta et al., 2004). For numerous species with higher assimilation and respiration Q10, mortality rate Q10 also must be higher to make the decrease of size with temperature optimal.
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There is no reason to assume that the temperature-size rule is caused either by the exponents of the production equation changing with temperature (as in Figs. 4 and 5) or by mortality changing with temperature (as in Fig. 6). It is possible for both mechanisms to work simultaneously, which makes the rule even more likely to appear, because the temperature sensitivity of mortality and bioenergetic parameters needed to produce a size decrease with temperature may be lower than when only one mechanism is engaged.
The effect of temperature in an aseasonal environment under a graded switch from growth to reproduction
As discussed earlier, an increase of both the production rate and mortality rate with body size is a necessary condition for the optimality of a graded switch from growth to reproduction in an aseasonal environment. The production rate is modelled by equation (8), with the temperature-dependence of parameters as shown in the insert in Figure 7B (exponents b1 and b2 decrease and constants a1 and a2 increase with temperature). Mortality is modelled by the equation
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where the first term represents size-dependent mortality caused by predators, and the second reflects mortality caused by aging, with T describing the maximum lifespan (Kozowski and Teriokhin, 1999). It is assumed that maximum lifespan T decreases linearly with temperature, reaching 496 units at 5°C, and 400 units at 25°C. Two cases of size-dependent mortality are studied: (i) with q invariant with respect to temperature and equal to 0.01, and (ii) with q increasing linearly with temperature and equal to 0.0085 at 5°C and 0.01 at 25°C. Optimal resource allocation in this case can be found using dynamic programming (Bellman, 1957).
Under the range of parameters studied, growth trajectories always have three phases: pure growth, mixed growth and reproduction (with slower growth as a result), and pure reproduction. The insert in Figure 7B shows the growth trajectories and duration of these phases in different temperatures. The growth rate in the first phase always increases with temperature, but prolonged growth in cold makes larger size optimal, expressed both as size at maturity and as final size (Fig. 7B). Thus the temperature-size rule can also be satisfied in an aseasonal environment under a graded switch.
The effect of temperature in a seasonal environment
To test how the proposed temperature-dependence of the production parameters shapes life histories in a seasonal environment, we use Kozowski's (1996a) resource allocation model. Briefly, the model organisms repeatedly experience summers of length 200 days (T) in which growth and reproduction is possible, followed by 165-day winters with zero production rate. Yearly survival is defined by the summer and winter mortality rates (ms, mw) according to the equation p = exp(–msT)exp[–mw(365 – T)]. The production rate at body size w expressed in energy units is defined by equation (8). The model finds the maximum body size wmax, beyond which it is never optimal to grow, and the lifetime pattern of resource allocation to growth and reproduction in previous seasons.
Figure 8 outlines the shape of the optimal growth curves (thick lines) generated from the model for three temperatures, and the potential growth trajectories if all production was devoted to growth only (thin lines). Temperature was assumed to affect parameters a1, b1, a2, b2 of production in the same way as in the example in Figure 4 for aseasonal conditions. The effects of temperature on the optimal growth pattern were analyzed across the mortality gradient (Fig. 8A, B, C). In general, a lower mortality rate extends the juvenile period, leading to larger size at maturity wmat and final size wmax, and it enlarges the fraction of final size attained at maturity (larger wmat/wmax). Under low mortality (Fig. 8A), a rising temperature, despite accelerating the growth rate in the juvenile period, selects for earlier maturation at smaller size and smaller final size. The fraction of final size attained at maturity rises with temperature; the ratio wmat/wmax is equal to 0.57, 0.59 and 0.66 at 5, 15 and 25°C. The figure clearly shows that rising temperature decreases the sizes wmat and wmax, leading to the "temperature-size rule," not because growth potential decreases with size but because increasingly fewer resources are devoted to growth to maximize the expected lifetime allocation to reproduction. Under high mortality (Fig. 8C), the temperature effect on life histories is just the reverse of the one observed under low mortality. Interestingly, under medium mortality (Fig. 8B), the temperature-size rule is not fulfilled for size at maturity, but does hold for final size.
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SYNOPSISINTRODUCTIONIF BERTALANFFY'S MODEL IS...IF BERTALANFFY'S MODEL IS...THE EFFECT OF TEMPERATURE...DISCUSSIONReferences |
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The use of Bertalanffy's (1957)
model of growth has been one of the main obstacles to a proper understanding of the factors responsible for the ubiquity of the temperature-size rule. This obstacle vanishes if Bertalanffy's growth equation is applied only as a phenomenological description of indeterminate growth. Growth pattern resembles Bertalanffy's curves because of optimal allocation of energy to growth and reproduction under some circumstances. Another obstacle lies in ignoring the basic fact that the adult size of an organism results not only from its bioenergetic parameters (the rates of anabolism and catabolism, or the rates of resource acquisition and metabolism), but also from the mortality rate: under the same physiological parameters, optimal size is lower when mortality rate is higher.
Taking that organisms are constantly selected for maximization of their reproductive output, we have shown that the allocation models can explain the adaptiveness of the decrease of size with temperature. This result was obtained for an aseasonal environment leading to determinate growth (Figs. 4, 5, 6), for an aseasonal environment with the mortality and production regime favoring growth after reproduction (Fig. 7), and finally for a seasonal environment causing optimality of indeterminate growth (Fig. 8). Importantly, the models for an aseasonal conditions extend also to the case of the short-lived multivoltine animals living in a seasonal environment. Furthermore, the models promise a more general explanation for life history reactions of ectotherms to temperature including the reported exceptions to the ‘temperature-size rule’ (Atkinson, 1994; Ashton, 2004). The models show that under a given selection context adaptive life history reactions follow the rule, whereas in other selection conditions opposite reactions to temperature are evolutionarily beneficial.
Optimization models predict a decrease of size with temperature under some temperature-independent mortality regimes, provided that the curves describing the dependence of production rate on body size cross. Such crossing can be obtained if the thermal sensitivity of the metabolic rate constant in equation (8) is higher than the thermal sensitivity of the resource acquisition rate constant. It is easier to meet the requirements for the optimality of size decrease with temperature if not only the constants but also the exponents of the resource acquisition and metabolic rates in (8) change with temperature (Figs. 4, 5, 7, 8). Available evidence shows that the body size exponents for production components do change their values with ambient temperature. In general, rising temperature has been demonstrated either to increase the coefficients of the resource acquisition and metabolic rates, simultaneously shallowing the scaling of both rates to body size, or to lower the exponent of the resource acquisition rate, while increasing the exponent for metabolism (and changing the respective coefficients the opposite way). In accord with the former scenario, Haure et al. (1998) found that in flat oyster Ostrea edulis the increase of temperatures across the range 10–30°C raised the coefficients of the clearance and oxygen consumption rates, decreasing both exponents. An inverse correlation between temperature and the value of the exponent for metabolism has also been found in a variety of ectotherms: in the slug Arion circumscriptus, barnacles, the earthworm Megascolex, the crab Uca (after Newell and Roy, 1973), sowbugs Porcellio scaber (Wieser and Oberhauser, 1984), and fishes Silurus meridionalis (Xie and Sun, 1990). The latter scenario of the temperature-dependence of the exponents is supported by data on the isopod Idotea baltica (Strong and Daborn, 1980). Its exponent for the ingestion rate decreased, whereas the exponent for metabolism increased and metabolism coefficient decreased in response to higher ambient temperature. A positive correlation between temperature and the exponent for metabolism has been found in the fish Oreochromis niloticus (Ross and Ross, 1983), the snail Littorina littorea (Newell and Roy, 1973) and other invertebrates (Rao and Bullock, 1954). Chappell and Ellis's (1987) evidence on 14 species of boid snakes suggests that the coefficients and mass exponents for the metabolic rate change with temperature in opposite directions depending on a species. The exact mechanisms driving the exponent changes and accounting for their opposite directions in different organisms still await explanation. The exponent for metabolism may be altered by temperature in both directions even in the same species, depending on the range of temperature considered (Chappell and Ellis, 1987). Liu et al. (2000) found a significant increase of the metabolic rate constant with temperature in mandarin fish Siniperca chuatsi and Chinese snakehead fish Channa argus. This increase was accompanied by a consequent, albeit non-significant, decrease of the exponent in the entire temperature range in Siniperca chuatsi and up to 25°C in Channa argus. It seems, therefore, that that the shape of metabolic rate allometries changes with temperature, as assumed in the model.
Although according to our analysis of optimization models the temperature-size rule can appear when the curves describing the size-dependence of the production rate cross, the reason for this crossing is important. If only the thermal sensitivities of the constants in production equation (8) differ, or if both constants increase and both exponents decrease with temperature (Fig. 4), the temperature-size rule may appear, but accompanied by a decrease of production efficiency with temperature, which is rare in nature (Angilletta et al., 2004). If the constant for the resource acquisition rate increases and the exponent decreases with temperature, while the parameters for the metabolic rate behave other way (Fig. 5), not only is the temperature-size rule satisfied for some mortality regimes, but additionally the production efficiency increases with temperature, which is a prevailing trend in nature (Angilletta et al., 2004).
Importantly, the nature of the temperature-dependence of the exponents assumed in our models is inferred from evidence on the resource acquisition and metabolic rates in organisms temporarily subjected to temperatures gradients. In organisms exposed to different temperature regimes for the entire lifespan, developmental effects of temperature, for example changes of cell size or membrane permeability, may have additional consequences on body-size scaling of production components different than those observed in organisms just acclimated to different temperatures (Kozowski et al., 2003).
An increase of the mortality rate with temperature can also produce the temperature-size rule in allocation models even when the curves describing the size-dependence of the production rate do not cross but mortality increases with temperature. Supporting such an assumption, Hirst and Kiorboe (2002) found that both total mortality and mortality caused by predators increases with temperature in marine planktonic copepods. Similarly, Houde (1989) describes an increase of mortality with temperature in fish larvae. Belk and Houston (2002) discovered increased longevity at higher latitudes in 10 of 13 studied species of fishes. It seems, therefore, that the phenomenon of increased mortality with temperature may be quite widespread, and can explain the temperature-size rule in many cases. However, our analysis reveals that to produce the temperature-size rule the thermal sensitivity of mortality must be higher than that of resource acquisition and metabolic rates.
Summarizing, the ubiquity of the temperature-size rule is understandable: it can be caused by changes of the parameters in the production equation with temperature, by changes of the mortality rate with temperature, and most likely by these two mechanisms operating together.
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ACKNOWLEDGMENTS |
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We thank M. J. Angilletta, D. S. Glazier, and two anonymous reviewers for comments on an earlier version of the manuscript, and M. Jacobs for helping to edit the paper. The work was supported partly by the Polish Ministry of Scientific Research and Information Technology, grant 448/P04/2003/24 and partly by the Foundation for Polish Science. Figure 1 is used with the kind permission of Blackwell Publishing.
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FOOTNOTES |
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1 From The Symposium Evolution of Thermal Reaction Norms for Growth Rate and Body Size in Ectotherms presented at the Annual Meeting of the Society for Integrative and Comparative Biology, 5– 9 January 2004, at New Orleans, Louisiana.
2 E-mail: kozlo{at}eko.uj.edu.pl
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