Individual Growth and Reproduction
Individual Growth and Reproduction
Abstract and Keywords
This chapter develops descriptions of how individuals grow and reproduce. More specifically, the chapter seeks to determine the growth and reproduction rates from the consumption rate, by developing an energy budget of the individual as a function of size. To that end, the chapter addresses the question of how an individual makes use of the energy acquired from consumption. It sets up the energy budgets of individuals by formulating the growth model using so-called life-history invariants, which are parameters that do not vary systematically between species. While the formulation of the growth model in terms of life-history invariants is largely successful, there is in particular one parameter that is not invariant between life histories: the asymptotic size (maximum size) of individuals in the population. This parameter plays the role of a master trait that characterizes most of the variation between life histories.
Developing a size spectrum theory for specific populations requires a more detailed description of the individual than I used in the previous chapter. In this chapter, I determine the growth and reproduction rates from the consumption rate, by developing an energy budget of the individual as a function of size. This chapter essentially seeks an answer to the question: how does an individual make use of the energy acquired from consumption?
Setting up energy budgets of individuals to determine growth rate is a tried and tested discipline, starting with Pütter in the early twentieth century, then developed into a robust tool by von Bertalanffy (1957) in his work on Quantitative Laws in Growth and Metabolism, and later extended to the type of biphasic growth model that I use (Ursin, 1979; Lester et al., 2004; Quince et al., 2008). Such energy budgets depend upon parameters that describe the individuals in the population. The simplest growth models, like the von Bertalanffy growth equation, rely on just two parameters, while the more complex models require more parameters. Clearly, the huge variety of life histories among fish—from small forage fish to large piscivores and from sluggish sunfish to highly active tuna—is better represented by a growth model with many rather than few parameters. However, from the perspective of building a simple theory, a large number of parameters makes it difficult to uncover simple relationships. I reduce the number of parameters by formulating the growth model using so-called life-history invariants, which are parameters that do not vary systematically between species. While the formulation of the growth model in terms of life-history invariants is largely successful, there is in particular one parameter that is not invariant between life histories: the asymptotic size (maximum size) of individuals in the population. This parameter plays the role of a master trait that characterizes most of the variation between life histories.
The energy budget accounts for all fluxes of mass and energy within the individual: assimilation losses, metabolic losses, as well as energy spent on reproduction and growth. The budget formalizes the decision made by the individual based on its state (hunger, size, maturation) and its life-history strategy. These decisions may be understood in a life-history optimization framework as those that optimize (p.39)
individual fitness. Fragments of such a theory have surfaced—for example, for the reproduction schedule (Charnov and Gillooly, 2004; Lester et al., 2004; Thygesen et al., 2005; Jørgensen and Fiksen, 2006; Quince et al., 2008)—but a complete theory has yet to emerge. Consequently, I will largely rely on empirical estimation of the life-history parameters in the growth model. The aims of this chapter are therefore (1) to develop descriptions of growth and reproduction of individuals; (2) to establish the asymptotic size as a master trait; and (3) to determine the parameters in the energy budget in terms of the individual size, the asymptotic size, and a set of life-history parameters.
3.1 The von Bertalanffy Growth Model
The growth model I develop belongs to the family of biphasic growth models (Ursin, 1979; Lester et al., 2004; Quince et al., 2008). These models divide growth into juvenile and adult phases. Juveniles use all acquired energy for somatic growth, while adults divide energy between growth and reproduction. In this manner, the biphasic growth model accounts for the energy spent on reproduction, which is needed to derive the population size spectrum in the next chapter. Biphasic growth models owe their fundamental concepts to the von Bertalanffy growth model, and it is instructive to first look at that model. (p.40)
The von Bertalanffy growth model describes growth rate dw/dt as the difference between two processes (von Bertalanffy, 1957):
The two processes represent acquisition of energy Awn and losses kw, or, in the words of von Bertalanffy, “anabolic” and “catabolic” processes. The coefficients A and k describe the overall level of the processes, while the exponents n and 1 describe how they scale with size. Regarding the exponent n, von Bertalanffy argued that acquisition was limited by anabolic processes—that is, those that involve absorbing oxygen or food across a surface (gills or the digestive system). Fish, he argued, are limited by the simple surface rule—that is, n = 2/3 (see p. 23). With that exponent, and the standard relation between length l and weight w = cl3, eq. 3.4 can be rewritten in the common length-based form (box. 3.1)
with the solution
where L∞ is the asymptotic length and K the von Bertalanffy growth constant with dimensions time−1
The resulting length-at-age curve in fig. 3.1a initially increases linearly with age with rate KL∞. This increase follows from eq. 3.5, where dl/dt ≈ KL∞ when l ≪ L∞. Combining with eq. 3.7, we see that the initial growth rate in length is proportional to the growth coefficient A. As length approaches the asymptotic length L∞, growth rate decreases until dl/dt = 0. The length-based von Bertalanffy growth equation has been very succesful in fisheries because it is simple, because it describes observed size-at-age curves fairly well, and because it is formulated according to body length that is easily measured. Consequently, almost all growth measurements of fish are reported via the two von Bertalanffy growth parameters K and L∞.
Though popular, the mathematical form of the length-based von Bertalanffy equation is unfortunate from a statistical point of view, because the two parameters K and L∞ are correlated (fig. 3.2 and eq. 3.7). Therefore, uncertainty in the estimation of one parameter will rub off on the other. When the growth function is fitted to data from commercially caught fish, there are usually only a few measurements (p.41)
(p.42) of the largest individuals, simply because these are fished out of heavily exploited populations. In that case, the estimation of L∞ becomes uncertain. Because of the correlation between K and L∞, this uncertainty leads to an uncertainty in the estimation of K: if L∞ is overestimated, K will be underestimated and vice versa. Therefore, the estimation of the initial growth rate KL∞ becomes more uncertain than it needs to be, and possibly with systematic bias depending on how L∞ is estimated. Had the relation eq. 3.7 been inserted into the von Bertalanffy growth equation (eq. 3.6) such that A and L∞ were estimated instead, the uncertainty would have been confined to L∞, whereas A would be reliably estimated from the data on juvenile fish.
The length-based von Bertalanffy growth equation (eq. 3.6) is based on n = 2/3. Von Bertalanffy should have read Haldanes (1928) essay “On Being the Right Size,” with the gifted insight that the 2/3 law is too simple for describing how anabolic processes limit uptake (see p. 23). Perhaps West et al. (2001) read Haldane, because in their reformulation of the von Bertalanffy growth equation they used the 3/4 exponent to represent the fractal nature of uptake surfaces (West et al., 1997). Anyway, whether one value of the exponent is used over the other is not crucial—though there are indications that the 3/4 exponent leads to a better description of growth than 2/3 (Essington et al., 2001). Following in the footsteps of metabolic ecology, I will use n = 3/4 as the metabolic exponent.
3.2 Asymptotic Size as a Master Trait
The von Bertalanffy size-at-age curve is shaped by the changing importance of the acquisition and loss terms, Awn and kw, as the individual ages and increases in size (fig. 3.1b). Because the two terms have different scaling exponents, n and 1, they will not be proportional to one another but losses will take an increasingly
large share of the available energy as an organism grows in size, leaving less and less for growth. At some size, all available energy is used for losses and growth stops. This size defines the asymptotic length L∞ or asymptotic weight W∞ of individuals in the species. The asymptotic weight can be derived from eq. 3.4 as the size where growth stops (dw/dt = 0)
This equation establishes a relation between the asymptotic weight W∞, the coefficient of acquired energy A, and losses k. It is a key relation because it shows the existence of a trade-off between A and k that determines the asymptotic size: large species (large W∞) either acquire more energy (higher A) or have smaller weight-specific losses (smaller k). In this way, the differences in growth between species are defined solely by the asymptotic size and the growth coefficient A. With the relation eq. 3.11, the von Bertalanffy growth model (eq. 3.4) can be rewritten as
The growth coefficient A represents processes related to energy acquisition and assimilation, and we can expect that these processes are unrelated to the asymptotic size of the species. To find the value of A, we have to rely on measurements of the von Bertalanffy growth parameters K and L∞ from the length-based size-at-age curves. The simple relation between the growth coefficient A and the von Bertlanffy parameters in eq. 3.7 is only valid for n = 2/3, and not for n = 3/4; however, a decent approximation of A for n = 3/4 is derived in box. 3.2. The compilation of growth data in fig. 3.3 shows how A is indeed roughly independent of asymptotic size, or perhaps slightly increasing. The data also reveal a substantial variation in growth rates between species with similar asymptotic size, by around a factor of 2 to either side of the mean. The slight increase of the growth coeffient with asymptotic size is often reported in other studies (Pope et al., 2006; Olsson and Gislason, 2016). It suggests that larger species tend to have faster growth than smaller species. Faster growth would be accompanied by elevated body temperature and higher metabolic rates, which is indeed found among the scombroids (tunas, swordfish, which), and so on are a dominant group of larger fish species (Killen et al., 2016). For simplicity, I will use A as a constant in the following.
Biphasic Growth Equation
The biphasic growth model is modeled on top of the skeleton provided by von Bertalanffy. The main differences is that life is divided into juvenile and adult stages: in the juvenile stage, all assimilated energy is used for growth, while adults (p.45)
also use energy for reproduction. We can write juvenile growth rate as
Individuals mature at a size wm. Size of maturation is roughly proportional to asymptotic size wm = ηmW∞, where the constant of proportionality is ηm ≈ 0.28 (fig. 3.4). Mature individuals invest some fraction of their acquired energy into reproduction, typically proportional to their weight. As the investment into reproduction scales linearly with weight, it belongs to the loss term in the von Bertalanffy equation, but only for adults. Adult growth then become:
Juvenile and adult growth are brought together by a maturation function ψm(w/wm), which switches smoothly between zero and 1 when the argument w/wm = 1 at the size of maturation
where the exponent u ≈ 5 determines the steepness of the function (see fig. 3.4b). Introducing this function, the combined growth equation is
where I use the subscript bp to signify the biphasic growth equation. Just as with the von Bertalanffy model, the biphasic growth model in eq. 3.16 can be formulated in terms of asymptotic size. Turning the relation between A, k, (p.46)
and asymptotic size (eq. 3.11) around reveals how total losses, and thereby the reproductive investment, is related to asymptotic size
The biphasic growth model does not allow an analytical solution for weight-at-age, but analytical solutions to juvenile growth are given in box. 3.3. fig. 3.6 shows numerical solutions to size-at-age from dw(t)/dt = gbp(w).
The main advantage of the biphasic growth model over the von Bertalanffy model is that it accounts for investment in reproduction (eq. 3.17) and shows how reproduction scales with asymptotic size. The exponent of the investment in reproduction is negative (−0.25), and the investment in reproduction per weight k is therefore decreasing with asymptotic size. This decreasing pattern is also observed in data on the reproductive output (fig. 3.5) (Gunderson, 1997; Charnov et al., 2001; Olsson and Gislason, 2016). The reproductive investment kw is not only spent on producing eggs, it also represents other aspects of reproduction such as a (p.47) spawning migration. Assuming that the additional energy used is proportional to the mass of eggs produced per time, the individual reproductive output Regg (mass of eggs per time per individual) becomes
The value of “reproductive efficiency” is estimated to be εegg ≈ 0.22 (fig. 3.5). (p.48)
The trait-based growth model in eq. 3.18 is formulated in terms of parameters that are expected to be roughly invariant between species, A, εegg, ηm and n, and with the asymptotic size as the main trait that characterizes growth and reproduction of a species. Formulating growth with a trait-based model makes it possible to make general statements about the differences between small and large species just by varying W∞. Of course, if some additional information about a specific species is available, such as the growth coefficient A, then this information should be used to described the species’ growth more accurately. The growth equation can therefore be used equally well as trait-based description of growth, with W∞ being the trait and all other parameters constant, or as a model of a specific species with all parameter values being specific to that species.
3.3 Bioenergetic Formulation of the Growth Equation
The biphasic growth equation does the job it was given: it describes growth and reproduction as a function of size. That in itself is sufficient for the single-species calculations in parts II and III. However, it is insufficient for a dynamic description of growth needed in part IV. Further, the central parameter, the growth (p.49)
coefficient A, was determined only from empirical data. How is the growth coefficient connected to the fundamental physiological assumptions developed in chapter 2? To answer this question, I will dig deeper into the metabolic processes by considering a complete energy budget of an individual.
The biphasic growth model developed is based on von Bertalanffy’s idea that processes can be divided into two parts: anabolic processes related to acquisition of energy (Awn) and catabolic processes associated with losses (kw). This led von Bertalanffy to conclude that respiration was associated with the catabolic kw processes. That interpretation of growth and metabolism in fish is a simplification: losses also occur during the acquisition of energy, notably during assimilation. Further, some of the losses in the kw term are not associated with metabolism but with the reproductive output. The development of an energy budget will clarify exactly where losses are occurring.
An energy budget states how consumption C(w) is used to fuel the processes of assimilation Massim, standard metabolism Mstd, activity Mact, reproduction Regg(w)/εegg, and growth g(w) (fig. 3.7)
All terms are mass rates with units of wet weight per time. Wet weight is strictly speaking not an energy, so how can this be an energy budget? The implicit assumption that allows equating mass with energy is that wet weight is proportional to energy (1 g of wet weight equals roughly 5.5 kJ), so if eq. 3.26 is divided by 5.5 g/kJ on both sides, it becomes an explicit energy budget. However, in the (p.50) remainder it is not necessary to distinguish between wet weight and energy, so this conversion is ignored.
Consumption and Assimilation
In chapter 2, consumption rate was described as C(w) = f0hwn (eq. 2.18), where the feeding level f0 is the consumption as a fraction of maximum consumption hwn. Not all consumed mass and energy is assimilated; some is lost during the assimilation process, Massim, owing to incomplete uptake (egestion and excretion), and owing to the metabolic expenditure of the uptake (the specific dynamic action). All of these processes can be taken to be proportional to the consumption. Kitchell et al. (1977) estimated the specific dynamic action to be 15 percent of food consumption and conservative estimates of egestion and excretion to be 15 percent and 10 percent, respectively. This results in assimilation losses Massim = (1 − εa)f0hwn, with the assimilation efficiency being 1 − 0.15 − 0.15 − 0.1—that is, εa = 0.6. The assimilated consumption is then
Standard Metabolism and Activity
The assimilated consumption partially describes the acquisition term Awn in the von Bertalanffy equation (eq. 3.4); however, there are also further losses to standard metabolism and activity. The standard metabolism is the energy required to maintain the basic metabolic processes that keep the organism alive, while activity is the energy spent on foraging, migration, and so on. The standard metabolism of fish was described in fig. 3.5 as being Mstd(w) = kswn, where again n is the metabolic exponent. Activity metabolism is difficult to measure because it depends on the level and frequency of activity. In the absence of information, I will assume that it is simply proportional standard metabolism and represented within the coefficient ks. The standard and activity metabolism is proportional to maximum consumption rate hwn. To reflect this relation, metabolism is represented as a fraction of maximum assimilated consumption, ks = fcεah, where fc is the critical feeding level —that is, the fraction of maximum assimilated consumption rate required to stay alive and feed
A reasonable value of fc is around 0.2 (Hartvig et al., 2011). (p.51)
The energy budget defined in eq. 3.26 can now be assembled and the growth rate g(w) isolated on the left-hand side
Collecting terms according to whether they are proportional to wn or w gives
This is the same as the biphasic growth eq. 3.16, if we define the growth coefficient as
Fig. 3.8 illustrates how energy is divided between the different processes as a function of size.
The relation between A and maximum consumption h was used in chapter 2 to estimate the maximum consumption used in fig. 3.5. The physiological parameters that make up A—the assimilation efficiency εa, the coefficient of maximum consumption h, and the critical feeding level (standard metabolism) fc—are constants (p.52) that are expected to vary little between species. The feeding level f0 will also on average be around 0.6. In part IV, however, I will consider dynamic models, and there the feeding level will no longer be constant but will depend on available food. For now, however, the feeding level is considered constant, so all these parameters can be combined into a constant growth coefficient.
3.4 Which Other Traits Describe Fish Life Histories?
I have used the asymptotic size W∞ as the master trait that determines the variable processes in the growth model. But are these processes really determined by W∞? For instance, eq. 3.11 related W∞ to the growth coefficient A and the investment in reproduction k as W∞ = (A/k)4. That relation could just as well be used to state that the decision about the reproductive investment k together with A determines W∞. That is correct, so why not choose the reproductive investment k as the master trait and let that determine W∞? The choice of W∞ is one of convenience: it is simply the easiest trait to determine in a population of fish. Even for the most data-limited populations, we have a reasonable guess of the asymptotic size, simply as the largest fish observed. Using W∞ as the master trait means that it will be possible to apply the theory to even very data poor stocks. Further, W∞ has a more intuitive meaning than k. Knowing for example that W∞ = 10 kg, I can immediately state that the stock in question is probably piscivorous, while that would be a harder statement to make on the basis of knowing that k = 0.44 yr-1.
The model of individual growth and reproduction suggests that many important life-history parameters correlate with asymptotic size W∞ and/or the growth rate coefficient A: age at maturation, size at maturation, reproductive investment, and life span. This implies that knowing only asymptotic size W∞ and the growth rate coefficient A, all other parameters can be estimated. While the relations between the parameters are borne out clearly in the theory, they are less clear in the data analyses shown in this chapter. How can alleged life-history “invariants,” such as the ratio between asymptotic size and size at maturation, be considered invariants when they vary considerably between species? There are three aspects of an answer to this question: semantic, empiric, and theoretic.
Calling a measure, such as the size of maturation relative to the asymptotic size, an invariant does not imply an absence of variation between species. Rather, it implies that the parameter does not co-vary with other traits, such as the growth rate, or with the environment—for example, temperature—or with phylogeny— that is, that related species have similar deviations from the mean. Using the term life-history invariant is therefore not a statement that life histories do not vary between species, but rather that the variation is random and unexplained, at (p.53) least according to our current level of understanding. To avoid implying that these parameters are fundamentally invariant, I refer to them as life-history parameters, and not as life-history invariants.
Regarding the empiric basis of the life-history parameters, it should be remembered that the quality of data is limited. For instance, some of the data are not direct measurements. An example is the growth coefficient A, which is only indirectly estimated from measured growth curves; this clearly generates extra noise and possibly even systematic biases, as discussed on p. 40. Further, measured growth curves are rarely growth curves of individuals, but are based on the size-at-age of average surviving individuals. If faster growth is correlated with higher mortality, then the size-at-age curves are biased toward having more slow growers in the higher age classes than faster growers. Other traits, such as reproductive investments, are very difficult to estimate reliably: gonad weight may be a proxy of reproductive investment, but the annual reproductive investment requires knowledge of the average number of spawning events during a year, how spawning varies with food concentration, the degree to which skipped spawning occur, and the fraction of the investment used for migration and spawning behavior. Nevertheless, while all these considerations are relevant, they are insufficient to explain the variation of the data around the theoretically predicted relationships.
There must therefore be other life-history correlates that explain some of the variation that we observe, we just have not yet uncovered them. Take the variation in the growth rate coefficient in fig. 3.3 as an example: there is almost a factor of 10 difference between the slowest and the fastest growing fish, even when correcting for temperature. It seems like a safe bet that such profound differences in growth rates are correlated with other traits. Even though we do not know which traits, we can form hypotheses. One obvious candidate for another “trait axis” would be a slow-fast life history continuum. Slower growing individuals are assumed to obtain less food, and have a correspondingly lower clearance rate. A lower clearance rate would imply a lower activity and thus a lower critical feeding level fc. However, clearance rates are difficult to measure directly and the activity coefficient is, in the words of my colleague and expert fish physiologist Niels Gerner Andersen, the “dark horse” of energy budgets—we know very little about in situ rates of activity. One way out of this problem is to assume that activity correlates with swimming speed, but then again, the fraction of time an individual is active versus passive also plays in. The difficulties of establishing credible data for the underlying processes for many species is a key reason why the trade-offs behind an obvious candidate for an additional trait like the slow-fast continuum have not been revealed.
Another, possibly related, trait is investment in defense. Fish live in an unforgiving environment where foraging implies an exposure to being eaten. The risk of (p.54) being eaten can be lowered by investing in defense. Defenses can be manifested as spines (sticklebacks or perches), by being cryptic (sculpins), or by hiding (many flatfishes or sand eel). All these strategies have costs in terms of creating the defense and in the defensive behavior itself. Hiding, for instance, comes at a cost in forgone feeding. In other words, defense implies slower growth. Defense traits can therefore easily be confounded with the “slow” end of a the fast-slow life-history continuum. Quantifying the cost and benefits of defense traits, the trade-offs, is even more difficult than for the fast-slow continuum because they require answers to tricky questions: How much feeding does a hiding individual forgo? How much reduced mortality does this imply? This quantification has been done for small animals such as copepods (Kiørboe, 2011) that can be studied easily in the laboratory, but not for fish. In the following chapter, I will include a linear trade-off between investments in defense and mortality: investment in defense comes at a cost of lower A, which translates into lower mortality.
If the trade-offs related to the slow-fast and defense axes could be quantified, some part of the unexplained variation in for example, the growth rate in fig. 3.3, will be explained by the extra dimensions added to the trait-space. The variation might not go away entirely, but the clever addition of even more traits might lower it. The addition of more traits complicates the analysis of models and should therefore be done only if they add significant insights. The art in formulating trait-based descriptions lies in an inspired choice of the smallest set of traits that describe the largest amount of the observed variation in life-history parameters. I will revisit this aspect in chapter 9; the minimal model developed here uses just one trait: the asymptotic size.
This chapter developed a simple von Bertalanffy–like biphasic model of growth and reproduction in fish. The model describes how growth and reproduction vary between fish species with different asymptotic sizes: species with large asymptotic size are expected to have a smaller reproductive output per body weight than smaller species. Other aspects of growth are determined by a set of life-history parameters. While these parameters are not exactly invariant between species, there is no systematic variation with asymptotic size. This establishes asymptotic size as the “master trait” to describe a fish stock.
The growth model predicts size-at-age curves that deviate from classical von Bertalanffy predictions. Notably, juvenile sizes-at-age are smaller than von Bertalanffy predicts. The deviations are unfortunate because the von Bertalanffy equation is generally perceived to be a fine representation of growth data (p.55) (von Bertalanffy, 1957; Ursin, 1967). The reason for the deviations stem from me using n = 3/4 and not n = 2/3 for consumption and for losses to standard metabolism and activity. These choices are not in perfect accordance with the current knowledge of how metabolism scales with size. There are few measurements of how standard metabolism scales with weight as individuals grow in size, but it does seems to scale with an exponent higher than 3/4; values between 0.8 and 0.9 are commonly reported (Killen et al., 2007). Losses to activity are harder to measure, but theoretical considerations indicate that they also scale with a higher exponent (Ware, 1978). Increasing the exponent of metabolism would introduce an extra term in juvenile growth (eq. 3.13) and bring it closer in form to the von Bertalanffy growth equation. An improved model would therefore include an explicit loss term in the growth equation with a different exponent from the acquisition, or, if the constraint of an equation with just two exponents (n and 1) were to be maintained, to include standard metabolic losses in the catabolic term.
There is one important reason for not including activity or other losses with an exponent higher than 3/4: loss rates scaling with a higher exponent than acquisition of energy—that is, higher than n, limits that maximum asymptotic size of all fish stocks. This limitation is illustrated in fig. 3.1b: higher asymptotic sizes means a lower coefficient of the rate in question. If the overall maximum size of any fish species is 1 ton, then the coefficient that scales directly with mass cannot be larger than A(1 ton)−0.25 ≈ 0.17 yr-1 (found using 3.17). Such a term would be insignificant for small fish species in comparison to the reproductive investment, which for a 10 kg fish is 0.54 yr−1 and 3.0 yr−1 for a 10 g fish. Alternatively, we could assume that this constant term also changes with asymptotic size, such that it is smaller for large species (Andersen and Brander, 2009; Calduch-Verdiell et al., 2014; Andersen et al., 2015, 2016). There are, however, no observations about how the level of standard metabolism or activity varies with asymptotic size—that is, do larger species have lower levels of standard metabolism and activity than smaller species? There is no a priori reason to expect that the levels should decline with asymptotic size. Why should a large species be less active or have lower size-corrected standard metabolic levels than a small species? On the contrary, it is prudent to assume that the metabolic rates correlate with other traits, as argued in section 3.4. If a size-based description is developed for a particular stock, the maintenance of the dependency with asymptotic size is less of a concern. In that case, a higher degree of accuracy is desirable, which requires a more complex growth equation where rates vary more accurately with size, as for example, in Ursin (1967, 1979).
The choices made in the growth equation are in line with my willingness to sacrifice some accuracy in the interest of developing a coherent and general theory. This sacrifice makes it possible to formulate growth and reproduction in terms of (p.56) asymptotic size and thereby make general statements about fish stocks broadly just by varying asymptotic size. Dividends on the sacrifice will be paid out in the following chapters, where the trait-based formulation of growth and reproduction reveals how density dependence and the impact of fishing vary across fish species broadly. In conclusion, the model of individual growth presented in eqs. 3.18 and 3.30 is far from perfect. It does, however, represent the best attempt at formulating a simple size-based growth equation where all the variation is represented by the variation in just one parameter: the asymptotic size W∞.