Our aim is to identify a generic specification for energy budgets that is both sufficiently complex and as simple as possible for representing individuals in population models (Fig. 1/Table 1). Our proposal is that the modelled animal forages as necessary to supply its energy needs for maintenance, growth and reproduction. If there is sufficient energy intake, the animal allocates the energy obtained in the following order: maintenance, growth, reproduction, energy storage, until its energy stores reach an optimal level. If there is a shortfall, the priorities for maintenance and growth/reproduction remain the same until reserves fall to a critical threshold below which all are allocated to maintenance. The maximum rates of ingestion and allocation depend on body mass and temperature. We make suggestions for how each of these processes should be modelled mathematically.
The energy budget
The resources acquired by an organism are generally considered to be allocated separately to maintenance, growth, reproduction or storage, as shown in Fig. 1 (Peters 1983; Sibly & Calow 1986b; Stearns 1992; Karasov & Martinez del Rio 2007). This is a diagrammatic representation of (eqn 1), omitting faecal and excretory waste. The total available for allocation is limited by the amount the animal eats, so if more is allocated to one function, less is available for others. This follows from conservation of mass and energy (eqn 1).
There is little information as to how priorities change when there is not enough food, but it is generally thought that until reproduction the first priority after maintenance is growth (see, e.g. Sibly & Calow 1986a). DEB and the MTE make different assumptions. DEB assumes that throughout life, a constant fraction of input is allocated to maintenance and growth, with the rest going in juveniles to maturation and in adults to reproduction, the ‘kappa rule’ (Kooijman 2010). MTE assumes that resources are allocated in fixed proportions to maintenance, growth and reproduction, the same in all species (Sibly 2012). Calculations are generally in units of energy per unit time, for example watts, even though acquisition and allocation of many specific nutrients subscribe to the same principles (see, e.g Kaspari 2012).
Scaling of the energy budget with body mass and temperature between and within species
Food acquisition, maintenance, growth and reproduction all require energy, and all scale in similar ways with body mass and body temperature (Brown et al. 2004). These scaling relationships underlie most of the processes in Fig. 1 and can be used in extrapolation between species when data for modelled species are not available directly. Fundamental to these scaling relationships is the way that animals' power consumption varies with body mass and body temperature.
The total power consumption of an organism is referred to as its metabolic rate. Technically, it is best measured as heat production by calorimetry in watts, but often is measured as rate of O2 consumption or CO2 production in animals. Animal physiologists distinguish basal or resting metabolic rate (BMR), the rate of metabolism of an inactive, starving animal measured over a relatively short period of time, typically minutes (McNab 1997), from the rate of metabolism in the field (FMR), which is of the order of three times BMR (Peterson, Nagy & Diamond 1990; Brown & Sibly 2012). Most but not all measurements of metabolic rate have been of BMR.
It has been known for at least a century that BMR varies with body mass and within limits with temperature. Although details are still debated, there is a large and longstanding literature showing that, across the diversity of living things and ecological settings, BMR scales with body mass as a power law and with temperature as an exponential (summarized by Peters 1983). More recently, the equation has been derived from first principles and biological mechanisms as the central equation of the MTE. The equation relates body mass M and body temperature T, measured in kelvins (=°C + 273·15), to metabolic rate B:
- (eqn 2a)
where B0 is a normalization constant that is independent of body mass and temperature, Mγ is how metabolic rate scales with M to a power γ, an allometric scaling exponent, and e−E/κT is the exponential Arrhenius function, where E is an ‘activation energy’ and κ is Boltzmann's constant (8·62 × 10−5 eV K−1) (Gillooly et al. 2001; Brown et al. 2004; Brown & Sibly 2012). Experimental studies have shown that the allometric exponent, γ, is usually between 2/3 and 1. Since the pioneering work of Kleiber (1932) and Brody & Proctor (1932), many empirical studies have obtained a value for γ close to 3/4, and M3/4 scaling of metabolic rate has often been referred to as Kleiber's law. Values for E have been reported in the range 0·41–0·74 eV, clustering around 0·65 eV (Gillooly et al. 2001; Brown et al. 2004). These considerations suggest the equation can be given in a more specific form in which γ = 3/4 and E = 0·65 eV for processes governed by respiration, so that:
- (eqn 2b)
In this form, variation between animal groups is only expected in the value of the normalization constant, B0. For homoeotherms, the Arrhenius term e−E/κT is unnecessary because body temperature is to first-order invariant and its value can be subsumed into the normalization constant. Values of γ and the normalization constant (taken as the intercept in a log–log plot) are given for 32 lineages of homoeotherms and 48 lineages of poikilotherm in Appendix III of Peters (1983). Some more recent estimates may be found in Glazier (2005).
So far we have considered the interspecific scaling of metabolic rate. There has been some debate as to whether the same scaling rules apply intraspecifically. This is expected if metabolic rate is determined by mechanistic constraints as many believe, and we suggest this be assumed in a minimum model. However, there is a suggestion that juveniles of large species have higher metabolic rates than same-size adults of smaller species (Makarieva, Gorshkov & Li 2009). Intraspecific scaling relationships of 218 species are tabulated in the study by Glazier (2005).
The importance of eqn (2) is that it represents how an animal's power consumption – measured as metabolic rate – scales with body mass and body temperature. Because power is needed for food acquisition, maintenance, growth and reproduction, these processes scale in similar ways with body mass and body temperature (Brown et al. 2004).
Food acquisition and digestion
Food resources are generally chosen from those available according to the principles of optimal foraging, that is, according to the net rate at which they provide energy per unit time (Davies, Krebs & West 2012). Thus, when foods vary in energy yield per unit time after allowing for energy costs of foraging, the animal selects the most profitable.
Generally, food resources vary both temporally and spatially. Variation in food density affects the rate of ingestion of food up to an asymptote, the form of this relationship being known as a ‘functional response’ (Fig. 2). Many functional responses have been proposed, all of which are at best approximations of reality. We will mention only one, the two-parameter Holling type 2 response (Holling 1959) (Fig. 2), which may be suitable for most purposes, as this response often approximates that observed in nature (Ricklefs & Miller 2000; Begon, Townsend & Harper 2006; Krebs 2009). The Holling type 2 functional response may be written as:
- (eqn 3)
where IGmax is the maximum ingestion rate in g or J per unit time, and k is a constant, inversely related to searching efficiency, which shows how quickly the response curve reaches its maximum as density increases.
Figure 2. The relationship between ingestion rate and food availability. The curves shown are Holling type 2 functional responses as specified by (eqn 3), with IGmax = 5.
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Maximum ingestion rates generally scale allometrically with body mass and temperature according to equations of the form of eqn (2). Values of normalization constants and body mass exponents are given for 10 lineages of homoeotherms and six of poikilotherms in Appendix VIIa of Peters (1983) (see also Clauss et al. 2007, for mammalian herbivores), but to our knowledge, no comparable data are available for temperature dependence.
The acquisition of food has energy costs, for example through locomotion, and these will sometimes be important (e.g. Bernstein, Kacelnik & Krebs 1991). An idea of their magnitude can be gained from a recent study of greylag geese, which swim at 2·2 × BMR, walk at 1·7 × BMR but fly at 10 × BMR (Kahlert, 2006). Useful allometries of the energy costs of running, flying and swimming are given in the study by Schmidt-Nielsen (1984) and discussed in Alexander (2005).
After ingestion, food is processed by the digestive system and a proportion becomes available for allocation to the various functions shown in Fig. 1. This proportion is called assimilation efficiency, defined as (energy obtained by digestion)/(energy ingested as food). Assimilation efficiency depends on diet and averages around 50–60% (Peters 1983) and appears not to vary with body mass (Hendriks 1999). Whereas flesh and seeds may be upwards of 80% assimilated, this falls to 40–70% for young vegetation and lower for mature vegetation and wood (Peters 1983). Hendriks (1999) gives the assimilation efficiencies of detritivores, herbivores and granivores/carnivores as around 20%, 40% and 80%, respectively.
Assimilated energy is available for distribution between the four destinations shown on the right of Fig. 1, described in detail in the next sections. We consider energy storage first because maintenance, growth and reproduction all draw on energy reserves when food is in short supply.
Energy reserves in terrestrial vertebrates are stored mainly as fat in adipose tissue or as carbohydrates in the liver. These reserves allow the animal to maintain its functions during temporary periods of starvation. If energy input from food exceeds the requirements of maintenance, growth and reproduction, then any excess is stored in the animal's energy reserves, the rate of storage being limited by ingestion or digestion rate. Conversely, reserves are used to supply energy requirements if the supply from feeding is inadequate.
Fat rather than carbohydrate is generally used for long-term energy storage because of its higher energy density: fat yields more than twice as much energy as carbohydrate (39·3 vs. 17·6 kJ g−1 dry weight) (Schmidt-Nielsen 1997); though there can be variation in the energy density of lipid stores between species that can be up to 40%, depending on the specific constituent triacylglycerides (McCue 2010). There are costs to energy storage, and the total cost of synthesizing and storing one gram of fat is about 54 kJ (Pullar & Webster 1977; Emmans 1994). Despite the attractions of fat, some animals use other fuels, for example sessile marine animals, for which carrying extra weight is not costly, use glycogen, while earthworms and flatworms use protein and degrow when starving.
Surplus energy from food is not added to reserves indefinitely. Instead, animals stop eating once reserves reach a certain level, presumably corresponding to an optimum compromise between the benefits of being able to survive a hunger gap and the costs of carrying extra weight, for example reduced ability to escape from predators (Witter & Cuthill 1993; Gosler, Greenwood & Perrins 1995; Lind, Jakobsson & Kullberg 2010). The optimum will vary with time and place, and prior to migration, animals may accumulate a fat store of 25–50% of body mass (Pond 1978; Peters 1983). While optimum values cannot be predicted a priori, information on natural fat content exists for many species (see, e.g. Pond 1978). Relative to energy expenditure, larger mammals carry more body fat than smaller ones [fat = 75 M1·19, fat in g and M in kg (Lindstedt & Schaeffer 2002, Table 3)] and so can survive substantially longer periods of starvation.
A more detailed model might divide biomass into irreversible mass, including compounds like bones and organs that cannot be starved away by the animal in time of need, and reversible mass, which includes energy reserves such as fat, muscle tissue and gonads (Persson et al. 1998). Persson et al. (1998) constrained the ratio of reversible mass to irreversible mass to be below a specified maximum, which differs between juveniles and adults, as the latter also allocate mass to gonads.
Maintenance and survival
Energy for maintenance is roughly equivalent to BMR, so its dependence on body mass and temperature is given by eqn (2). Energy allocated to maintenance fuels the basic processes of life essential for survival, and these have first call on energy obtained from feeding and on an animal's energy reserves when food is short. Energy is allocated to maintenance as long as energy is left in the reserves. For modelling purposes, the animal may be considered dead when the reserves are exhausted. After this point, muscle protein is consumed, but it is unlikely the animal could then recover if fed. Starvation refers to the process during which an animal requiring food is unable to eat for lack of food and should be distinguished from hibernation and aestivation, which are not considered here but have been reviewed elsewhere (see references in McCue 2010). There have been reports that metabolic rate decreases with prolonged fasting, but this may be simply a result of decreased body mass (McCue 2010).
Growth at constant body temperature
If energy is available after the costs of maintenance have been paid, juveniles allocate energy to somatic growth. The energy content of wet flesh is about 7 kJ g−1 (Peters 1983), and to this is added the costs of synthesizing flesh, which are of the order of 6 kJ g−1 for mammal embryos and 2 kJ g−1 for embryos of birds and fish developing in eggs (Moses et al. 2008). After hatching/birth, the costs of synthesis are of the order of 6 kJ g−1 for all three taxa (Moses et al. 2008). Taking a different approach, Sibly & Calow (1986b, pp. 54–55) estimate the efficiency of synthesis in juveniles (J flesh/(J flesh + J synthesis)) as 40–50% for homoeotherms and somewhat higher for poikilotherms (the mean of 26 species was 66%, range 30–89%). As 1 g wet flesh contains 7 kJ, this gives the energy cost of synthesis as a little over 7 kJ g−1 for homoeotherms and around 3·6 kJ g−1 for poikilotherms.
Growth is, however, not just a matter of supplying energy. Molecules have to be precisely assembled in appropriate order, and so there are limits to the rate at which new flesh can be synthesized. These limits are implicit in the relationship of maximum growth rate with juvenile body mass, m. A frequently used relationship for maximum growth rate has the form:
- (eqn 4a)
where a, b and g are parameters and dm/dt denotes growth rate at body mass m (Reiss 1989; Kerkhoff 2012). If body mass can be assumed proportional to the third power of body length, l, then (eqn 4a) can also be written as:
- (eqn 4b)
where a′ is a new parameter. MTE suggests an exponent of g = ¾ (Moses et al. 2008; Kerkhoff 2012), which perhaps fits the data a little better than the exponent of 2/3 suggested by Von Bertalanffy (1957). However, in describing growth curves, it makes little difference which exponent is used (Kerkhoff 2012). An additional consideration is that a 2/3 exponent allows (eqn 4b) to be rewritten in the simple form:
- (eqn 4c)
where l∞ denotes maximum body length. (eqn 4c) can be integrated and expressed as:
- (eqn 4d)
- (eqn 4e)
where l0 and m0 are neonate length and mass at t = 0 and m∞ denotes maximum body mass. (eqn 4e) is commonly referred to as the von Bertalanffy equation. The parameter b can be obtained by fitting (eqn 4d) or (eqn 4e) to data recording increase in body length or mass with age in ideal conditions. (eqn 4a) can now be written as:
- (eqn 4f)
Equation (4) shows how the maximum rate at which resources can be allocated to growth changes as the juvenile increases in mass. If more is available from digestion than can be consumed by maintenance and growth, then any surplus goes into energy reserves.
In this section, we have considered the maximum rate of allocation of resources to growth in juveniles. It has been implicitly assumed that body temperature is constant, but this is not necessarily true in ectotherms; the effects of rearing temperature on growth in ectotherms are considered in the next section. The case of growth continuing after first reproduction is more complicated and is considered below in 'Indeterminate growth: where growth continues after the age of first reproduction'.
Complications of temperature-dependent growth in ectotherms
Ectotherm metabolic and juvenile growth rates depend not only on body mass but also on body temperature, and ectotherm body temperatures are affected by ambient temperature. It is known that many ectotherms emerge smaller at higher temperatures [the temperature–size rule (Atkinson 1994)]; though, the adaptive reasons for this are not well understood (Atkinson & Sibly 1997; Kingsolver & Huey 2008). Phenomenologically, it seems ectotherms initially develop faster if it is warmer but then mature at a smaller final mass. In at least some species, the process can be described by a negative linear relationship between the logarithms of parameters l∞ and b in (eqn 4c) (Charnov 1993; Pauly, Moreau & Gayanilo 1996; Atkinson & Sibly 1997). For example, in the fish Merlangius merlangus, the relationship has been estimated as ln l∞ = −½ ln b + 3 with l∞ in cm and b in y−1. Using this, (eqn 4c) can be rewritten as:
- (eqn 5)
Initial growth rate is proportional to the Arrhenius function (Gillooly et al. 2002, using data from terrestrial invertebrates and zooplankton), and this corresponds approximately to the first term in (eqn 5) when l is small, that is , giving:
- (eqn 6)
where c is a constant that has to be determined. (eqn 5), (eqn 6) together represent a first attempt to show how the maximum rate at which resources can be allocated to growth varies with body size and temperature.
Reproduction does not occur until the animal has attained a certain size and assembled the bodily structures necessary for reproduction. These structures (e.g. gonads, oviduct and uterus) themselves require resources and some models account for this explicitly (e.g. Kooijman 2010). We suggest that this is not necessary provided a minimum size (or age) of reproduction is included. Allometric coefficients for age at maturity are given for seven vertebrate lineages in Peters (1983) Appendix VIIIb, and further information is available for mammals and birds in the study by Calder (1984) and for mammals in Ernest (2003).
Reproduction, like growth, requires that molecules be precisely assembled in appropriate order, and this imposes limits on the rate at which new flesh can be synthesized. Several offspring may be synthesized simultaneously as a ‘litter’ or ‘clutch’. The maximum rates of production are implicit in the allometric coefficients for numbers and sizes of offspring given for a number of lineages in Appendix VIIIa of Peters (1983). Data on the timing of the phases of reproduction in some vertebrate lineages are given in Appendix VIIIb of Peters (1983). The energy cost of synthesizing flesh for reproduction is the same as for growth, see section 'Growth at constant body temperature'.
Food supply and in some species temperature affect when an animal reaches the size required for reproduction. For determinate growers, that size would be adult size. However, while this approach may suffice for many vertebrates, some invertebrates respond to food shortage/stress in more complex ways, by decreasing size of first reproduction and clutch size, and in some species by increasing neonate mass. Some of these invertebrates are indeterminate growers, and these are dealt with in the next section.
Indeterminate growth: where growth continues after the age of first reproduction
Although in some species, somatic growth stops when reproduction starts, as in most birds and mammals, in other species, it continues, as in some fishes, reptiles and invertebrates. These strategies are referred to as ‘determinate’ and ‘indeterminate’ growth, respectively. In general, allocation follows the rules indicated in Fig. 1, but there is a complication: How should resources be partitioned between growth and reproduction in the case of indeterminate growth? Evolutionary theory provides only limited insight (Perrin, Sibly & Nichols 1993; Ejsmond et al. 2010), and the process is generally modelled phenomenologically using von Bertalanffy's eqn (4e) with modification if temperature varies (see, e.g. Fontoura & Agostinho 1996; Jager, Reinecke & Reinecke 2006; Kooijman 2010). Equation 4f shows as before how the maximum rate at which resources can be allocated to growth changes with body mass and ambient temperature. When food is abundant, then energy is allocated to reproduction and growth as fast as it can be used, and any surplus goes to reserves. When there is not enough food, reproduction likely has priority over growth, because early reproduction is in general strongly favoured by natural selection (Sibly & Calow 1986a).
Other approaches to modelling indeterminate growth are possible. Many DEB models assume the animal allocates throughout life a fixed fraction of energy to somatic maintenance plus growth, the rest being allocated to reproduction and the bodily structures necessary for reproduction and their maintenance (Kooijman 2010). Quince et al. (2008a,b) developed a fitness-maximizing model of biphasic somatic growth in fish, in which they distinguished between pre- and postmaturation growth with an explicit description of energy allocation within a growing season, and tested predictions against growth data from lake trout (Salvelinus namaycush).
Allometry of mortality rate
Some ABMs require specification of the background mortality rate. If this is not known directly for the modelled species, an estimate can be obtained from the allometric relationships governing energy budgets (eqn 2). Mortality rate is strongly linked to the energy budget because mortality rates must equal birth rates long term, so that populations do not indefinitely increase or decrease (Peters 1983; Sutherland, Grafen & Harvey 1986; Sibly & Calow 1987). Equalizing of birth and death rates comes about through ecological density-dependent processes that regulate the population (Sinclair 1989). The processes that produce density dependence may be direct (animals interfering with each other) or indirect (mediated by a factor such as food availability) and should be part of the ABM, but an estimate of background mortality rate may be obtained if necessary from MTE's suggestion that per capita mortality rates should, like birth rates, be proportional to M−1/4e−E/kT (Brown et al. 2004; Brown & Sibly 2006). Allometric coefficients for mortality rates overall follow MTE predictions: values are given for mammals, birds, fish, invertebrates and phytoplankton in the study by McCoy & Gillooly (2008, 2009).