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A fundamental issue in foraging theory is whether it is possible to find a simple currency that characterizes foraging behaviour. If such a currency exists, then it is tempting to argue that the selective forces that have shaped the evolution of foraging behaviour have been understood.
We review previous work on currencies for the foraging behaviour of an animal that maximizes total energy gained. In many circumstances, it is optimal to maximize a suitably modified form of efficiency.
We show how energy gain, predation and damage can be combined in a single currency based on reproductive value.
We draw attention to the idea that hard work may have an adverse effect on an animal's condition. We develop a model of optimal foraging over a day when a forager's state consists of its energy reserves and its condition. Optimal foraging behaviour in our model depends on energy reserves, condition and time of day. The pattern of optimal behaviour depends strongly on assumptions about the probability that the forager is killed by a predator.
If condition is important, no simple currency characterizes foraging behaviour, but behaviour can be understood in terms of the maximization of reproductive value. It may be optimal to adopt a foraging option that results in a rate of energy expenditure that is less than the rate associated with maximizing efficiency.
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In the simplest setting, the fitness of a behavioural strategy is the mean number of offspring produced over the lifetime of an organism following the strategy. Natural selection produces behavioural strategies that approximately maximize fitness. Behavioural studies are often concerned with short periods of an organism's life. It might not be obvious how performance over such a period is related to lifetime reproductive success. A common method of analysis is to suggest a performance criterion that involves the maximization (or minimization) of a suitably defined currency. If the currency is well chosen, then maximizing (or minimizing) the currency will maximize fitness. The currencies chosen are usually relatively simple; in the context of foraging they often just depend on time and energy (Stephens & Krebs 1986; McNamara & Houston 1997). In some cases, there may be no simple currency that is related to fitness. In these cases, it is of interest to establish the extent to which a simple currency can provide reasonable predictions about behaviour (Schmitz et al. 1998). In this paper, we review currencies for foraging, stressing the importance of the forager's condition.
Time and energy
Investigations of foraging (Kacelnik 1984; Ydenberg et al. 1994; Nolet 2002) have often concentrated on two currencies: net rate of gain γ and efficiency q. If b is the forager's gross rate of gain and c is its rate of energy expenditure then
Note that γ is a power with S.I. unit Watts = J/s, whereas q is dimensionless. The first of these currencies is the net rate of energetic gain. The second is the gross rate of gain per unit energy expended. Note that maximization of this latter currency is equivalent to maximization of the net rate of energy gained per unit energy expended; that is, maximization of γ/c. The maximization of efficiency provides a better account of the foraging in honeybees than does the maximization of net rate (Schmid-Hempel, Kacelnik & Houston 1985; Kacelnik, Houston & Schmid-Hempel 1986; Houston 1995). Furthermore several studies of bumblebees that have been taken to support net rate as a currency are also consistent with the maximization of efficiency (Charlton & Houston 2010).
We now look at these currencies for foraging in more detail.
Maximizing energy gain in a given time
Given what we said about currencies, it is necessary to relate a possible currency to the forager's reproductive success. A foraging animal might be selected to obtain as much energy as possible from a given time foraging (energy maximization) or to minimize the time to obtain a given amount of energy (time minimization; Schoener 1971). We concentrate on energy maximization.
If foraging is deterministic and is not subject to any constraints, then obtaining the maximum possible energy from a given time that can be devoted to foraging is achieved by maximizing the net rate of energetic gain. We denote this maximum rate by γnet.
Now assume that the animal is constrained by the amount of energy that it can spend during the time interval (Drent & Daan 1980; Hammond & Diamond 1997). This constraint may mean that the option with the highest net rate of gain cannot be adopted for the whole of time interval available to the animal. Assume that the lowest rate of energy expenditure occurs when the animal is resting. Let the rate of metabolic expenditure under this option be cr. Let Omin be the foraging option at which the modified efficiency b/(c−cr) is minimized and let γmin be the resulting net gain rate under this option. McNamara & Houston (1997) show that if the constraint still allows the animal to use option Omin for the whole time interval, then the animal should forage for the whole interval at a particular rate of gain that lies between γmin and γnet. If the constraint prevents the animal from using Omin for the whole interval, the animal should spend part of its time resting and part using Omin.
Foragers may also be constrained by the amount of energy they can process, for example because of digestive constraints (Kersten & Visser 1996; Fortin, Fryxell & Pilote 2002; Van Gils et al. 2005; Quaintenne et al. 2010). A similar analysis to the above shows that the modified form of efficiency can again be used to characterize the foraging behaviour that maximizes energy gained subject to this constraint (Houston 1995; McNamara & Houston 1997).
We have focused on an organism that consumes the food that it obtains. The appropriate currency for an animal that is delivering food to its young is considered by Houston (1987) and McNamara & Houston (1997). If the parent is not up against a constraint on its energy expenditure then it should maximize a form of delivery rate that includes the time for the parent to replace the energy that it spends. If the parent's energy expenditure is constrained then it should maximize a form of efficiency. In this case, the best choice of option when collecting food for the young depends on the foraging options the parent has for feeding itself.
In the above, the objective is to maximize energy gain, but constraints mean that this is achieved by maximizing various form of efficiency. Energy expenditure may, however, not only reduce the net energy gained but also have other costs. We discuss possible costs later. Here, we consider a scenario in which a cost means that it is optimal to maximize efficiency.
Minimizing energy expenditure to achieve a given objective
Consider a foraging animal that stores energy on its body as fat. The animal must get its energy reserves to a critical level L in order to reproduce. Its current level of reserves is x. There is no predation, and foraging is deterministic. If the animal chooses a foraging option with net rate of gain γ and rate of expenditure c then the time required to reach L is t = (L−x)/γ.
In general, an animal's fitness will depend on both time at which it reaches L and the energy that it has expended in doing so. We start with a very simple case in which there are no time constraints, that is the time at which L is reached has no effect on fitness. Suppose that fitness is a decreasing function of the total amount of energy spent ct. Then in terms of the foraging parameters, the total energy spent is
which is minimized when γ/c is maximized. It follows that the best foraging option is the option that maximizes efficiency. Now suppose that in addition to the effect of total expenditure, fitness also decreases with t. Maximizing γ/c minimizes ct, but maximizing γ minimizes t so, we expect the optimal option to lie between the option that maximizes efficiency and the option that maximizes net rate. As an example, let fitness = exp (−αct) exp (−ϕt), where the constant α measures the effect of total metabolic expenditure on fitness and the constant ϕ measures the effect of time delay on fitness. For this fitness measure, fitness is maximized by minimizing
which is equivalent to minimizing
The option that minimizes this expression can be found graphically (Fig. 1a). The ratio Φ/α is a measure of the cost of being late in reaching L relative to the cost of metabolic expenditure. When the ratio is zero, being late is unimportant, and it is optimal to maximize efficiency. As the ratio tends to infinity, metabolic expenditure becomes unimportant, and it is optimal to maximize net rate of gain.
Optimal behaviour associated with migration has been explored in terms of currencies based on time and energy. For example, a migrating bird might be selected to minimize journey time or minimize energy spent (Hedenström & Alerstam 1997). In the latter case, this is achieved by minimizing the energy expenditure per unit distance travelled, again a form of efficiency.
In our discussion of foraging, we have concentrated on the energetic content of food, but food may contain other important components, such as protein and minerals. This may mean that no single food type is optimal. For an analysis of the maximization of energy subject to a constraint of obtaining a critical amount of a mineral, see Belovsky (1978). For the case of minimizing the total cost of deviating from the ideal intake, see Simpson & Raubenheimer (2012).
Energy and predation
Early work on optimal foraging ignored the fact that the foraging animal might be killed by a predator. Subsequent work has included this factor. If animals have options that differ in energetic gain and the risk of predation, high energetic gain is likely to be associated with high predation risk. This can occur because the forager can achieve a high gain by devoting its time to foraging as opposed to vigilance. The available foraging options can be characterized by the net rate of gaining energy γ and the rate of predation M (Fig. 1b). Clearly, if an option has the same or lower gain and a higher predation rate than another option, then this option will never be optimal, and can be eliminated from an optimality analysis. The remaining set of options are such that each is not dominated by another; if one option has a lower gain rate than another it also has a lower predation risk (Gilliam & Fraser 1987). This set of options is referred to as the Pareto front (Fig. 1b). From any point not on the front the forager can achieve a lower M for given γ, or can increase γ for given M, or do both. It follows that optimal behaviour is predicted to be on the front. Exactly where on the front depends on the animal's state and future prospects.
In order to provide a general analysis of the trade-off between energy and predation, it is necessary to quantify the value of extra energy and the cost of loss of life in a single common currency. To do so, we define V(x) to be the expected future lifetime reproductive success of an animal with energy reserves x; that is, the function V describes what is known as the reproductive value of the animal. This function provides the necessary common currency (McNamara & Houston 1986). The derivative V′(x) gives the rate at which future reproductive success increases as reserves increase. Houston & McNamara (1989) show that a foraging animal with energy reserves x should choose behaviour, u, to maximize
where γ(u) is its net rate of energy gain and M(u) is its rate of mortality. In this formula, γ(u)V′(x) is the rate at which future reproductive success is increasing as a result of energy gain, and since the value of the animal's life is V(x), the term M(u)V(x) is the rate at which future success is lost as a result of predation.
The behaviour, u, that maximizes expression (2) depends on , which is the marginal rate of substitution of the value of food to the value of life. This optimal choice is illustrated graphically in Fig. 1b.
It is of course not easy to know the exact value of . If behaviour is observed (and lies on the front) then given that behaviour is optimal, we can infer (Schmitz et al. 1998).
Gilliam (1982) and Werner & Gilliam (1984) assumed that a foraging animal must get its reserves to a critical level L in order to reproduce. The time at which the animal reaches L has no effect on fitness. Foraging is deterministic and the animal can choose between options that differ in rate of gain γ and rate of predation M. Gilliam shows that it is optimal to minimize M/γ at every level of reserves. The optimal choice of option is illustrated in Fig. 1b.
The situation is more complex if the time at which L is reached has an effect on fitness. If the time penalty is exponential with parameter ϕ, then it is optimal to minimize (M + ϕ)/γ. This modification can be used to investigate optimal migration (Houston 1998). In this model, the fuel load of a migrating bird determines its speed of migration and its fitness depends on the time at which the destination (rather than the critical level) is reached.
Vrugt, van Belle & Bouten (2007) use the Pareto front approach to analyse the autumn migration of warblers. In the context of reaching the wintering ground, various migratory strategies are assessed in terms of time taken and energy spent. Vrugt et al. claim that the two components of fitness are incommensurable. This is not correct; reproductive value provides a common currency based on all the relevant components. In the case of energy and predation, expression (2) shows that the two components of fitness are not incommensurable, but can be combined. The marginal rate of substitution is the conversion factor.
It has been observed that different animals in the same population may be at different locations on the Pareto front. It has been suggested that the Pareto front approach provides an account of why behaviour is variable (Mesterton-Gibbons 1989; Schmitz et al. 1998; Vrugt, van Belle & Bouten 2007). This is because the analysis predicts a range of possible positions. However, we emphasize that for a given state and ability of an animal all points on the front are not equivalent – there is just one point that maximizes fitness. So, if variability is observed and animals are optimal, they must differ in state or ability or both. Thus an analysis based on a Pareto front is not a new explanation for variability, it is just saying that animals differ in what is best for them.
We end our discussion of variability by noting that it is unrealistic to expect natural selection to produce the exact outcome predicted by optimality models. It is more reasonable to expect variation about the prediction with costly departures being rare (McNamara & Houston 1987).
If the forager is not subject to predation, it is often assumed that fitness is determined solely by the net amount of energy gained. In the absence of constraints, options with the same rate of gain are thus equivalent. There are, however, reasons to believe that there may be costs in terms of fitness that depend on the foraging option chosen. For example, the probability that an oystercatcher damages its bill increases with the size of prey item (Rutten et al. 2006). Many costs increase as energy expenditure increases (McNamara & Houston 2008). For example, the build up of lactic acid associated with high rates of exercise may temporarily reduce the ability of an animal to compete with conspecifics or escape from predators. When there are such costs, options with the same net rate of energy gain are not equivalent (Houston & McNamara 1999; Yearsley et al. 2005).
Such effects can be handled by means of state variables (Houston & McNamara 1999; Clark & Mangel 2000). A state variable has an effect on expected future reproductive success and may be influenced by behaviour. We have used energy reserves as a state variable earlier in this paper. Other possible state variables include body size, body temperature (Clark & Dukas 2000; Pravosudov & Lucas 2000; Welton et al. 2002; Dell et al., 2014), ambient temperature (Reuman et al., 2013), cellular nutrient concentration (Reuman et al., 2013), location (Barta et al. 2008), parasite load, dominance status (McNamara & Houston 1996), number of dependent young (Houston & McNamara 1999), time since a predator was last seen (McNamara et al. 2005) and condition of a bird's feathers (Barta et al. 2006; Barta et al. 2008). Population level variables, such as the current population density (DeLong et al., 2014) may also be considered as state variables if the animal has knowledge of these quantities. To take account of the deleterious effects of hard work, lactic acid and damage might be appropriate state variables. When we considered the energy vs. predation trade-off, the animal's reproductive value quantified how an animal's expected future lifetime reproductive success depended on its energy reserves. To deal with any of these extra state variables, it is necessary to extend the idea of reproductive value to quantify the dependence of expected lifetime reproductive success on the combination of state variables which are under consideration. Reproductive value will also typically depend on time of day and time of year (Houston & McNamara 1993; McNamara, Houston & Lima 1994).
General equations for the effect of condition
We now introduce a simple modelling approach that captures the idea that metabolic expenditure not only depletes energy reserves but has other fitness costs. As before, we let x denote the energy reserves of an animal. We now introduce a second state variable y which we refer to as condition. This variable is a schematic representation of physiological variables such as lactic acid, damage or parasite load that are affected by hard work. At a given time, the animal's expected future lifetime reproductive success can be expressed as a function V(x, y) of its reserves, x, and its condition, y.
We consider an animal that must choose an option at a particular time. Options differ in their associated metabolic expenditure. If the animal chooses an option with metabolic expenditure c then its rate of increase of reserves is γ(c) and the rate at which condition decreases is κ(c). The rate of increase in future reproductive success with increasing reserves is . Similarly, the rate of increase in future reproductive success with increasing condition is . Thus, if the animal chooses an option with metabolic expenditure c its reproductive value changes at rate [cf. expression (2)]
Houston & McNamara (1999) show that in the absence of predation, it is optimal to choose the foraging option that maximizes this quantity. The trade-off between γ and κ and the optimal option can be illustrated graphically in a similar manner to that used to represent the trade-off between γ and M in Fig. 1b. If κ(c) is a linear function of the metabolic expenditure then the total change in condition during a time interval depends only on the total expenditure; the bout structure is irrelevant. In contrast, if κ(c) is nonlinear, then the total change in condition depends on the bout structure of metabolic expenditure.
If predation is present as well then it is optimal to choose the value of u that maximizes
Expression (3) tells us the optimal trade-off between energy gain and condition given that the dependence of reproductive value on energy and damage is known. We now present a model of the daily routine of foraging which shows how optimal behaviour depends on time of day, reserves and condition for an animal during a nonreproductive period.
A model of daily routines
The model is based on a series of days. During the hours of daylight, the animal can forage, whereas at night it must rest. In the baseline case, both day and night last 40 time units. The animal's decision is how hard to forage during daylight. The animal is characterized by two state variables, energy reserves and condition. Energy reserves lie in the range 0 ≤ x ≤ 1. If the animal has maximal energy reserves, it is unable to store any additional energy obtained from food. If its reserves fall to zero, it dies of starvation. Condition also lies in the range 0 ≤ y ≤ 1. The probability of death from disease increases with an increasing slope as condition decreases (see Appendix S1, Supporting information for details).
The mean change in an animal's energy reserves depends on its foraging intensity. Foraging intensity lies in the range 0 ≤ u ≤ 1. As intensity increases, both the animal's probability of finding food and its metabolic expenditure in a time interval increase (Appendix S1, Supporting information). We refer to this expenditure as the metabolic rate. As intensity increases, the probability of finding food increases at decreasing rate, whereas metabolic rate increases at a constant rate from basal metabolic rate (BMR) when u =0 to 5 BMR when u =1. This means that there is an intensity unet at which mean net rate of gain is maximized and a lower intensity ueff at which efficiency is maximized. For the parameters used in our computations, an animal that forages with intensity unet during the whole of the daylight period would on average gain more energy from food than it expended during a 24-h period. Thus, there is a foraging intensity ueven during daylight, that is less then unet, and is such that the average net gain in energy over 24 h is zero.
Metabolic rate determines how mean condition changes. This relationship is given in the Appendix S1 (Supporting information). There is a critical level of intensity uc below which there is a mean increase in condition, and above which there is a mean decrease in condition. For the parameter used in the Fig. 2, we have ueven < uc < ueff < unet. Exact values with their associated metabolic rates and mean state variable changes are shown in Table 1.
Table 1. The metabolic rate, mean net rate of energy gain and mean net rate of change in condition (all times 100) for the four benchmark foraging intensities together with u =0 and u =1
u = 0
ueven = 0·2515
uc = 0·3125
ueff = 0·6884
unet = 0·8736
u = 1
Mean net rate of energy gain
Mean net rate of condition change
In the baseline version of this model, there are two sources of mortality, starvation and disease. We find the state-dependent strategy that minimizes total rate of mortality. The procedure involves working backwards over days using dynamic programing to find the optimal strategy at convergence. We then follow this strategy forward over days until expected behaviour settles down to a routine that is independent of initial state. This procedure of backward and forward convergence is described by Houston & McNamara (1999).
For the parameters shown in Table 1, it can be seen that an animal that has constant foraging intensity just greater than ueven will not only be on a positive energy budget on average but will also gain condition on average. Thus, it would seem that it is easy for an animal to survive. This, however, ignores stochasticity. The change in energy reserves is stochastic due to good and bad luck in finding food. In addition, there is stochasticity even when resting (see Appendix S1, Supporting information for details). The motivation for this is that the animal will experience variation in cooling due to temperature and wind fluctuations. Thus, an animal that forages with intensity just greater than ueven will occasionally be unlucky and have low reserves. It will then have to forage more intensively to increase reserves, and this will tend to reduce condition. Added to this, there is also stochasticity in the dynamics of condition (see Appendix S1, Supporting information). As a consequence, the animal's probability of surviving 100 days if it follows the optimal strategy is only 0·708.
Figure 2 shows how the optimal foraging intensity depends on state and time of day. The general pattern is that intensity increases with condition and decreases with reserves. (There are some circumstances in which intensity increases slightly with reserves.) There are many circumstances in which intensity is less than the intensity ueff that maximizes efficiency. The advantage of foraging at an intensity less than ueff can be understood in terms of the increasing damage that results from foraging at high intensity. It can also be seen from the figure that optimal intensity is sometimes slightly greater than the intensity unet that maximizes net rate of gain. If all that is important to the animal is energy expended and expected energy gained then there is no reason for intensity to be greater than intensity unet that maximizes net rate of gain. The fact that the intensity is higher than unet can be explained in terms of risk-sensitive foraging (McNamara & Houston 1992). Because the probability of finding food increases with intensity, the forager can control both the mean and variance of energetic gain. By adopting a high intensity, the animal can reduce the variance in gain at a cost of a reduction in mean gain. High foraging intensity is also costly because it reduces condition. When condition is good, it can be advantageous to accept these costs in return for the benefit of a reduction in variance.
By following a cohort of animals adopting the optimal foraging behaviour forward over days, we can find the expected distribution of states and the expected foraging intensity. Reserves tend to increase over the day. Condition tends to be high especially at the start of the day because condition recovers overnight. Some idea of the distribution of foraging intensity is given in Fig. 3. In this figure, we show the proportion of animals resting and those actively foraging (u > 0) with intensity less than various benchmark intensities. These are in increasing order of magnitude uc, ueff and unet. The figure shows that the proportion of animals resting increases as the day progresses. Just after dawn, there is a wide range of foraging intensities. For those animals that are actively foraging at a given time of day, the proportion with intensity less than ueff first increases and then decreases as a function of time of day and tends to zero as dusk approaches. Thus, near dusk almost all animals are either resting or foraging at an intensity that is greater than ueff. Only a small proportion of animals adopt a foraging intensity greater than unet. This proportion increases from the middle of the day to the end of day. These trends seem robust; they are found when the relationship between condition and mortality is changed.
Figure 4 shows how the dependence of intensity on time of day depends on day length. In the figure, we compare a day that is shorter than baseline (T = 32, short day) and a day that is longer than baseline (T = 48, long day). The long day results in higher levels of resting at the start of the day. When faced with the short day, a higher proportion of animals has an intensity above unet. The proportion of animals with intensity less than ueff is always greater when days are long.
We have investigated how the above conclusions are changed if foraging incurs a risk of death from predation, so that there are three sources of mortality: starvation, disease and predation. An optimal strategy now minimizes the rate of total mortality from all three sources. We model predation in two ways. Figure 3a shows activities under the optimal strategy when predation risk is taken to be an increasing and accelerating function of foraging intensity. In comparison with the baseline without predation, it can be seen that there is less resting and a greater proportion of animals actively foraging at intensities less than ueff. This effect occurs because high foraging intensities incur a disproportionate predation risk compared with intermediate intensities. We have also analysed the case when predation risk while actively foraging is independent of foraging intensity (and is zero when resting), that is, predation is a step function. Figure 3b shows activities in this case. Now compared to the baseline without predation, there is more resting and less individuals actively forage with intensities below ueff. Figure 5 compares both forms of predation with the baseline. The figure shows the proportion of those individuals that are actively foraging that forage with an intensity greater than ueff. As can be seen, this proportion is greatest when predation has the step-function form and is least when predation is accelerating.
We use condition as a general term for a state that is influenced by hard work. In the previous section, condition tends to improve if the animal does not work hard. We use the term irreversible damage to refer to a condition variable that cannot improve; instead the level of damage either remains the same or increases. McNamara et al. (2009) model irreversible damage that builds up over an organism's lifetime for an animal that does not grow after maturity. Damage is envisaged to be a combined measure of physical (e.g. wear and tear) and physiological (e.g. DNA and cellular protein) deterioration (Beckman & Ames 1998; Sohal, Mockett & Orr 2002; Monaghan, Metcalfe & Torres 2009). Let D denote the rate at which damage builds up, r denote the animal's rate of reproduction and m denote its instantaneous mortality rate. All of these quantities can depend on the animal's current level of damage. They can also depend on the animal's strategy, that is, the animal's behaviour and its internal allocation of resources to repair, reproduction, etc. McNamara et al. show that under an optimal strategy the quantity
is maximized at every stage of the animal's life after maturity, where V denotes the animal's reproductive value at that stage. This expression integrates reproduction, damage and mortality into a single currency. If death only occurs once damage builds up to a critical level, so that m is identically equal to zero below this level, the optimal strategy maximizes the amount of reproduction per unit of damage accumulated (the Gilliam criterion, Werner & Gilliam 1984).
Currencies for metabolism
Currencies are usually used to predict behaviour, but they can also use to analyse physiology. Assume that natural selection is able to act on metabolic rate m and that a high metabolic rate enables the forager to achieve a high rate of energetic gain b(m). Houston (2010) shows that if it is optimal to minimize the time spent foraging then
is a currency that should be maximized by natural selection. This is the net rate of gain divided by the metabolic rate, which is a form of efficiency. If c = βm, then this currency simplifies to
The ultimate currency for natural selection is fitness. In the simplest case, fitness is the lifetime production of offspring. In more complex cases involving intergenerational effects or environmental stochasticity, it is a suitable measure of the growth rate in the number of descendants left far into the future. Even if individuals are genetically identical, they can differ in their state, and this can result in differences in the ability to leave descendant far into the future. Reproductive value measures how this ability depends on an animal's state. As we have pointed out, both in this paper and elsewhere (McNamara & Houston 1986), reproductive value provides a common currency for comparing the value of alternative actions; a strategy that maximizes reproductive value at each stage of an animal's life maximizes fitness. Reproductive value is not in general a tractable currency for predicting moment-to-moment decisions. In order to make predictions at this level, a relatively simple surrogate currency is often used. The appropriate currency to use may depend on the time-scale under consideration. For example, for a small bird in winter, a suitable currency that evaluates behaviour over the whole of the winter might be the probability that it survives the winter, whereas a suitable currency that evaluates behaviour over an hour during the winter might be the bird's net rate of energetic gain. In this paper, we are concerned with various timescales and how they are related.
Maximizing efficiency results in a lower rate of energy expenditure than maximizing net rate of energy gain (McNamara & Houston 1997). It is therefore of interest to establish the conditions under which it is optimal to maximize efficiency or a modified form of efficiency known as the foraging gain ratio (Hedenström & Alerstam 1995). One of the central themes of this paper is the extent to which optimal behaviour maximizes net rate of gain or efficiency.
We have reviewed a simple case in which fitness is maximized by maximizing the amount of energy gained when there is no predation and no state-dependent effects. In the absence of constraints on energy expenditure, it is optimal to maximize net rate. When there are constraints, it is never optimal to adopt a foraging intensity that is less than the intensity associated with maximizing the foraging gain ratio. An animal's intake rate can be limited by their ability to digest food. In this context, a distinction is sometimes made between maximizing short-term and long-term intake rate (e.g. Fortin, Fryxell & Pilote 2002; Quaintenne et al. 2010). A digestive constraint constitutes a constraint on energy intake. In our terminology, maximizing short-term rate corresponds to maximizing the gain rate while foraging and maximizing long-term rate corresponds to maximizing total energy gained subject to constraints. Fortin et al. go beyond our analysis in that they consider diet choice; different foods differ in their energy density and digestibility, with a constraint on daily consumption. They derive a currency for energy maximization under these circumstances. For a graphical analysis see Hirakawa (1995, 1997).
When there is a trade-off between energetic gain and predation risk a general analysis involves comparing the value of gaining energy with the cost of being killed. Reproductive value acts as a common currency which makes this comparison possible [see expression (2)]. One of the problems with this formulation, however, is that reproductive value is not known a priori. When all that matters is that the animal reaches a critical level of energy reserves, optimal behaviour can be given in terms of a simple alternative currency: the mortality risk per unit of net energy gain is minimized. Including predation reduces foraging intensity below the intensity unet that maximizes net rate. If predation rate is proportional to rate of energy expenditure, the optimal foraging intensity maximizes efficiency. If these rates are not proportional, intensity may be either greater or less than the intensity ueff that maximizes efficiency. If instead of accumulating energy, the forager converts energy into reproductive success, lifetime reproductive success is maximized by minimizing the mortality rate per unit reproductive success (Houston & McNamara 1999; see also Houston & McNamara 1986).
Foraging at high intensity may reduce an animal's condition and/or cause irreversible damage. [Another possibility, which we do not consider here, is that condition is determined by the allocation of resources (Cichon 2001; McNamara & Buchanan 2005)]. As in the case of predation, reproductive value provides a common currency for comparing actions when there are such costs [see expression (3)], but once again reproductive value may not be easy to establish. When a forager has to raise its reserves to a critical level (cf. Gilliam 1982), a simple alternative currency can again be used. Suppose that fitness decreases as the total energy spent increases. If the time at which the critical level is reached has no effect on fitness, then it is optimal to maximize efficiency. If the time at which the critical level is reached is important, the optimal foraging intensity u* lies between ueff and unet (Fig. 1a) and is given by a modified form of efficiency when the time cost is exponential [see expression (1)].
In this paper, we present a model of the optimal daily routines of foraging of an animal that faces a trade-off in which a high rate of energy gain results in a loss of condition. We evaluate performance over a period of many days. At this time-scale, the animal minimizes its mortality rate. At the level of moment-to-moment decisions, we compare optimal foraging intensity with the intensity predicted by the maximization of various simple currencies. Our results show that the optimal foraging intensity depends on state and time of day (Fig. 2) and produces a daily routine (cf. McNamara, Houston & Lima 1994). Foraging intensity is often less than unet but is not necessarily equal to ueff and may be less than it, particularly early in the day (Fig. 2a). At the end of the day, if reserves are low and condition is not very low, then u* tends to be between ueff and unet (Fig. 2b).
Herbers (1981) showed that many species spend a significant fraction of their time resting. A range of explanations has been offered, including satiation (Sutherland & Moss 1985; Jeschke 2007) and predation (Houston & McNamara 1993). Our results show that in the absence of predation, resting can be favoured because it avoids the energy expenditure that would reduce condition (see the interaction of reserves and condition for u* = 0 in Fig. 2).
The requirement that an animal is in energy balance over a day is often used in models of time budgets (Houston, Thompson & Gaston 1996; Gorman et al. 1998). In our model, the foraging intensity that achieves energy balance over 24 h results in a tendency for condition to improve; a higher intensity is often adopted in order to avoid starvation in a stochastic environment.
Our computational model based on reserves and condition looks at routines over a day. Welham (2002) investigated annual routines in a model with the same state variables. He found that mean foraging intensity depended on time of year and was usually less than ueff. Tolkamp et al. (2002) claim that efficiency is a generally valid currency when costs result from an increase in metabolic rate. Our results do not support this view.
We have considered two state variables, energy reserves and condition. Environmental temperature may also be a relevant state variable. Models of the foraging behaviour of birds have included the effect of environmental temperature on optimal energy reserves given that body temperature is maintained (e.g. McNamara & Houston 1990; Houston & McNamara 1993; McNamara, Ekman & Houston 2004). For many species, body temperature will also be an important state variable (see Dell et al., Reuman et al.); it may be optimal for an endotherm to allow its body temperature to drop during the night (Clark & Dukas 2000; Pravosudov & Lucas 2000; Welton et al. 2002). In their analysis of ectotherms, Dell et al. and Reuman et al. make the key features of their models depend on temperature in order to obtain predictions about the temperature dependence of interactions between species. The same approach could be used to extend the analysis of currencies to ectotherms, but if temperature fluctuates, it might not be adequate just to apply the existing analysis with the rates of gain and expenditure depending on temperature; results from models in which food or predation fluctuates show that the rate of fluctuation can be important (Higginson et al. 2012).
In our model of daily routines, the change in the forager's condition is determined by its rate of energy expenditure. Because condition decreases as an accelerating function of energy expenditure, the bout structure determines the change in condition. In particular, for a given amount of energy expenditure over an hour, the loss in condition is minimized if the expenditure is spread evenly over the time interval. Whether the change in condition is determined in this way is an important empirical question. Time-scale might be critical. For a given total energy expenditure, it might be that within a day it is better to alternate between foraging and resting, whereas over a period of several days, it is better to expend the same amount of energy on each day. We have modelled condition as a single state. It is likely that there are several relevant variables that change on different time-scales (McNamara & Houston 2008).
The standard assumption regarding predation is that an animal is at less risk of being killed by predators when resting than while actively foraging. Our results illustrate that the assumptions about risk of predation while actively foraging are crucial. An animal may be able to reduce its predation risk at the cost of reducing its intake by increasing its level of vigilance. Under these circumstances, predation is likely to be increasing and accelerating function of foraging rate (Gilliam & Fraser 1987). Alternatively, it may be that the animal cannot significantly reduce its predation risk while actively foraging. In the absence of a condition variable, it is then always optimal to forage at maximum intensity while foraging in this case. This is no longer true when high foraging intensity is deleterious to condition. Although the optimal intensity is then less than unet, our results (Fig. 5) show that there is a stronger tendency to be above ueff than in the case of accelerating predation risk.
DeLong et al. and Reuman et al. explore the effects of the presence of conspecifics. We have not explicitly included such density-dependent effects. Various density-dependent effects of the presence of conspecifics can be captured in simple currencies. For example, the number of conspecifics could influence the focal animal's gross rate of gain b. If the behaviour of conspecifics does not depend on the behaviour of the focal animal, we can find the optimal behaviour of the focal animal by using the appropriate simple currency with density dependence incorporated in b. When conspecifics interact, a game-theoretic approach is required. As an illustrative example, consider a group of animals that can choose between two patches of food. At a stable distribution of animals between the patches (an ideal free distribution), no animal can increase its reproductive value by moving (Fretwell & Lucas 1969; Milinski & Parker 1991). When reproductive value corresponds to rate of energetic gain then at equilibrium rate of gain is the same for all animals, that is, rate is equalized across the two patches. Moody, Houston & McNamara (1996) extend this approach to include predation by finding the distribution for which expression (2) is equalized across the two patches. McNamara & Houston (1990) and Houston & McNamara (1997) analyse the effects of density dependence when animals choose between patches on the basis of their level of energy reserves. Under these conditions, the stable distribution does not equalize a simple currency across patches.
AIH was supported by the European Research Council (Advanced Grant 250209). There are no conflicts of interest.