Optimal diet choice for large herbivores: an extended contingency model

Authors


Abstract

1. A more general contingency model of optimal diet choice is developed, allowing for simultaneous searching and handling, which extends the theory to include grazing and browsing by large herbivores.

2. Foraging resolves into three modes: purely encounter-limited, purely handling-limited and mixed-process, in which either a handling-limited prey type is added to an encounter-limited diet, or the diet becomes handling-limited as it expands.

3. The purely encounter-limited diet is, in general, broader than that predicted by the conventional contingency model.

4. As the degree of simultaneity of searching and handling increases, the optimal diet expands to the point where it is handling-limited, at which point all inferior prey types are rejected.

5. Inclusion of a less profitable prey species is not necessarily independent of its encounter rate and the zero-one rule does not necessarily hold: some of the less profitable prey may be included in the optimal diet. This gives an optimal foraging explanation for herbivores’ mixed diets.

6. Rules are shown for calculating the boundary between encounter-limited and handling-limited diets and for predicting the proportion of inferior prey to be included in a two-species diet.

7. The digestive rate model is modified to include simultaneous searching and handling, showing that the more they overlap, the more the predicted diet-breadth is likely to be reduced.

Introduction

The classical model of prey choice (named the ‘contingency model’ (CM) by Belovsky 1984) describes the diet that would maximize the long-term intake rate of a mobile predator whose foraging time is divided between two alternative activities: searching for and handling prey items (MacArthur & Pianka 1966; Charnov 1976). Handling time is defined as the time taken for an animal to pursue, capture and consume its prey. This use of carnivore-centric language is merely historical and the foraging terminology throughout this paper is meant to apply equally to herbivory. The contingency model has been progressed in a number of ways (reviewed by Stephens & Krebs 1986), by changing the assumptions about the constraints under which the forager operates, for example that the forager has complete information, encounters prey sequentially and recognizes them at once. The assumption that searching and handling are mutually exclusive and competing activities originated with the use of Holling's disc equation (an economic model of diminishing returns) to describe multispecies functional responses (relationships between food abundance and intake rate; Holling 1959), and has persisted in standard descriptions of the prey model (e.g. Krebs & Davies 1991). The CM continues to be the most frequently used model of diet selection.

There are many foraging circumstances when exclusivity of searching and handling will not necessarily hold. Vertebrate herbivory offers an important example where certain aspects of feeding do not exclude searching for food, others include the short-term feeding of Baleen Whales on zooplankton and birds and bats that feed by flying through clusters of insects (Stephens & Krebs 1986). Recent empirical studies have supported the view that the handling time of large grazers is partly exclusive to searching and partly inclusive (Spalinger, Hanley & Robbins 1988; Gross et al. 1993; Laca, Ungar & Demment 1994; Ginnett & Demment 1995).

Vertebrate herbivores are frequently generalists, but feed selectively. The way the vertebrate herbivore's functional response varies with prey characteristics and foraging circumstances is important to optimal diet choice (Lundberg, Åström & Danell 1990; Spalinger & Hobbs 1992). The contingency model of optimal diet choice has been used in an attempt to explain the proportions of food types in the diet of large herbivores (e.g. Owen-Smith & Novellie 1982). It has been contrasted with linear programming models (Belovsky 1978, 1984, 1986) in Belovsky (1981) and Belovsky (1984) and nutritional wisdom models (Westoby 1974; Rapport 1980) in Dearing & Schall (1992) and in both cases, the classical CM matched observed diet choice less closely. In the above studies, herbivore searching and handling were assumed to be mutually exclusive in the CM.

Spalinger & Hobbs (1992) derived functional responses that explicitly account for the overlap in handling and searching times that mammalian herbivores can exhibit (following the observations of Spalinger et al. 1988). Recently, Farnsworth & Illius (1996) extended Holling's disc equation to include overlapping of searching and handling, and showed that grazing by mammalian herbivores is consistent with it. Overlapping searching and handling undoubtedly affects the functional response of predators and will have an impact on the make-up of optimal diets (Spalinger & Hobbs 1992) and hence on diet breadth.

In this paper we will derive optimal prey choice rules from the more general foraging model of Farnsworth & Illius (1996) which allows searching and handling to overlap, building on the theoretical developments of Spalinger & Hobbs (1992). We will go on to enumerate the effect of competition between the searching and the prehension and mastication components of handling on diet choice in grazing and browsing herbivores using an extended contingency model. We shall conclude that this has significance for plant–herbivore interactions. It is important to note that our theory refers to within-patch foraging, rather than issues of choices between patches, thus we are considering situations in which different prey types are encountered at random during simultaneous searching. Also, in common with other models of this type, the values of foraging variable referred to in our analysis are the expected values of random variables whose time series is assumed to be stationary so that there is no change in expected values as foraging progresses (thus we do not consider patch depletion, for example).

Variable simultaneity of searching and handling

Let the proportion of handling time Th (s) which is exclusive to time spent searching Ts (s) be η, then the total time spent in foraging can be accounted for as follows:

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Let D (m–2) be density of prey of size S (kg) and A be the predator's searching rate (m2 s–1), so that the encounter rate is AD. The number of prey consumed is given by one of two possible calculations, whichever gives the smallest value. The first refers to foraging limited by encounters and is the product of total search time and the encounter rate AD. The second refers to foraging limited by handling, and is the total time of foraging T divided by total handling time Th. Assuming that intake rate I is proportional to the number of prey of size S consumed per unit of time, according to Farnsworth & Illius (1996), it can be calculated as:

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where h (s) is the handling time for a single prey item. Equation 2 expresses a more general prey model than the classical model of Holling (1959), by relaxing the assumption that searching and handling are mutually exclusive.

THE MULTISPECIES DIET

Consider a range of diet options (species) to choose from, each denoted by the subscript (i = 1…N). Conventionally, λι is defined as the encounter rate with the ith prey type during searching which is equal to AD for that prey type. We shall briefly introduce the term λri which is the realized encounter rate with the ith prey type (i.e. the rate of encounter over all foraging time). During foraging, λri is diminished from the short-term maximum attainable during searching (equal to λi), by the time during which handling excludes searching:

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The term Tlost represents the total time lost by exclusive handling to searching in the total foraging time T and is found from

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where pi is the probability that an encountered prey item of the ith type will be included in the diet and N is the number of prey types to choose from in total.

Accordingly, by substituting eqn 3a into 3b and solving for Tlost/T, the realized encounter rate can be written:

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from which the energy intake rate across the encounter-limited diet (Iencounter) of N prey types can be calculated, assuming the energy of the ith prey item is ei:

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Equation 5 is the multispecies case of the upper part of eqn 2.

In handling-limited foraging the intake rate across the whole diet (Ihandling) is simply the mean intake rate among the prey items that are included:

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So eqn 2 can be written for a multispecies (also called mixed) diet as:

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There are now three imaginable cases in foraging: (1) all prey items are subject to encounter-limited foraging, (2) all prey types are subject to handling-limited foraging, or (3) the ‘mixed case’ where some items are encounter-limited and the rest are handling-limited.

ENCOUNTER-LIMITED FORAGING

Following the arguments used by Stephens & Krebs (1986), if eqn 7 for encounter-limited foraging (the upper part) is maximized with respect to pi, a rule determining diet choice for a rate-maximizer under encounter-limited foraging is obtained.

If the prey types {1,2…iN} are ordered in rank of profitability (e/h), then prey type i is rejected if

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where j = 1,2,3…i– 1,i + 1,i + 2…N.

This result follows several steps of algebra that are exactly the same as those given in Stephens & Krebs (1986) and therefore not repeated here. Equation 8 is the same as the classical contingency model, with representing exclusive handling time, but it is now only one part of a fuller solution. It is possible to carry on the arguments used in Stephens & Krebs (1986) to show that pi = 0 or 1 is the only possible solution in pi for eqn 8 (this is the zero-one rule). Note that the replacement of h with ηh (when η≠ 1) has the effect of broadening the diet by reducing the effect of handling time. The term η has introduced a new degree of freedom into the model. In the extreme, if the predator can search throughout handling for all i (ηi = 0), ηihi = 0 and the prey selection rule becomes:

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If ei are all positive (energy gained is greater than energy lost in handling), condition 9 is never true, hence when searching can take place throughout handling, none of the prey types should be rejected during pure encounter-limited foraging.

HANDLING-LIMITED FORAGING

A prey type is subject to handling-limited foraging if hi≥ 1/λi (i.e. λihi≥ 1). If all the prey items are subject to handling-limited foraging, then only the most profitable prey type should be included in the diet, since by the time the predator has finished handling any item of this type, it will have encountered another of the same type, rejecting all others.

MIXED PROCESS FORAGING

Suppose that foraging is encounter-limited for a diet including the k– 1 most profitable prey types (i.e. highest ei/hi) and a further prey type (the kth) is added to the diet, and that foraging for the kth prey type alone were handling-limited. Under these circumstances, the diet of k prey types would be handling-limited. In this ‘mixed’ case, all subsequent prey types would be rejected by a rate maximizer.

Mixed process foraging does not necessarily mean foraging for prey types, some limited by encounters and others limited by handling ‘in their own right’. As more prey items are added in a diet of encounter-limited prey, the diet will tend towards handling-limited even if the prey items constituting it are all individually encounter-limited. If the kth prey type is sufficiently frequently encountered in relation to the remaining time available for handling, all inferior prey k + 1…N will be rejected, some of the kth type will be rejected and pk will not in general be 1. Therefore the zero-one rule would not apply in such circumstances. The case where handling-limited foraging emerges from a diet made-up of encounter-limited prey types may be called contingent handling-limited foraging because the handling limitation is contingent on the inclusion of other prey types in the diet. This, together with the case when a handling-limited prey type, in its own right, is included in the diet (discussed above), we call mixed process foraging, after Spalinger & Hobbs (1992), who referred to encounter-limited foraging as processes 1 and 2 and handling-limited as process 3 foraging.

Two questions arise concerning mixed process foraging: (1) what criterion indicates its onset and (2) how many of the kth prey type should be included in a mixed process diet by a rate maximizer? The answers to both questions can be derived from the following statement. If the predator can find more than a threshold number (which we shall call nlim) of the kth prey type in the total time available for searching, then the kth prey type can make the diet handling-limited. The threshold is simply the total time available for handling additional prey (equivalent to the total time in ‘pure’ searching (T*s)) divided by the time taken to handle an item of prey type k:

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Let nk be the number of the kth prey type actually taken per any other prey item and assume that the inclusion of these in the diet makes it handling-limited. This diet will give maximum intake rate only if the intake rate, for handling-limited foraging (eqn 7) (for nk > 0) is greater than the rate for encounter-limited foraging (eqn 7) (for nk = 0). That is,

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In the case of a rate-maximizer, if inclusion of the kth prey item caused the diet to become handling-limited (i.e. in mixed-process foraging), then pj = 0 for all j > k. The rate-maximizer would take no more than nlim of the kth prey type under any circumstances, since any addition of these beyond that number would displace the handling of more profitable prey types. However, unless ηk = 0, search time will be lost to competition with the search-exclusive handling of prey items of the kth type and the optimum number taken will be less than nlim. In the limit, if ηk = 1, any time devoted to handling the kth prey type would be lost from searching for superior prey, so pk = 0 in this case. Thus pk strongly depends on ηk.

TWO-PREY TYPE EXAMPLE

Like Stephens & Krebs (1986), we illustrate this acceptance rule with a two-prey choice example. Ranking the prey by profitability, denoting them as 1 and 2, respectively, after rearrangement, inequality 11 reduces to

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where λ’ = λ1(1 –η2n2).

Further rearrangement yields

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SOLUTION BOUNDARIES

The solution, inequality 13, is not valid over the whole range of n2 and h2: it is constrained by three boundaries. The first of these is simply that n2≥ 1 for a mixed-process diet, since fractional prey are meaningless in the present context (n2 is not treated as a statistical quantity). The left-hand side of inequality 12 assumes that the diet is handling-limited and must therefore be subject to the criterion defining handling-limited foraging, which, in this case, is the first to occur of:

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The number of prey type 2 taken per prey type 1 cannot exceed the number encountered between items of prey type 1, thus

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which resolves into

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Thus if

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n2 has no upper bound (this is when prey type 2 alone would be handling-limited). However, taking more than nlim of prey type 2 would be non-optimal because fewer than all encountered prey type 1 would be taken. The optimal number of prey type 2 per prey type 1 is, then, equal to nlim, which in this case is

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Note that if η2≠ 0, then handling prey type 2 during searching for type 1 will extend the search time by n2η2h2. In the limit as η2→ 1, available search time will fall to zero so the time to the next encounter with prey type 1 will tend towards infinity, predicting that n2→∞. This is the reason why n2 may have no upper bound in inequality 13. Taking more than nlim(2) of prey type 2, delaying encounters with prey type 1 in this way, is always suboptimal since intake rate during handling of prey type 1 is, by definition, greater than during handling of prey type 2 and in mixed process foraging all time is taken up by handling.

An example solution to inequality 13 is shown graphically in Fig. 1 as the intersection (line A) of the two surfaces representing handling-limited and encounter-limited foraging, respectively. The solution boundaries (n2≥ 1, inequality 14 and nlim(2)) are drawn on the handling-limited surface as lines B, C and D, respectively.

Figure 1.

. Intake rates for two foraging strategies when two prey types are encountered varying with h2, the handling time of an inferior prey type, and n2, the number of the inferior type taken per superior type. The solution to inequality 13 (line A) is the line of intersection between the curve, which represents intake rate from handling-limited foraging (including an inferior prey type), and the plane which represents intake rate from encounter-limited foraging (rejecting the inferior prey type). The solution is valid within the boundaries (n2≥ 1; line B) and inequality 14 (line C). The maximum number of prey type 2 per prey type 1 is nlim(2) (from eqn 17), which is represented by line D. In this example, e1 = 2.0, e2 = 1.0, h1 = 1.0, η1 = 0.1, η2 = 0.2, λ1 = 0.5.

At least one item of prey type 2 per item of prey type 1 is included in the rate-maximizing diet, given the conditions that inequalities 14 and 15 are true and nlim(2) ≥ 1, if h2h2(1) where

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The solution boundaries discussed for this two-prey type illustrative case also apply, with some modifications, to the general multispecies diet case at the boundary between encounter- and handling-limited foraging.

INCLUDING DIGESTIVE HANDLING

It is argued that in vertebrate herbivores, digestive handling often exceeds ingestive handling and therefore overrides it in governing diet choice (e.g. Doucet & Fryxell 1993; Owen-Smith 1994; Wilmshurst, Fryxell & Hudson 1995). For this reason, Verlinden & Wiley 1989 developed the digestive rate model (DRM) to explain the effect of digestive rate constraints on optimal diet choice, but their model does not allow for simultaneous searching and handling. In a recent modification of the DRM, Hirakawa (1997) explicitly stated that his and Verlinden and Wiley's model assumed exclusivity between search and handling processes and was itself a modification of the contingency model. We can therefore consider what effect simultaneous searching and handling of prey items has on the solutions to the DRM. For this, we shall adopt Verlinden and Wiley's notation (defining Qi to be the quality of food type i in J ml–1 min–1).

If the total time available for foraging Tf is more than the time taken to fill the gut with the highest-quality food (T1), then as Verlinden & Wiley (1989) conclude, no other food types should be included in the diet and the rate of energy absorption would be E = Q1Vc (J min–1) as they state, where Q1 is the quality of the best food and Vc is the capacity of the gut (ml). However, where in their analysis T1 = Vc(s1 + h1), we now have T1 = Vc(s1 + η1h1) (where symbols have their previous meanings). For η1 < 1, i.e. as searching and handling overlap, T1 gets shorter and the optimal diet is more likely to be limited to the highest-quality food item.

If Tf < T1 then additional food types may be added to the optimal diet under the DRM, in the notation of Verlinden & Wiley (1989) if E(1) < E(1,i). If foraging for the food with highest is itself handling-limited, then our previous conclusion that under this circumstance no other prey types should be included in the diet still holds – only take the best food type.

If foraging for the food with highest on its own is encounter-limited, then we can calculate:

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We have demonstrated that under the CM, in these circumstances, the diet should be expanded until it just becomes handling-limited as a whole diet. Let us therefore assume that adding an amount V2 (ml) of a second food type with quality causes the diet to just become handling-limited. In this case, total time foraging is accounted for as follows:

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since there is now no pure searching (recall that this is only possible because searching can take place during handling) and note that V2 may represent a ‘partial preference’.

The time available for handling food type 2 is the time that would otherwise be spent in pure searching, but this searching must still be accommodated in the same total foraging time; thus we can equate total pure search time (without food type 2) with the total time handling type 2 that overlaps with searching:

[s1 + (η1– 1)h1]V1 = (1 –η2)h2V2,

giving

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Combining eqns 20 and 21, we obtain

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An amount V2 of food type 2 should be included if

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To illustrate the meaning of inequality 23, we will briefly present four special cases.

If η1 = η2 = 1, there is no overlap of searching and handling and inequality 23 becomes

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which is effectively the same as eqn 2 in Verlinden & Wiley (1989).

If η1 = 0, η2 = 1, inequality 23 becomes

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which is a more severe limitation on the diet. This makes sense because we have now allowed searching to overlap with all of handling in the highest-quality food type, so it is less likely that a lower-quality food type will improve the diet's energy intake rate.

If η1 = 0, η2 = 0, then inequality 23 becomes

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giving

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This result is significant for the fact that the criterion to include the inferior prey type on the grounds of digestibility and passage times, under these circumstances, is independent of those qualities.

Finally, if η1 = 1, η2 = 0, then inequality 23 becomes

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Assuming Q2 > 0, this gives

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The meaning of inequalities 26 and 27 is that the animal should take a quantity (V2 ml) of food type 2 to top up the gut, irrespective of its quality (as long as it is positive). Note that this is an example of the partial preference which is a natural outcome of Verlinden & Wiley's (1989) model and also of ours.

Discussion

We have derived a prey-choice algorithm based on the contingency model, for which exclusive searching and handling is not a required assumption. In common with the CM, we have assumed that (1) predators have complete knowledge of the relevant properties and abundance of available prey types, (2) they maximize the rate at which some currency (usually energy) becomes available by ingestion, and (3) searching for each prey type is carried out simultaneously. From this, three distinct cases of prey-choice dynamics have been shown: handling-limited, encounter-limited and mixed-process foraging. This extends the work of Spalinger & Hobbs (1992) by integrating their theory with the CM to give a more general theory of optimal prey choice. Only in the case of encounter-limited foraging are optimal diets found to follow the same rule as in the classical CM, with the substitution of ηihi for hi. The classical CM is, then, a special case of a more general selection algorithm and applies over a limited range of conditions. As n→ 0, the optimal diet expands to the point where it is handling-limited. Both increasing encounter rate and reducing η make handling-limited foraging more likely. As n→ 1, the diet tends towards that predicted by the classical CM. Real diets generally lie between these limits.

Our modified CM shows what happens when a relatively rare high-profitability prey is mixed with an abundant lower profitability prey. If the latter gives rise to handling-limited foraging when included in the diet, the zero-one rule of the classical CM no longer holds, but the ratio of numbers of inferior to superior prey that should be taken can be calculated using inequality 13 subject to various boundary conditions. This ‘take some of the encountered prey’ strategy is only possible because simultaneous searching and handling enable handling-limited foraging. Intuitively, it may be more profitable to handle continuously than to spend time searching for a superior prey type, during which instantaneous intake rate is zero.

In large animal grazing, where exclusive handling time could be interpreted as the prehension (cropping) component and inclusive handling time as the mastication component of feeding (Shipley et al. 1994; Newman, Parsons & Penning 1994; Laca et al. 1994; Ginnett & Demment 1995), foraging is likely to be handling-limited with searching substantially overlapping handling time. This is far from the assumptions of the classical contingency model, but within the scope of our extended version. If a predator is able to continue searching for the more profitable prey type while handling the less profitable, then it may be optimal to take a certain number of these less profitable prey between encounters with the most profitable prey. This conclusion provides a possible optimal foraging explanation for the ‘mixed diet’ of large herbivores.

Our modified CM theory predicts a greater diet breadth than the classical theory in encounter-limited foraging, but, compared with the classical CM, in handling-limited foraging it predicts a narrower diet. Belovsky (1981) concluded that the CM predicted that the Moose in his study would optimally reject prey of less than 0·84 kJ/item, whereas they were observed to reject all prey less than 1·3 kJ/item – the predicted diet was too narrow. This result may be due to overlapping of handling and searching putting the Moose into processing, not encounter-limited foraging. Dearing & Schall (1992) found the CM predicted too narrow a diet in a herbivorous lizard (attributing this to the importance of a variety of nutrients being ignored by single-nutrient maximizing models), but allowing for simultaneous searching and handling (if the diet remains encounter-limited) would give a broader diet prediction. (Unfortunately, insufficient data were published with these two studies to allow us to test such assertions.)

One criticism of the classical CM is that it does not take digestive rate into account, though this may be a limiting constraint (Belovsky 1978). The alternative of linear-programming modelling (Belovsky 1978, 1984) suffers from the drawbacks that (1) it can only resolve choices between n prey types if there are at least n + 1 different constraints, (2) it has a poor taxonomic resolution, identifying an optimal mixture of food types rather than a ranking of prey and (3) it does not lend itself to direct comparison with the CM (Verlinden & Wiley 1989). A possibly better alternative is the digestive rate model (DRM) (Verlinden & Wiley 1989) which is the logical complement of the CM and has been supported by evidence from measurements of cattle diet selection (Vulink & Drost 1991). The DRM does rank prey choices and can generate equivalent predictions to the contingency model. Our extension of the DRM to allow overlapping of searching and handling has sometimes profound effects on the conclusions of the DRM. The result is generally to reduce diet breadth relative to the original DRM prediction, unless the highest-quality food type is encounter-limited and searching can take place throughout a substantial part of its handling time, in which case the diet breadth may be increased relative to the original DRM prediction. It is likely that where ingestion rate is high enough to meet digestive constraints, the DRM will explain diet choice better than the CM with our modification, but where this is not the case, the reverse might be true. Unfortunately for modelling, many predators may have sufficiently flexible physiological constraints that the border between ingestive and digestive control of diet choice is in practice very hard to predict (Owen-Smith 1994).

Our solutions have assumed stationary distributions of food parameters (encounter-rate, handling-time, etc.) because constant expectation values have been used throughout. Where food items are aggregated it is more likely that handling-limited foraging will alternate with encounter-limited foraging, as the animal moves between patches. The optimal solution would then need to consider patch exploitation and depletion which is beyond the scope of this paper. In large animal herbivory, there are many examples where the most profitable prey type is relatively rare and distributed among a more common (even ubiquitous) prey type of less profitability. The classical contingency model predicts that inclusion of the less profitable prey is independent of its encounter rate. If it is rejected from the optimal diet, the zero-one rule dictates that none of the inferior prey type is eaten, irrespective of its abundance, or of the time spent searching for the more profitable prey type. We have shown that by allowing searching to carry on through part of handling, some of the less profitable prey type will be included in the optimal diet even under the CM. This gives an alternative optimal foraging explanation for herbivores’ mixed diets. We have recognized that many vertebrate herbivores are constrained by digestive rate rather than ingestive handling time and we have shown that in this case as well, more of the most profitable prey can be taken because the animal can be more selective if searching can continue through handling.

Acknowledgements

Gary Belovsky and Norman Owen-Smith made helpful criticisms of an earlier version. The studies were funded by grant in aid from the Scottish Office Department of Agriculture, Environment and Food.

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