Sequential-encounter prey choice and effects of spatial resource variability

J. theor. Biol. (1989) 139, 239-249

Sequential-Encounter Prey Choice and Effects of Spatial Resource Variability GREGG M. RECER AND THOMAS CARACO

Behavioral Ecology Group, Department of Biological Sciences, State University of New York, Albany, New York 12222, U.S.A. (Received 23 August 1988, and accepted in revised form 27 February 1989) Models of diet choice generally assume that resource attributes such as encounter rates, energetic values and handling times are fixed constants. We present a model where either encounter rates or profitabilities of one food type vary randomly through space according to a Markov process. As in the standard diet model, we assume sequential encounter with prey items, mutually exclusive search and handling and we equate enhanced fitness with maximization of the long-term rate of energy gain. The ability of a forager to perceive and respond to the variability determines whether the forager tracks variability or averages over it. The results of this model differ from those of the standard diet model. In particular, the overall proportion of a given food type in the diet does not always match the proportion suggested by the "zero-one" rule of the standard diet model. Also, under some conditions, a forager is predicted to specialize on a prey type of lower average profitability, and to decide on the inclusion of a given prey type based on the abundance of a lower profitability prey. These are never predictions of the standard model. In general, tracking variability results in a greater weighted long-term rate of energy intake than averaging when encounter rates are variable. However, when profitability varies stochastically, a trade-off occurs between the cost of recognizing different kinds of variable items and the increased foraging efficiency gained when tracking variability. This trade-off determines whether tracking or averaging yields the greater long-term rate of energy gain for the forager.

Introduction Several authors have independently presented the standard diet model (e.g. Schoener, 1971; Pulliam, 1974; Werner & Hall, 1974) that describes the diet maximizing a forager's expected long-term rate o f energy gain. The models assume that food items are encountered sequentially and that search time and handling time are mutually exclusive. The results of these deterministic models suggest that ratemaximizing foragers should follow the so-called "zero-one" rule. That is, a food type should either always be taken upon encounter or always ignored. The classical diet model also predicts that the decision o f whether or not to include each lower profitability food type in the optimal diet is independent of the rate at which those items are encountered. Although the model has received some qualitative empirical support (e.g. Werner & Hall, 1974; Krebs et al., 1977), quantitative agreement with the all-or-nothing predictions of the model has seldom been observed. Several generalizations o f the optimal diet model, such as allowing simultaneous encounters (Engen & Stenseth, 1984; Stephens et at., 1986) and examining the effects 239

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of recognition constraints (Elner & Hughes, 1978; Hughes, 1979; Erichsen et al., 1980; Houston et al., 1980) make more realistic predictions concerning diet choices by foragers (reviewed by Stephens & Krebs, 1986). In this paper we present the results of two models of diet choice that explore the decisions made by a forager maximizing its long-term rate of energy gain when the quality or abundance of a food type varies across space or through time. These models maintain the original diet model's constraints of sequential encounter and exclusivity o f search and handling. However, the effects of resource variability produce results that differ significantly from those of the original model. In an earlier paper (Recer et al., 1987) we demonstrated that temporal variability in the rates of encounter with food items had a significant effect on the dispersion of competing foragers between two patches of food. Here we similarly allow the rate of encounter with a food type, or the profitability of that food type, to vary spatially. The forager is assumed to track the variability in resource abundance or quality whenever it can perceive it. Otherwise the forager is assumed to average over the variability. Variability in Encounter Rates For simplicity consider two food types. The net energetic gains per item, e~ and e2, for each type are fixed as are h~ and h2, the handling times for each type. The profitability of type one exceeds that of type two (e~/h~ > eE/h2). The standard diet

model predicts that type one items should always be taken by a forager and type two items should always be ignored if

;~,>

e2

et h2 - e2hl

(1)

where A~ is the encounter rate with the more profitable item. Consider a larger foraging area where these two food types occur. Type two is assumed to be spread homogeneously in a random and independent manner throughout the area such that a single probabilistic encounter rate, Ac, suffices to describe its distribution. However, type one items are distributed in a heterogeneous manner such that the encounter rate of type ones, Av, alternates randomly between two values, Ah and ;t~ (where h and I refer to high and low rates, respectively). The pattern of variability in the density of type one items governs the frequency of transitions between A~ and Ah. The spatial variation experienced as a mobile forager moves through high and low density patches is equivalent to the temporal variation experienced by a sit-andwait predator when input rates for a prey type vary through time. We can model the spatial variation in encounter rates experienced by a forager as a Markov process (e.g. Parzen, 1962). During periods of searching for food a forager experiences transitions in the rate of encounter with type one items. When Av = Ah, a transition to the lower encounter rate At occurs at constant probabilistic rate Yht while the forager searches. Transitions from At to A h O c c u r at constant probabilistic rate ~/th when the animal searches at the lower encounter rate.

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At any time t during the searching process the state probability for )to = through time according to dPr[Ao(t) = Ah] -- 3"thPr[Av(t) = A t ] - 3"htPr[Ao(t) = Ah]. dt

Ah

varies

(2)

Reversing the 1 and h subscripts in eqn (2) gives the dynamics of the state probability for the lower type one encounter rate. At equilibrium, d P r [ A o ( t ) = At]/dt = dPr[Ao(t)= Ah]/dt = 0. The equilibrium distribution 7r gives the long-term proportion of time spent searching at the respective encounter rate pairs. After a large number of transitions the forager will have spent a proportion wt of its search time at combined encounter rate )to + At, and a proportion 7rh o f its search time at combined encounter rate Ac+A h. Using eqn (2) and the identity 7rh = 1 -- 7rt, we obtain the equilibrium distribution

~ = ( ~r, ~rh) = (

Yht

Y,h .)

(3)

\ ")/hi "~- 3"lh 3"hi "~- 3"lh ] "

If 3'm = 3'lh then ~r = (0.5 0-5). The average duration o f a state of Ao relates the spatial dispersion of type one items to a forager's perception and, in turn, its dietary choice. The duration of a state is the random length o f time spent searching for food (which may be interrupted by one or more handling events) between consecutive transitions in Ao. Let d represent the temporal duration of a particular realization of a state of Ao. Then d is a continuous random variable that has an exponential distribution fo(d). For the lower density o f type one items, fo(d) is f t ( d ) = 3',h e -v'ha,

d > 0.

(4)

The expectation and variance are

I~, = 1/3"th

(5)

o'~= 1/3"28.

(6)

Reversing the subscripts on the 3' entries gives the mean and variance for the duration o f a state when ho = hh. Any f o r , g e r capable of phenotypically plastic dietary choice will require a certain length o f time experiencing a particular )to before it can possibly respond to that encounter rate. Consequently, if transitions between )t~ and Ah occur too frequently, the forager will have difficulty tracking the local density of type one prey. Gillespie & Caraco (1987) discuss the way perceptual limitations may constrain foraging efficiency in a rapidly fluctuating environment. We hypothesize that a discrimination time D characterizes the ability to detect transitions between levels of ho. The length of D, relative to the properties of the duration o f a state of Av, determines whether our model forager either tracks variation in the local abundance o f type one items or averages over that variation (see below). Our models assume that the forager makes an assessment of the pattern of encounter with type one items early in the foraging process. Thereafter, the animal

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tracks or averages according to its discrimination abilities and corresponding assessment of the abundance of type one items. Consequently, the local rate of energy gain during the initial assessment period does not influence the asymptotic rate of gain for either tracking or averaging. To examine our hypothesized constraint o n discrimination, a s s u m e / ~ -> ~h- Symmetric arguments for all of the cases we discuss hold if the reverse is true. If the duration of the higher encounter rate with type one items exceeds the discrimination time D with a sufficiently high probability, the forager will track variation in Av. Given that Ah has the shorter expected duration, assume the animal tracks if

Pr[d > Dlfh(d)] = e-V"'° > p (7) where 0 < p < 1. Then the forager tracks if D < - (In P)t.th. Tracking occurs if the discrimination time is shorter than some multiple o f the mean duration of Ah; otherwise the forager averages. When a forager tracks the variability in encounter rates with type one items, three different results are possible depending on the magnitudes of Ah and At. Type one items are always more profitable (by assumption) so they should always be taken regardless o f their abundance. Tracking implies always rejecting type two items if the lower density of type one items is sufficiently great to predict specialization. That is, a tracker never accepts a type two item if condition (1) holds with A~ set equal to A~. If, however, e2

Ah<

(8) e l h 2 - e2hl then the forager should always take type two items along with type one items since even the higher density of the more profitable items is insufficient to produce specialization. Finally, if Ah >

e2

-~ Ai (9) elh2 - e2hl then the optimal diet under tracking includes both prey types when Av = At but excludes type two items when the forager encounters type one items at the greater local rate Ah. The long term proportion of type one items in the diet, under condition (9), can be calculated using a weighted mean of the proportions of type one items taken under each of the two different encounter rates. When A, = Ah this proportion equals 1. When Av = At the proportion of type one items in the diet equals their local, proportional abundance; = At/(A~+Ac). Given the equilibrium distribution of high and low encounter rate states described by eqn (3), the overall proportion of type one items in the diet, qb~, under condition (9) is At q~ = (¢rh)(1) + (m) At + A-----'--~"

(10a)

Rearranging gives qb~

A I "~ "FFhAc

Al+Ac

(lOb)

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Clearly, although this is a generalist diet (i.e. both types are taken), the more profitable food type is taken in a proportion greater than its overall average environmental frequency. This is a prediction of the standard diet model only under complete specialization on the more profitable item. Equation (10b) predicts a diet consistent with the c o m m o n observation that foragers eat a mixed diet, but prefer certain prey types (e.g. Chesson, 1978). If the discrimination time ( D ) is sufficiently long, relative to/zt (recall that/zt >/Zh), the model forager does not detect transitions in the local rate of encounter with type one items. Given this constraint on discrimination, we assume that the initial assessment process yields an average encounter rate for the entire environment, and that the forager selects a diet based on this expectation. The unconditional expected encounter rate with type one items is simply ~rtAt+ "IThAh. Again, type one items should always be accepted, regardless of their encounter rate, due to their greater profitability. The following condition must be met if type two items are to be excluded from the diet (~tAt + ~h,'Xh)>

e2

et h2- e2hl"

(11)

When a forager perceives only the average abundance of a food type with a variable density it responds as if that food type had a homogeneous distribution. However, this may lead to suboptimal behavior relative to the predictions o f the standard diet model. Suppose, by condition (11), an averager should specialize, but At is small enough that, if Av = At and At could be perceived, the standard diet model would predict generalizing. Then for some parts of a foraging bout the forager's energy intake rate would not be maximized. A similar argument follows when A h is large enough to require specializing on type one items part of the time but eqn (11) requires the averager to generalize. Next, suppose that the discrimination time (D) takes a value intermediate to the previous two cases. Specifically, let the forager detect patches characterized by the local encounter rate with the greater average duration (A/by assumption). However, the forager does not detect patches where type one items are encountered at rate Ah before a transition to At occurs. For this case we assume that the forager selects its diet as if type one prey items were encountered at rate At throughout its environment. As discussed previously, the more profitable type one items are always taken. Once again, the forager always rejects a type two item upon encounter if condition (1) holds with At set equal to At. The constraint on the ability to discriminate the varying encounter rates with type one prey again induces what would be termed errors under the classical diet model, as was true for averaging. When condition (1) holds, with At = At, no errors occur b e c a u s e Ah > h. I. That is, if the lower encounter rate with the more profitable prey is sufficiently high to induce specialization, no error occurs when that prey type is encountered even more rapidly. However, if condition (1) does not hold, with A~ = At, errors may occur. The forager now generalizes because At is too low to

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induce specialization. A tracker might do better by specializing on type one items when Av = h.h if '~h is sufficiently large to satisfy condition (1) with AI = h.h.

Variability in Profitability Here we consider a situation where the encounter rates for the two food types remain constant, but the net energetic content o f one of the types varies randomly between two different values. Take the encounter rates for each type to be At and )t_,, both constants. Each type has a constant handling time as before. However, the energy content of type one items, eo, varies randomly between eh and el while type two items have a constant energetic content, ec. Instead of discriminating between high and low rates of encounter a forager now has to tell two different kinds of type one items apart. Stephens & Krebs (1986) use the phrase " s u b - t y p e " to refer to different items which are all of a single prey type but, within that type, are not recognizable upon encounter. Following Hughes (1979) we take the cost of recognition when encountering either type one sub-type as a constant r units of time. r is, in effect, an additional c o m p o n e n t of handling time. Therefore, when type one sub-types are recognized more time is expended on each prey-capture event. Time is also expended with each encounter with a type one item even if it subsequently rejected. We will assume, for simplicity, that discriminating type one from type two items is without cost; " t y p e s " are defined (Stephens & Krebs, 1986) as recognizable upon encounter. In this context choosing to pay the cost associated with recognizing which type one sub-type has been encountered is equivalent to tracking the variability in type one items. Conversely, if a forager does not pay the recognition cost it, in effect, perceives all type one items as being identical. We assume that the forager will then respond only to the average energy content of the items, although we could just as easily use the expectation of a nonlinear function of energy content (Caraco et al., 1980). We will assume that transitions in the profitability o f the type one items follow the same Markov process as in eqn (2) above. The equilibrium distribution, 7r, of low and high profitability items encountered by the forager is given by eqn (3). First assume that a forager tracks variation in profitability. Take type one items to be more profitable, on average, than type two items. But let el/h~ < ec/h2, so that type two items are the more profitable type in some patches. A tracker, by our definition, pays the time cost r on every encounter with any type one item regardless of whether that item is accepted or rejected. The information thus gained concerning the relative profitability of type two items c o m p a r e d to either of the type one sub-types results in two different rules for deciding what items will be included in the tracker's diet. At the higher profitability level of type one items (eo = eh) type two items should be excluded from the diet if ec

hi > ehh2-- e,.(ht + r)"

(12)

However, if eo = el type two items b e c o m e the more profitable type. Again, the recognition cost is paid each time a type one item is encountered, even if that item

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is subsequently rejected. Therefore, the condition for specializing (now on type two items) becomes A2>

el(1 + A i r ) echl - elh2 "

(13)

As can be seen, tracking can result in s~ecializing on the food type which is, on average, less profitable. Further, from condition (13), the encounter rates of both type one and type two items become important, in terms of choosing items to include in the diet, when a tracker is encountering type one items with the low energy value. This is a direct result of assuming a recognition cost (Hughes, 1979). When no recognition cost is paid by the forager we assume that the only information used by the animal is the average profitability of the variable food type. Again, assuming that type one items are, on average, more profitable than type two items, E [ e v ] / h ~ > ec/h2, condition (1) can be directly applied to express the condition where type two items should be excluded from the diet: AI >

ev E[ e~]h2- echl

(14)

where E [ ev] = oriel+ 7rheh. This condition appears to be essentially the same as the one derived in the original diet model. However, if et/h~ < ec/h2 it then predicts that an averager will exclude the more profitable item (in this case type two items) from the diet whenever eo = e~ and condition (14) is met. If condition (14) is not met, then whenever eo = eh, inclusion of type two items in the averager's diet would also result in not maximizing the long term rate of gain if e h was large enough so that a tracker would specialize on type one items at the higher profitability. Discussion

How do trackers and averagers compare in terms of their long-term rate of energy gain? Suppose the encounter rate with type one items fluctuates between At and Ah. Further suppose that a tracker always specializes on type one items (because the lower encounter rate )t~ is large enough for specialization), or that a tracker always generalizes (because the higher encounter rate Ah is insufficient for specialization). In both cases trackers and averagers select the same diet and consequently attain the same long-term average rate of energy gain. Now suppose condition (9) applies, so that a tracker should specialize in response to Ah, but should generalize in response to At. In this case tracking and averaging result in different long-term rates of energy gain. If E[Ao]<

e2

(15)

e l h 2 - e2hl

then tracking yields a higher long term rate of energy gain than averaging if ZrhAhel ---4 l+Ahht

"n't(Atel + Ace2)

t-

"rrh(Abel + Ace2) zrt(Atel + Ace2) +

l+Athl+Ach2 l+Ahht+Ach2

1 +Athl + Ach2"

(16a)

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This reduces to hi, >

if2

(16b)

el h 2 - e2hl

which, by condition (9), is always true. Therefore when the mean rate of encounter with type one items is low enough so that averagers always generalize, and condition (9) is true, tracking the variability in encounter rates yields a higher long term rate of energy gain than does averaging. We obtain the same conclusion when averagers specialize on type one items. It can be shown that tracking yields a higher long-term rate than averaging when both conditions (9) and (11) hold if At<

e2

(17)

el h 2 - e, hl"

But this is always true when condition (9) is satisfied. Hence, when encounter rates vary randomly across space, tracking the variability always yields a long-term rate of energy gain equal to or greater than averaging. This is intuitively appealing as there is no cost, in our model, associated with the gain of additional information about the resource under tracking. This result is consistent with the results of Recer et al. (1987). When profitability varies the situation is more complicated. Tracking variability is a result of the forager having more information about the resource. This generally leads to greater foraging efficiency for trackers compared to averagers. However, in our model there is a cost associated with gaining this additional information; the recognition cost, r. As a result, a trade-off between the increased handling time associated with recognizing different type one sub-types and the increased energy per item selected under tracking determines whether tracking or averaging yields the higher net long-term rate of energy gain. As an example, suppose the energetic content of type two items is much lower than the energetic content of the more profitable type one sub-type while still retaining the assumption that the type two items are more profitable than items of the less profitable type one sub-type. For a given r, At and A2 are most likely to be sufficiently large to satisfy conditions (12), (13) and (14) when this is the case. When conditions (12) and (13) are met a tracker will switch between two specialist diets. When the forager encounters type one items of the more profitable subtype, along with type two items, only the type one items will be selected. However, when type two items are encountered in patches with the less profitable type one sub-type only type two items will be accepted. In contrast, under condition (14) an averager only takes type one items. A proportion 7r~ of all items in the averager's diet are the low profitability type one sub-type while the remainder (Trh) are of the high profitability sub-type. Tracking will result in a higher long-term rate of energy gain when the above conditions apply if "lrhA I 8h

l+A~(hl+r)

+

"rl'lA2ec

l+A2h2+A~r

>

7rhA I eh

"rl'lA~et

~ - l+Alhl l+Aihl"

(18a)

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Rearranging gives ~rt l+A2h2+Alr

l+A~h~J >~rh

l+A~h~

l+A~(h~+r)

"

(18b)

If the increased efficiency gained by specializing on type two items when ev = e~ [the left side of condition (18b)] outweighs the cost of gaining information about type one items [right side of condition (18b)] then tracking will yield a higher net long-term rate of energy gain. Similarly, suppose an averager should, by condition (14), take only type one items. However, suppose type one items are encountered more frequently (or type two items are encountered less frequently) than assumed above so that a tracker should reject type two items only when encountering the more profitable type one sub-type and should accept both food types when encountering type two items along with low profitability type one items. Under these conditions it can be shown that the condition

l+Aj(h~+r)+A2h2

l+A~h~.j >rrh

l+A~h~

l+A~(h~+r)

(19)

must be satisfied if tracking is to yield a higher net long-term rate of energy gain than averaging. As in condition (18b), if the extra efficiency gained by the tracker, by generalizing when e~ = e~, exceeds the cost associated with tracking then trackers will do better than averagers in terms of their energy intake rate. A consequence of the trade-off between the cost of recognition and the increased foraging efficiency under tracking is that when conditions (12), (13) and (14) all predict generalizing, averagers always do better than trackers. This is because, following a generalist diet, trackers and averagers take the same kinds of food items in the same proportions. However, trackers still pay the time cost associated with recognizing the different type one items and so gain the same total amount of energy as averagers but in a longer period of time. In this case the information gained under tracking does not result in any increase in foraging efficiency (see Stephens, 1987). Many workers have changed the constraint assumptions of the standard diet model in order to increase the realism of the model (reviewed by Stephens & Krebs, 1986). These changed assumptions, concerning both attributes of the forager and its environment, can result in predictions of partial preferences for, or even specialization on, low profitability items, unlike the original model. Most of this work has maintained the original model's assumption of an unchanging environment. However, Stephens (1987), like our paper, explores the effect of variation in profitability on diet choice. Stephens' work concerns how to sample food items which have stochastically varying profitabilities. In his model the forager is incapable of discerning good from bad sub-types of the variable food type until the forager has consumed each item. This contrasts with our model where the forager can tell variable food type items apart with certainty, upon encounter, when the recognition cost is paid. Although the two models are concerned with different questions the results of each are

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complimentary. Stephens finds that tracking variation in profitability is less worthwhile when obtaining information about the variable food type is diMcult [in Stephens (1987) this is when q is near 0.5] or when the energetic value (i.e. profitability) of the alternative food type (our type two) is small compared to the average value of the variable food type. We have shown similar results here (r high or e~./h2 low makes tracking less favorable). However, when the profitability of the alternative food type is large compared to the average value of the variable food type, Stephens' model also suggests that tracking becomes less advantageous. When ec/h2 increases, relative to the average profitability of type one items in our model it becomes easier to satisfy the conditions that favor tracking over averaging [i.e. conditions (18b) or (19)]. This difference occurs because, in Stephens (1987), the forager attempts to gain information about the variable food type by sampling. As the energetic content of the alternative food type increases the cost of the errors incurred, due to this sampling, increases and so specialization on the alternate food type is favored. Increasing the profitability of the constant food type in our model, on the other hand, does not increase the cost of gaining information. Tracking, therefore, becomes more likely, compared to averaging, as ec/h2 increases. However, tracking variation in profitability still results, in our model, in specializing on the alternative prey type for part of the time. Tracking the variation in resource abundance can lead to switching between a specialist and generalist diet. This kind of response could suggest another mechanism for dietary partial preferences (see McNamara & Houston, 1987). Although our model predicts all or nothing decisions, under tracking, for each state of Av, the overall proportion of low profitability items in the diet of a tracker is not necessarily 0 or [ zr~hc/(hc + hi)] + [~hAc/(hc + Ah)]. The simplest, and perhaps, most likely explanation of observed partial preferences (e.g. Krebs et aL, 1977) is variance about a threshold (Stephens, 1985). However, responses to more complex environmental conditions such as spatial resource heterogeneity or simultaneously encountered prey items could also lead to partial preferences. Our model has demonstrated that stochastic variation in resources can lead to prey choice behaviors more complex than those predicted by the standard model when animals track the variation. By recognizing perceptual constraints (as in Recer et al. 1987) that might limit the ability of the forager to respond to spatial resource variation, we also have shown that averaging over resource variability generally results in all or nothing predictions, concerning diet choice, similar to those of the standard model. However, the lack of information, implied by averaging, which results in these predictions may often lead to reduced efficiency compared to the predictions for trackers under the same condition. We thank David W. Stephens and an anonymous reviewer for their comments. B. V. White provided thoughtful discussions on an earlier version of the manuscript. We appreciate the support from NSF grant BNS-8616736 and from the SUNY Albany Benevolent Fund. REFERENCES CARACO,T., MARTINDALE,S. 4&Wl-.IIT'TAM,T. S. (1980). An empirical demonstration of risk-sensitive

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