Evolutionary Ecology, 1995, 9, 18-29

The effect of costly information in diet choice

B U R T P. K O T L E R and W I L L I A M A. M I T C H E L L * Mitrani Center for Desert Ecology, Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus 84990, Israel

Summary

We distinguish three cases which consider the effect of information on animal behaviour: static information, obligate information and facultative information. Static information deals with the case in which the animal does not acquire additional information; it starts with enough information to discriminate options. Obligate information deals with the case in which the animal acquires information at no additional cost. Facultative information is when the animal may choose to pay a cost in order to acquire information. We illustrate the differences among these three situations by analysing the optimal diet problem subject to the different information regimes. Compared to the case with static information, obligate recognition time narrows the range of prey densities over which an optimal forager feeds selectively, and facultative recognition time reduces it further still. The three models yield qualitatively different predictions regarding how the optimal diet varies with relative abundances of alternative resources. In the space of resource densities, the line separating the optimal behaviours of selectivity and opportunism is straight for both the perfect and obligate information cases. In the case of facultative recognition time this line or isoleg is part of a quadratic curve. This non-linearity yields two completely new predictions: a less profitable resource may be lost from the diet after becoming more abundant and the poor resource may be included in the diet as a result of the rich resource becoming more common.

Keywords: diet choice; static information; obligate information; facultative information

Introduction

Animals make decisions based on their information about available options. Often, better information can result in decisions generating higher fitness. The quality of information possessed by an animal may be improved in two distinct ways. One way is that the animal acquires the information at no additional cost while performing some necessary task. For example, a forager may update its estimate of patch quality even as it exploits the patch (Iwasa et al., 1981; McNamara, 1982; Valone and Brown, 1989). The improved estimate derives from the discovery of food items and the estimate of time spent foraging in the patch. Similarly, a forager may not be able to recognize the quality of a food item until it has spent some time preparing the item for ingestion. Recognition time here is part of normal handling time (Hughes, 1979). In these cases, the acquisition of information can be considered as free because the decision-maker did not have to pay any special cost to obtain it.

Another way to obtain information is to pay a special cost of acquiring it. If this is an option, then the animal is faced with an additional choice of whether that information is worth paying for. The animal should not pay the cost of improving the state of its information unless the expected improvement in the benefit of making the bet ter decision is greater than this cost. In these cases, the behaviour can be viewed as involving two different but interdependent decisions. First, how much information is worth buying and, second, given the information state, which option should be chosen (e.g. which mate choice or foraging strategy).

* Present address: Department of Zoology, University of Wisconsin, Madison, WI 53706, USA.

0269-7653 t~) 1995 Chapman & Hall

The value o f information to diet choice 19

The problem of costly information is a general one that should arise in a variety of contexts in behavioural ecology (Stephens, 1989). Here, we examine the consequences of costly information in the context of a forager choosing its diet. We select this model to illustrate the potential importance of costly information because the basic model is widely known in behavioral ecology, as testified to by its appearance in a number of texts (e.g. Perrins, 1980; Huntingford, 1984; Alcock, 1990; Gill, 1990). In addition, this model has been the focus of a considerable amount of empirical testing (see a review in Stephens and Krebs, 1986). A number of modifications of the basic model have already been developed; indeed, one of these modifications considers the effect on diet choice of acquiring information while foraging (Hughes, 1979; Houston et al., 1980; Lessells, 1985). However, our examination of the effect of costly information differs from these treatments in a very general and important way. The essential distinction is that these models deal with information acquired in an obligate sense through the normal course of handling or preparing food for consumption, whereas we also consider information that is optionally paid for, i.e. information that may be acquired in the facultative sense.

Examples of facultative information might include insect parasitoids inspecting potential host larvae for signs of previous parasitism. Host type (quality) is determined by the number of times the host has already been parasitized (Bakker et al., 1967, 1972). This may be determined by the parasitoid by careful and sometimes lengthy inspection and probing of the potential host (Bakker et al., 1967, 1972; van Lenteren, 1976; Vinson, 1976). Oviposition itself can be accomplished rapidly and independently of the inspection and probing (van Lenteren and Bakker, 1978). When the frequency of parasitism is low, parasitoids inspect potential hosts and avoid laying eggs on previously parasitized ones, but when most potential hosts are already parasitized, a parasitoid may lay eggs on all encountered hosts regardless of whether they have been parasitized before (Salt, 1961; Bakker et al., 1967). Mate selection may also involve facultative recognition time, with individuals having the option of using 'recognition time' to assess mate quality (e.g. looking for quality cues, performing ritualized mating behaviours, etc.) or choosing mates at random (O. Hasson, personal communication). Despite the variety of contexts in which costly infor- mation can be important, we think that the following simple model captures some of the essential features and results.

Models

We start by recalling the results of the diet selection model with instant (static information) and obligate recognition. Throughout, we consider the case of two prey types randomly distributed in the environment. We assume that the prey are not depletable, i.e. prey encounter rates do not decline as prey are eaten. The encounter rate with prey type i is the product of its density (Ni) and a parameter relating its 'detectability' in the dimensions of area/time (ai). Each prey type is characterized by a per capita energetic value (El, i = 1,2) and a per capita handling time (hi, i = 1,2). There are two kinds of foraging tactics, opportunistic and selective. An opportunistic forager is one who handles and eats every food item that it encounters. A selective forager takes only the food type characterized by the higher ratio of per capita value to handling time (E/hi). We refer to this food type as the more profitable and index it by i -- 1. The rate of energy yield to a selective forager (assuming no recognition cost) is

alN1E1 Ys~e°ln- 1 + alN~h 1 (la)

and the yield to an opportunistic forager is

20 Kotler and Mithcell

aIN1E 1 + a2N2E 2 Y~g = (lb)

1 + alNlh 1 + a2N2h2

where the superscript, non, refers to the fact that this is the yield when there is no recognition time involved.

The well-known result for this case is that the forager should be selective unless the profitability of the poor resource exceeds the rate of yield experienced by a selector (Pulliam, 1974; Sih, 1979):

alN1E1 E2 > (2) h 2 1 + alNlh 1

In this case the decision rule is quite straightforward, but when we introduce the effects of information the decision rule becomes more complicated. Therefore, we introduce a graphical representation of the decision rules to facilitate the presentation. Following Holt (1983) we represent the decision rule by an isoleg in the space of resource densities or, more precisely, the resource densities multiplied by their respective per capita probability of encounter (Fig. 1). The isoleg is the line that separates the densities where selectivity is favoured from the densities where opportunism is favoured. This line is called an isoleg because all along it the forager is indifferent between the tactics of selectivity and opportunism (Rosenzweig, 1981). In the case of no

Instant Recogntion Isoleg

I00

O

O 0

O

90

80

7 0 o

°° i 50

O

30

20

10

0 0 10

s e l e c t i v e

i

20 30 40 50 60 70 80 90 100

D e n s i t y of R i c h R e s o u r c e

Figure 1. The optimal strategy of diet choice for a forager with instant recognition. The dashed line indicates density combinations where opportunism and selectivity yield the same rates. Parameter values: E 1 = 0.2, E 2= 0.1, h l = h 2= 1, a l = a 2= 1.

The value of information to diet choice 21

recognition cost (neither obligate nor facultative) the isoleg is perpendicular to the axis of density of the rich resource, illustrating that the optimal behaviour (selective or opportunistic) is independent of the density of the poor resource.

To understand how the pay-off to the optimal forager depends on resource densities, we can represent the yields to selectors and opportunists in the same space. Equal yield lines for the selector (defined by Equation 1) are perpendicular to the axis denoting the density of the rich resource because the unutilized poor resource neither interferes with nor contributes to yield. Equal yield lines of the opportunist (defined by inequality 2; shown in Fig. 2), with one exception, have a finite slope. The exception is where the equal yield line of the opportunist coincides with the isoleg, as well as with one of the equal yield lines of the selector. By definition, at each point on the isoleg the yields of selector and opportunist are equivalent. In the case of no recognition time, it is also true that the yield is identical all along the isoleg. To the left of the isoleg where opportunism is optimal, the equal yield lines of the opportunist have a negative slope. The negative slope is a consequence of the fact that densities of both resource types enhance the yield to an opportunist in this part of the graph. To the right of the isoleg where opportunism is not optimal, the slopes of the opportunist equal rate lines are positive. The slopes are positive in this region because consuming more of the poor resource reduces the yield to an opportunist.

O p p o r t u n i s t ' s Y ie ld L i n e s

1 0 0

90

80 0

o 70

60

0 o 50 ~w

o 4 0

• .-, 30 m

~ 2 0

10

0 0 i0 20 30 40 50 60 70 80 90 100

D e n s i t y of R ich R e s o u r c e

Figure 2. Equal yield lines for an opportunist. The dashed line indicates density combinations where opportunism and selectivity yield the same rate in the case of instant recognition.

22 Kotler and MithceU

Obligate recognition cost

When recognition cost is obligate, the forager has 'no choice' but to learn what food type it has encountered. Upon an encounter, the forager begins by 'handling' the item just as it would if it had instantaneously recognized the item and decided to consume it. In the process of handling the resource item, the forager is able to distinguish whether it is rich or poor in quality. Only after recognizing an item can the forager decide whether to reject it. As a result, the selector's yield is reduced by the density of poor quality resource because the recognition time, r (r < h2) , must be spent on even unexploited prey (Hughes, 1979). The yield for the selector in this case is thus

yobl alN1E1 el = 1 + alNlh 1 + azN2r (3)

which differs from Hughes' (1979) presentation in that here handling time subsumes recognition time (r < hi, h2) for a resource that is exploited; hence, there is no separate recognition cost for the profitable resource. Because the density of the poor resource occurs in the denominator, equal yield lines of the selector have a positive slope (Fig. 3).

The yield for the opportunist is identical to the case with no recognition cost (because recognition cost is subsumed by handling time). Hence, the opportunist's yield is

O b l i g a t e Case S e l e e t o r ' s Y i e l d L i n e s

100

90

80 0 ~w

o 70

60 Iw 0 o 50

o 40

• ,. 30

~ 20

10

0 0 I0 20 30 40 50 60 70 80 90 100

D e n s i t y of R i c h R e s o u r c e

Figure 3. Equal yield lines for a selector with obligate information. The slopes are positive because the time spent recognizing poor resources reduces the rate of consuming rich resources.

The value of information to diet choice 23

NoON __-- aaN1E1 + a2N2E 2 PP 1 + alNlh I + a2N2h 2 (4)

The forager will be opportunistic when the energetic value of the poor resource divided by the difference between handling and recognition time is greater than the the rate of energy intake from selecting only the richer resource, i.e. when

E2 alN1E 1 > - - v o b l

h 2 - r 1 + alNlh I + aeN2r -sel (5)

This inequality can be satisfied by increasing the density of the less profitable resource. Consequently, the obligate recognition model predicts that the less profitable resource can become included in the diet when it becomes more common (Hughes, 1979). We represent this result graphically as an isoleg with a positive slope (Fig. 4). This prediction distinguishes the obligate recognition time model from the instantaneous recognition model discussed in the previous section in which the isoleg was vertical.

And, as in the case with no recognition cost, the isoleg is the locus of an equal yield line for both the selector and opportunistic tactics. Hence, the forager achieves the same yield at any point on the isoleg (Appendix I).

Obligate Recognition Isoleg

0 In ID

O O

O

100

90

80

70

60

50

40

30

20

10

0 0 I0

/~ selective 20 30 40 50 60 70 80

Density of Rich Resource

90 100

Figure 4. The optimal strategy of diet choice for a forager with obligate recognition. The solid line indicates density combinations where selectivity and opportunism yield the same rate. For the sake of comparison, the dashed line shows where the isoleg would be if recognition were instantaneous. Parameter values: E I = 0.2, E 2 = 0 . 1 , h I = h 2= 1, a I = a 2= 1, r=0 .02 .

24 Kotler and Mithcell

Facultative recognition time

Now we consider the case in which a forager can choose between recognizing or not recognizing the type of resource encountered. If it chooses to recognize the encountered resource item, the forager must pay a recognition cost, R. Only after paying the recognition cost is the forager able to decide between accepting or rejecting the encountered resource item. If the forager decides to accept the resource item, then it must pay the handling time cost, hi, in addition to the initial recognition cost, R. Should the forager choose not to pay the recognition cost, then it simply accepts every encountered resource and pays the appropriate handling cost. Because the recognition cost is optional, the forager should not pay this cost unless the benefit is even greater. Clearly, the cost of recognizing encountered resource items to type depends on their profit- abilities and abundances. The facultative case was noted by Holt (1983) and by Stephens and Krebs (1986), but not analysed.

The selector's yield includes a term in the denominator that represents the time spent recognizing encountered resource items to type. Consequently, the selector's yield declines with increasing density of the less profitable resource.

alNIEI (6a) y~a~ = 1 + alNlh I + R(alN 1 + a2N2)

The opportunist's yield contains no term for recognition cost and, hence, yield is identical to that of the opportunist with instantaneous recognition.

alN1EI + a2N2E2 (6b) Y~°~P~P = 1 + alNlh I + a2N2h 2

We use Equations 6a and 6b to find the equal yield lines and the isolegs. For the selector, the family of equal yield lines is given by

N2 = Nlal E 1 - Y~ae](h 1 + R) 1 a2 l~a~ R - a2 R (7a)

and the equation for the family of opportunist's equal yield lines is:

N2 = Ula~ E1 _ y~apCph 1 _ ~apCp (7b) Y ,apCph2 - E2 a (Vg%h -

As always, the isoleg is determined by where the pay-offs to the two tactics, selector and opportunist, have identical pay-offs, with a different tactic better on either side of the line. Recall that when recognition cost is obligate or absent, the tactics have identical pay-offs where their equal rate lines corresponding to the same value coincide over the entire line. But with facultative recognition, the selector and opportunist equal yield lines do not ever coincide, but intersect at a point. The isoleg is generated by an infinite number of such intersections; consequently, the yield changes over the length of the isoleg.

The isoleg for the facultative case constitutes one leg of a hyperbola. This can be seen by setting the yields in Equations 7a and 7b equal and then rearranging the terms into a standard quadratic form.

N22a22E2 R + N2a2E 2 + N2N 1 (E2(h I + R) - E l ( h 2 - R)) + (Nlal)2E1R = 0 (8)

For values of search time, handling time, recognition time and energetic value of the prey where the isoleg exists in the positive quadrat, the discriminant (Thomas, 1972) of Equation 8, is > 0; thus Equation 8 represents a hyperbola (when Equation 8 represents an ellipse, it is entirely

25

Seleetor's Yield Lines

¢9

O

0 O

0

.,,,w

0 10 20 30 40 BO 90 100

100

90

80

70

60

50

40

30

;80

10

0 50 60 70

The value of information to diet choice

Faeultative Case

D e n s i t y of R i c h R e s o u r c e

Figure 5. Equal yield lines for a selector with the ability to facultatively acquire information. The slopes of the lines are positive because the selector has chosen to spend time recognizing resources and poor resources increase the time spent in recognition.

in the negative quadrat and does not represent an isoleg). The hyperbola is rotated and translated with respect to the origin; one of its legs runs through the origin, is to the left of the no- recognition-time isoleg and is not itself an isoleg. The other leg of the hyperbola is the isoleg, exists in the positive quadrat and is always found to the right of the no-recognition-time isoleg.

The curve of the isoleg separates resource densities at which the tactics of selectivity and opportunism are optimal. Selectivity is optimal within the isoleg's curve, and opportunism is optimal outside of the curve (Fig. 6). From this shape we get three predictions, two of which are not made by either of the previous models. The old prediction is that the poor quality resource may become included in the diet as a result of becoming more abundant. This is similar to the case of obligate recognition (Hughes, 1979; see the previous section). As in the case of obligate recognition, an increase in the encounter rate with poor resource items reduces the encounter rate with rich items. Consequently, the missed opportunity cost of handling an encountered poor quality item is reduced sufficiently so that the opportunistic tactic is better than the selective tactic.

The major new prediction of facultative recognition is that, when the poor resource is rare it may become excluded from the diet as it becomes more common. How can this happen? When

26 Kotler and Mithcell

Facultative Recognition Isolegs

I,w

0

~J C~

o o

o

100

90

80

70

60

50

40

30

20

10

0

/ se lec t ive a t R=.O04

se l ec t i ve at R=.O02 . . . . . . . .

. .

, ° , . . ° . . - . ' " " " " •

• . . . .

• . . . . . . . ° - . .

o p p o r t u n i s t i c

0 !O 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

D e n s i t y o f R i c h R e s o u r c e

Figure 6. The optimal strategy of diet choice when recognition time is facultative. The hyperbolic isoleg is shown for the case of two different values of recognition time. Selectivity is optimal within and opportunism is optimal outside the hyperbola. For the sake of comparison, the corresponding isoleg from the case of instant recognition is represented by a vertical dashed line. Parameter values: E1 = 0.2, E 2 = 0.1, hi = h2 = 1 , a 1 = a 2 = 1 .

the poor resource is rare the forager rarely suffers the cost of wasting effort on it. But as the poor resource becomes more common, the forager wastes too much time handling it and is willing to pay the cost of recognition. By combining these two predictions we can see that as the abundance of the poor resource increases from rare to very common, the optimal forager may change its tactics twice, first from opportunistic to selective and then back to opportunistic (Fig 6).

There is a second new prediction when the cost of recognition is quite high. At higher costs of recognition the isoleg shifts up and to the right and the lower boundary of the isoleg curve is tilted up. This means that an increase in the density of the preferred resource can shift the optimal tactic from selective to opportunistic. Why should encountering more preferred items induce a forager to start accepting less preferred items? At higher densities of the rich resource the relative frequency of encounters for which the information (recognition) is actually used to reject an item decreases. Meanwhile the cost of acquiring this information increases because more resource items are encountered. Hence, the forager would pay more to get less and instead opts out of spending the time to recognize items. And, when the forager does not recognize different resource types, it accepts all encountered items.

The value of information to diet choice

Discussion

2'7

If information could be had at no cost, then natural selection would produce organisms that always acquired and used perfect information. Clearly this is not the case and organisms often must deal with imperfect information (Iwasa et al., 1981; McNamara, 1982; Valone 1990). Selection must then be for evolutionary strategies that balance the benefit of additional information against its cost. The benefit of additional information depends on what the organism can do with it. As Stephens (1989) notes, if having more information has no effect on the optimal behaviour, then the benefit of information is zero and the cost of it will not be paid. If having additional information were to change the subsequent behaviour, then the improvement of fitness attributable to the change is the benefit of information. This benefit must exceed the cost of information for selection to favour the strategy of acquiring and using the information. The challenge to behavioural ecologists is to understand how the benefits and costs of information vary with environmental and resource parameters.

We have analysed the effect of costly information in the context of one of the most basic theoretical models in behavioural ecology. Costly information changes the predictions not just quantitatively, but qualitatively as well, because the facultative use of costly information introduces a new trade-off. The cases of no recognition cost (static information) and obligate recognition cost produce a simple trade-off between time spent handling a poor quality resource item (once it has been recognized) and searching for and handling better quality resources. The result is a linear isoleg separating resource densities at which opportunism and selectivity are optimal. Facultative recognition introduces the new trade-off between time spent gathering information and time spent searching for and consuming resources. The value of this information depends largely on the relative abundances of the resources. When the poor resource is relatively rare, the value of recognizing and rejecting it is small. Hence, the forager should not pay the cost of recognition and it will be opportunistic. When the poor resource is relatively common, paying the cost of recognition produces a situation similar to that with obligate recognition - the high cost of recognizing frequently encountered poor resource items reduces the encounter rate with rich items to the point that opportunism is optimal in any case. Hence, the optimal forager will not pay the cost of recognition and will simply be opportunistic.

We have assumed for this analysis that recognition time (either obligate or facultative) is constant. But recognition time may shorten with experience if the forager gets better at correlating environmental cues with resource quality (Papaj and Vet, 1990; Krebs~ and Inman, 1992). Learning by the forager will make the isolegs shift in time. This kind of dynamic interplay between learning and the willingness to pay for information as it becomes less costly may be very important for foragers encountering new environments.

The value of information in a facultative sense will be relevant to non-foraging behaviours as well. Mate choice requires that the choosers have some means of discriminating among potential mates (upon whatever basis is relevant). Information is rarely static. In both the resource exploitation case and the mate choice case the object being chosen may attempt to influence the assessment of quality by the chooser.

Acknowledgements

J.S. Brown, M.L. Rosenzweig, E.A. Sandlin and P. Smallwood all improved this paper through their careful reading and commentary. This paper is Publication no. 160 of the Mitrani Center for Desert Ecology.

28

References

Kotler and Mithcell

Abrams, P. (1987) The functional responses of adaptive consumers of two resources. Theor. Pop. Biol. 33, 262-88.

Alcock, J. (1989) Animal Behavior (4th edn.) Sinauer Associates, Sunderland, MA. Bakker, K., Bagchee, S.N., Zwet, W.R. and van Meelis, E. (1967) Host discrimination in Pseudiocoila

Bachni (Hymenoptera: Cynipidae). Entomol. Exp. Appl. 10, 295-311. Bakker, K., Eijsackers, H.J.P., van Lenteren, J.C. and Meelis, E. (1972) Some models describing the

distribution of eggs in the parasite Pseudeucoila bochci (Hym: Cynip.) over its hosts, larvae of Drosophila melanogaster Oecologia 10, 29-57.

Brower, L.P. and Glazier, S.C. (1975) Localization of heart poisons in the Monarch butterfly. Science 188, 19-25.

Brown, J.S. and Mitchell, W.A. (1989) Diet selection on depletable resources. Oikos 54, 33~3. Elner, R.W. and Hughes, R.N. (1978) Energy maximization in the diet of the shore crab, Carcinus maenus.

J. Anim. Ecol. 47, 103-16. Gilbert, L.E. (1983) Coevolution and mimicry. In Coevolution (D. J. Futumya and M. Slatkin, eds), pp.

263-81, Sinauer Associates, Sunderland, MA. Gill, F.B. (1990) Ornithology. W.H. Freeman, New York. Hollings, C.S. (1959) Some characteristics of simple types of predation and parasitism. Can. Entomol. 91,

385-98. Holt, R.D. (1983) Optimal foraging and the form of the predator isocline. Am. Nat. 122, 521-41. Holt, R.D. and Kotler, B.P. (1987) Short-term apparent competition. Am. Nat., 130, 412-30. Houston, A.I., Krebs, J.R. and Erichsen, J.T. (1980) Optimal prey choice and discrimination time in the

great tit (Parus major). Behav. Ecol. Sociobiol. 6, 169-75. Hughes, R.N. (1979) Optimal diets under the energy maximization premise: the effects of recognition time

and learning. Am. Nat. 113, 209-21. Huntingford, F. (1984). The Study of Animal Behaviour. Chapman & Hall, New York. Iwasa, I., Higashi, M. and Yamamura, N. (1981) Prey distribution as a factor determining the choice of

optimal foraging strategy. Am. Nat. 117, 710-23. Krebs, J.R. and Inman, A.J. (1992) Learning and foraging: individuals, groups and populations. Am. Nat.

140, $63-84. Lessells, C.M. (1985) Parasitoid foraging: should parasitism be density dependent? J. Anim. Ecol. 54, 27-

41. McNamara, J. (1982) Optimal patch use in a stochastic environment. Theor. Pop. Biol. 21, 269-88. Mangel, M. and Clark, C.W. (1988) Dynamic Models in Behavioral Ecology. Princeton University Press,

Princeton, NJ. Murdock, W.W. (1969) Switching in general predators: experiments on predator specificity and stability of

prey populations. Ecol. Monogr. 39 335-54. Papaj, D.R and Vet, L.E.M (1990) Odor learning and foraging success in the parasitoid, Leptopilina

heterotoma. J. Chem. Ecol. 16, 313-50. Pulliam, H.R. (1974) On the theory of optimal diets. Am. Nat. 108, 59-75. Rosenzweig, M.L. (1981) A theory of habitat selection. Ecology 62, 327-35. Salt G,. (1961) Competition among insect parasitoids. Syrup. Soc. Exp. Biol. 15, 96-119. Sih, A. (1979) Optimal diet: the relative importance of the parameters. Am. Nat. 113, 460-3. Stephens, D.W. (1989) Variance and the value of information. Am. Nat. 134, 128-40. Stephens, D.W. and Krebs, J.R. (1986) Foraging theory. Princeton University Press, Princeton, NJ. Thomas, G.B. Jr (1972) Calculus and Analytic Geometry. Addison-Wesley, Reading, MA. Valone, T.J (1990) Information and exploitation: patch assessment strategies in birds and mammals. Ph D

Dissertation, University of Arizona, Tucson. Valone. T.J. and Brown, J.S. (1989) Measuring patch assessment abilities in desert granivores. Ecology 70,

1800-10.

The value of information to diet choice 29

van Lenteren, J.C. (1976) The development of host discrimination and the prevention of superparasitism in the parasite Pseudocoila bochei (Hym: Cynipidae). Neth. J. Zool. 126, 1-83.

van Lenteren, J.C. and Bakker, K. (1978) Behavioural aspects of the functional response of a parasite (Pseudocoila bochei Weld) to its host (Drosophila rnelanogaster). Neth, J. Zool, 28, 213-33.

Vinson, S. B. (1976) Host selection by insect parasitoids. Ann. Rev. Entomol., 21, 109-33.

A p p e n d i x 1

We show in this appendix that the yield does not vary over the isoleg in the obligate recognition case. This is because the isoleg in this case coincides with an equal rate line that is common to both the selector and opportunist tactic.

We derive the equal yield lines for the selector tactic by rearranging Equat ion 3a for the selector's yield,

_ hlYCsel )N 1 _ vobl N 2 al (E 1 obl " sel - - ~ - v o b l a 2 , ~t sel

The opportunist 's equal yield lines come from opportunist 's yield

al (E 1 obl _ -- hlYOpp)N1 rff "°bl - opp N2- a2 Vo l h2 -

(A1)

a rearrangement of Equat ion 3b for the

(A2)

At any point on the isoleg the yield to the opportunist equals the yield to the selector. By setting the yields in Equat ion A1 and A2 both equal to Y, we can determine how the yield on the isoleg (I 0 varies with the parameters. After a rearrangement we obtain

y _ E2 (A3)

h 2 - r

This is the value of the yield on the obligate recognition isoleg. It is not a function of the densities of resources, when those densities are constrained to fall on the isoleg. Hence, the yield is constant all along the obligate isoleg.