And the Palatability Spectrum* (Unpalatable/Predator/Gregariousness/Prey) F

Proc. Nat. Acad. Sci. USA Vol. 70, No. 8, pp. 2261-2265, August 1973

Theoretical Investigations of Automimicry: Multiple Trial Learning and the Palatability Spectrum* (unpalatable/predator/gregariousness/prey) F. HARVEY POUGHt, LINCOLN P. BROWERt§, HAROLD R. MECKI, AND STEPHEN R. KESSELL§ t Cornell University, Ithaca, New York 14850; § Amherst College, Amherst, Massachusetts 01002; and I Shillington, Reading, Pennsylvania 19607 Communicated by G. E. Hutchinson, May 3, 1973

ABSTRACT We previously explored automimicry as- Case 1 and develop two additional models. Case 2 will con- suming that a species of prey was so unpalatable as to sider a situation in which palatable individuals of the prey promote conditioned avoidance for a period of time after a predator encountered a single individual (Case 1). In this species can intervene between the two unpalatables, and paper, we assume that the prey is less noxious and that Case 3, the most restrictive, requires that the predator must two encounters are required. Case 2 allows the two en- encounter the two unpalatables consecutively. counters with unpalatables to be separated by any number of palatables, while in Case 3 the predator must encounter two unpalatables, consecutively. MATHEMATICAL MODELS The general relationships in the three cases are similar, The mathematical models for all these cases assume that but the automimetic advantage is reduced moderately in en- Case 2 and greatly in Case 3. To attain the same auto- palatable alternative prey are available in the natural mimetic advantage as in Case I requires an increase in the vironment of the predator and that an attack by a predator proportion of unpalatables, or in the induced rejection on an individual of the automimetic species is lethal for the period, or both. Consequently, selection will tend to prey. increase the unpalatability so that Cases 2 and 3 converge The variables we consider are the following: n is the number to Case 1. Species that are uniformly and highly unpalatable can of prey that each predator would eat if none of the prey afford to be more dispersed than automimetic species. were unpalatable, m is the number of prey available per Case-2 and -3 automimetic species will benefit greatly from predator, and k' is the frequency of unpalatable prey in a gregariousness, while in Case-i automimicry situations predator's sample. We have chosen n values ranging from 2 this is less important. to 100 and m values from 0.05 to 10,000. The limits upon and Brower, Pough, and Meck (1) explored mathematically relationship between the variables were discussed in our the theory of automimicry resulting from the discovery previous paper (1). that some monarch butterflies (Danaus plexippus L.) are The mathematics of Case 2 follow from Eqs. 1-3 in our palatable and others are not (2). Because the palatable and previous paper (1). unpalatable individuals are members of the same species they At the beginning of the ith predation the fraction of preda- are visually identical and predators cannot distinguish them tors that have not yet had an emetic experience is (1 - k')i-i. on the basis of sight. Our mathematical model assumed that The fraction of predators that have had one emetic experience an unpleasant experience with a single unpalatable individual is (i - 1)k'(1 - k') -2. Hence, the fraction of predators still would be sufficient to condition a predator to avoid several eating at the beginning of the ith period is subsequent individuals. The analysis led to the conclusion that automimicry enables a remarkably low proportion of un- (1 kl)'-' + (i 1)kl(l kl) i-2 [4] palatable prey to confer a substantial immunity from predation to the entire population. and this expression is equal to the average number of prey A recent study (3) has shown that monarch butterflies eaten by one predator during the ith period. The average from natural populations exhibit palatability spectra and total number of prey eaten by one predator in n prey rejection include individuals that are only moderately unpalatable. A periods is predator might have to experience more than one of these before it was conditioned to avoid subsequent individuals. n n 1: (1 -k')'- 1 + k' E (i -1)(1 -k')i [5] In this paper, we explore the extent to which the auto- i=l i=l mimetic advantage accruing to a population is changed when each predator must encounter two unpalatable prey instead The first series is the same geometric series that appeared of one to avoid subsequent prey on sight alone. We shall refer in the analysis of Case 1. The summation is repeated below: to the former mathematical model, which involved prey sufficiently unpalatable to produce single trial learning, as n 1 (1 - kl')-' = - [1 - (1 - kt)n]. * This is paper no. II of the series. The first paper is ref. 1. si=1o I Requests for reprints may be addressed to this author, Depart- ment of Biology, Amherst College, Amherst, Mass. 01002. The sum of the second series can be found by differentiating 2261 2262 Zoology: Pough et al. Proc. Nat. Acad. Sci. USA 70 (1973) 1.0 PREY TOO COMMON PREY TOO RARoEthe foregoing equation with respect to k'. Thus

-,CASE 2 + 1

W K [7 0 1 (k+)2-[l-(1-ka)k].O.Oa/ K F~~~~~~~~~~~~~~~~~~~~~~~~(\)-n( - -'k-+-[1 - (1 - k l)n-1 o/.I The expression for the average total number of prey eaten by one predator in n rejection periods now becomes 4~~~~~~~~~~~~~~~~~~~~ W 04 k 0.5000 § W W jut = 1 _ [2 - (nk' + 2 - 2k')(1 - k')"-']. [7]

0.2~~~~~~~~.9 XZ 05 al QSOL 1.0 5S 10 50 100 mk' 1.0- PREY TOO COMMON K'9s PREY TOO RARE, Case S. In this case the direct method of solution used n/m n/m Tpreviouslyfr co cannot be applied. A finite difference equation 0.50004 must be used. The symbol xi will be used to denote the 0.8- number of predators that still retain their appetites after < CASAE22 the ith eating period. The number of predators susceptible to - b 0.2500 a)thloss of appetite in the (ixi+ period is minus the number predators that did not get an unpalatable prey in the ith z 0-6- l\ period, i.e., (\k'10 = 1--, [2-k)X].)xf1.+2-2k)(1- [9] 0.48 The number of predators that stop eating during the 0~~~~~~~~~)0b 0.2500of/aploss(i + 1)thpetitperiodinisthequal to Eq. 9 multiplied by k'. This result risequaltoxt-xf+n.Therefore, 0.0 1\\\ - - (1 - k')x1..i = xi+, (1~~~~~~~~~~~~~~x [9])j- 0.2- ;!11 -tso~~~~~~~~~~of xi f-X+lx -= k'[x - (1 k')xi-l]. [10] Li l a \ \ \ ~~~~~~~~~~Afterrearranging terms, the finite difference equation for xi 0~~~~~~~~050

°'01 05 01 0.5 1. 5 10 50 100 X~-1k)fk(-'x_=° [l K' A trial solution of the form 0o PREY TOO COMMON PREY TOO RARE n/m < I o0sffoO°/2\ n/m >1 xi =co,f [12] 0.2500\ is assumed where C is an arbitrary constant. It is found by 0.8- 0.1000-isubstitutione t x of- the trial solution into the finite difference W C ||\ \ \\equation that A is given by the quadratic equation <0.62 x|j\\(1- - k') -k'(1 - k') = 0. [13] 0.0~ AfThisequationhastheroots

/ + 2[(1-kl)(1 + 3k')]l/ [14]

0.22D/ \ X X X 2 = -~~~~~~~~~~~~22 [(1 - k)(1 + 3k')]1' ///o\ The equation for xf now becomes ooL~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~X=\X 1 tctC202iC11 [1s5 .01 05 0.1 0.5 1.0 5 10 50 0 abundance (m) andfrequency of unpalatable prey in a predator's PREDATION POTENTIAL n/m sample (k') on the automimetic advantage (Aj). The induced FIG. 1. (a) Autom2m cry, Case 2: Two unpalatable prey are prey rejection period (n) is 5. (b) Same as (a), except n = 25. required to condition subsequent avoidance. Effect of prey (c) Same as (a), except n = 100. Proc. Nat. Acad. Sci. USA 70 (1973) Theoretical Investigations of Automimicry 2263 1.0- PREY TOO COMMON PREY TOO RARE The constants C1 and C2 are evaluated from the conditions that xi = Xowhen i is equal to zero or 1. Then, it is found that n/rn

0 X a- 0.4-_/_05030 13i - 132 [(1 - 132)13I - (1 - 131)021 1. [17] < / a~~~~~~~~~500>011- 2 0.2- The total number of prey eaten during n trials is

0.2500air [ -132) jal l (1 -1) j132'-]. [18]

0.0-I F .01 .05 0l 0.5 1.0 5 10 56 I0 When the two geometric series are summed, this becomes 1.0 PREY TOO COMMON K' PREY TOO RARE). - 132)(1 - ), - - < 0~~~.99 xo #2i - I')l (1 13)(1 - 132n)1 1] n/m <1' n/m >1 132L 1-1 [19]

0-8 CAE 3 0 The fraction of prey eaten is equal to this result divided by \ the initial number of prey: b 1 ~~~~~~~~~~~~~~~~~~~~~[(1-#2)2(l _-11 Z 0.6- / (1 - $2)(1 - 1)(1 - 32) ll- (1 - #1)2(1 - d32n)] [20] 2 o/.4X and the fraction of the population surviving is 0<,IA m(k')2(1 [- 12) 0.2- -(1- 1)2(1-_ 2)] [21] 000 0// The equations for Cases 1-3 break down when k' = 0. However, if no prey is unpalatable, it is immaterial which case 0.o05o'sapplies, and Eq. may be used. This can be verified by .01 0.5 N 50 1 i0.11.0K` i00 applying a simple limiting process to the three cases to find 1.0 PREY TOO COMMON PREY TOO RAE the value of j when k' = 0. It is found in each case that the result is identical to Eq. 1. n /rn I The mathematical models were analyzed by computer as before. 0.88 i 0.2500 CASE 3 RESULTS - C There is no change in the general relationship of automimetic Z 0.6 // \ \ \ advantage (Aj) to the frequency of unpalatables (k') in the predators' samples at different values of n/m for Cases 1, 2, and 3 (Figs. la-c, 2a-c, and ref. 1). In all three cases the 2 maximum benefit to the population occurs at the point at a .O/40 which the predators could eat the entire population (i.e., when a 01000 n/m = 1). The requirement of encountering two unpalatables in Cases 2 and 3 lowers the automimetic advantage for all 0.2 // / | | | \ levels of k' and n (Figs. 3, 4, and ref. 1). 0.2 / / \ l lWith n values >25, k' values of >0.50 give similar auto- 0o500 mimetic advantage in Cases 1, 2, and S. The critical k' value for Cases 1 and 2 is 0.25, but for Case 3 it is 0.50; below those O0000 values Aj drops sharply. At very low k' values, automimetic .01 .05 0.1 0.5 1.0 5 10 50 100 advantage virtually disappears. For example, for k' = 0.01 PREDATION POTENTIAL n/m FIG. 2. (a) Automimicry, Case 3: Two consecutive unpalatable as in Fig. 1. (a), n = 5. (b) Same as (a), except n = 25. (c) Same prey are necessary to condition subsequent avoidance. Symbols as (a), except n = 100. 2264 Zoology: Pough et al. Proc. Nat. Acad. Sci. USA 70 (1973)

CASE 2 K' CASE 3 K' -:0.99990.5000 - 0.2500

w 0 w I-. 0I- z

0 I w

0 I-- 0 I-

50 0.0 25 50 75 100 n n FIG. 3. Case 2: Automimetic advantage at optimal conditions FIG. 4. Case 3: Automimetic advantage at optimal conditions (n/m = 1). (n/m = 1). and n = 100 the automimetic advantage for Cases 1, 2, and 3 havior of the prey would make it possible for predators to in- is 0.36, 0.10, and 0.01, respectively. gest several individuals in rapid succession. Gregariousness When I' = 0.25, critical values for n in the three cases are may also increase n by visually reinforcing the conditioned about 20, 40, and 60, respectively. In other words, to achieve rejection responses; the sight of the prey species may remind a given similar automimetic advantage the number of condi- a predator of a previous unpleasant experience, and thus tioned rejections must be twice as great for Case 2 and three prolong the induced prey rejection period (1). times as great for Case 3. The characteristically nonrandom distributions of many unpalatable insect species achieved by gregarious behavior DISCUSSION probably reflect the advantage gained by increasing n. If so, The introduction of levels of palatability does not change the moderately unpalatable species should have more clumped relationships of k', n, and m in automimicry or the analogous populations than those in which there is a clear-cut Case 1 situation of perfect Batesian mimicry. Lower levels of unpalatability dimorphism. Extending this reasoning, those palatability change only the degree of protection gained from a species in which all individuals are highly unpalatable should particular frequency of unpalatable individuals and limit be the most dispersed. Benson's (7) finding that gregariousness automimetic advantage to a smaller range of n/m ratios, i.e., and unpalatability in Heliconiine butterflies are positively n/m must be closer to 1. correlated might be taken as evidence against our conclusion. The greatest difference in Cases 2 and 3 compared to Case 1 However, the palatability experiment upon which he based is the effect of changes in the frequency of unpalatables (k') his correlations was not of sufficient scope to distinguish in a predator's sample. In Case 1 very low k' values are effec- Case-i from Case-2 and -3 species, and gregariousness would tive in conferring a large automimetic advantage on a popula- be expected of species conforming to the two latter cases. tion even with very low n values. When a predator must Presumably one of the major selective advantages of auto- sample two unpalatables, either n, or k', or both, must be mimicry in monarch butterflies is that it allows the utilization large to produce an automimetic advantage comparable to of several larval foodplants, or different parts of the same those of Case 1. plant which contain different cardenolides (8). This would If a species is to maintain a population with low k' values, permit the butterflies to maintain larger populations per unit n must increase. There are several mechanisms by which area than would be possible if the insects were limited to larger n values could be achieved by the prey. One way is to glycoside-containing plants or tissues. Recent data suggest evolve greater unpalatability (4). However, a continuation that the poison may have deleterious physiological effects on of this trend would eventually transform Cases 2 and 3 into the butterflies themselves (3). If this is correct, automimicry Case 1. In fact, it seems likely that a predator's behavior will would have the additional advantage of allowing a portion of modify Cases 2 and 3 to Case 1 even without changes in the the population to avoid the handicap of containing glycosides palatability of the prey, because subsequent reinforcement while benefiting from the reduced predation that results from may require only occasional single samples. The learning the presence of some unpalatable individuals in the population. curves of the scrub jays in J. Brower's experiments indicate Undoubtedly, many natural populations of unpalatable this (5). insects represent complex mixtures of Cases 1, 2, and 3 as well Another mechanism by which moderately unpalatable prey as other levels of unpalatability. The situation will be further can produce high n values is, paradoxically, through being complicated by individual variation in the sensitivity of vulnerable to predation by gregarious behavior (6). This will predators to the unpalatable substance (9), temporal provide the opportunity for rapid education of all the preda- changes in the predators' behavior as the result of hunger and tors in an area. In addition, when more than one experience is previous experience, alternate prey species, and different needed to effect rejection by the predator, gregarious be- species of predators. Some species of birds are more repelled Proc. Nat. Acad. Sci. USA 70 (1973) Theoretical Investigations of Automimicry 2265

than others by the same distasteful or emetic experiences (10); common patterns. Moderately unpalatable prey species thus, a prey species might fit Case 1 for some predators and would benefit from resembling highly unpalatable ones. Case 2 or 3 for others. However, the addition of moderately unpalatable species to One natural situation appears to approach Case 1 very the complex could be extremely disadvantageous to the closely: overwintering monarch butterflies from the California highly unpalatable species if it lowered the overall unpala- coastal aggregations show a bimodal palatability distribution. tability level of the complex and caused predators to treat the About half of the individuals contain sufficient cardiac entire complex as Case 2 or 3 instead of Case 1. In that situ- glycoside to cause extreme emesis in a blue jay while the other ation, it would become advantageous for a highly unpalatable half contain virtually no glycoside. In contrast, monarch species to evolve a new pattern, different from that of the butterflies from Massachusetts contain glycosides that are complex. In turn, moderately unpalatable species would less emetic and may approach Case 2, at least for some preda- benefit from evolving to resemble the new pattern, and a new tor species (manuscript in preparation). complex would develop. This reasoning may explain why Cases 1, 2, and 3 indicate that the evolution of auto- there are several Mullerian mimicry complexes in a given area. mimicry or Batesian mimicry can be a very complex process. We dedicate this paper to the memory of Robert H. Mac- The introduction of mimics into a population of models Arthur. We thank Helen Sullivan for her patience and care in causes simultaneous changes in k', n, m, and n/m, all of which preparing the manuscript, Betty Steele for help with the pro- affect Aj. In addition, the evolution of Batesian mimicry gramming, and Edward Francolini for the drawings. The research proceeds by small steps so that initially the predators will was supported by National Science Foundation Grant GB- 35317X, L.P.B., principal investigator, and by the Director of usually be able to distinguish between models and mimics Research, New York State College of Agriculture and Life (11). (In our Cases 1-3 automimicry and Batesian mimicry Science at Cornell University. are analogous only when the latter is perfect, ref. 1.) 3 1. Brower, L. P., Pough, F. H. & Meck, H. R. (1970) Proc. Cases 2 and cast additional light upon the relationship of Nat. Acad. Sci. USA 66, 1059-1066. numbers of models to numbers of mimics in an established 2. Brower, L. P., Brower, J. V. Z. & Corvino, J. M. (1967) Batesian mimicry situation. Our Case 1 and theoretical Proc. Nat. Acad. Sci. USA 57, 893-898. analyses of other workers have shown that in an established 3. Brower, L. P., McEvoy, P. B., Williamson, K. L. & Flan- Batesian mimicry the mimic can be considerably more abun- nery, M. A. (1972) Science 177, 426-429. 4. Alcock, J. (1970) Anim. Behav. 18, 592-599; Duncan, C. J. dant than a highly unpalatable model (12). For example, in & Sheppard, P. M. (1963) Proc. Roy. Soc. Ser. B. 158, 343- Case 1 if n = 100, mimics can be nine times as common as 363, Duncan, C. J. & Sheppard, P. M. (1965) Behaviour 24, models (k' = 0.1) with only a 10% reduction in the protection 269-282; O'Donald, P. & Pilecki, C. (1970) Evolution 24, of the model-mimic complex compared to a population com- 395-401. models = Alcock 5. Brower, J. V. Z. (1958) Evolution 12, 32-47, 123-136, 273- posed entirely of unpalatable (k' 1.0). (13) 285. predicted that a moderately unpalatable prey would not 6. Rothschild, M. (1972) in Phytochemical Ecology (Academic support as high a frequency of mimics as a very unpalatable Press, New York), pp. 1-12, and ref. 1. one. Cases 2 and 3 support this prediction. When the prey is 7. Benson, W. W. (1971) Amer. Natur. 105, 213-226; Brower, only moderately unpalatable, Aj is very sensitive to changes in L. P., Brower, J. V. Z. & Collins, C. T. (1963) Zoologica 48, 65-84. k'. Consequently, the presence of a Batesian mimic could be 8. Duffey, S. S. & Scudder, G. G. E. (1972) J. Insect Physiol. very detrimental to the model by lowering k' for the model- 18, 63-78. mimic complex. For example, for Case 2 when n = 100, nine 9. Arnold, G. W. & Hill, J. L. (1972) in Phytochemical Ecology times as many mimics as models would reduce the protection (Academic Press, New York,) pp. 71-101. of the complex by In Case 3 the reduction would be 10. Alcock, J. (1971) Behaviour 40, 1-9; Pilecki, C. & O'Don- 18%. ald, P. (1971) Evolution 25, 365-370. 66%. In this situation there would be selection for increased 11. Morrell, G. M. & Turner, J. R. G. (1970) Behaviour 36, distastefulness of the model, tending eventually to Case 1, 116-130; Brower, L. P., Alcock, J. & Brower, J. V. Z. (1971) as well as the evolution of distastefulness in the mimic. in Ecological Genetics and Evolution (Blackwells, Oxford), Huheey (14) originally proposed that one route to Mullerian pp. 261-274. 12. Brower, J. (1960) Amer. Natur. 94, 271-282; Holling, C. S. mimicry was by first becoming a Batesian mimic which is (1965) Mem. Entomol. Soc. Can. 45, 3-60. supported by Cases 2 and 3. 13. Alcock, J. (1970) Anim. Behav. 18, 733-739. In the tropics large mimicry complexes center around 14. Huheey, J. E. (1961) Evolution 15, 567-568.