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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 num- ber of palatables, while in Case 3 the predator must en- counter 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 preda- tion 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 <I n/rn al the number of predators that are still eating at the end of the ith eating period is CASE 3 J = [16] w Xi [(1 - 2)31'- (1-1)1i] a K' 0 -0 1- 0.9999~~~~~~K.9 Z z0.6-0.6~ ~ ~ ~ ~ This is equal to the number of prey eaten during the (i + 1)th period. The number of prey eaten in the ith eating period is 2 \ found by rewriting the foregoing results with i replaced by W 0 (i- 1). The result is 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.
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