1
Supplementary Material
A)Descriptive statistics for Models and Mimics separately for variation in frequency of Mimics. Note there is an increase on mean Model attack rate, even though there was near-significant discrimination at frequency of 0.5.
Model? / MimicFrequency / Mean / Std. Deviation / N0 / .25 / .28 / .452 / 40
.50 / .41 / .495 / 80
.75 / .34 / .476 / 120
1.00 / .57 / .497 / 160
Total / .44 / .497 / 400
1 / .00 / .16 / .364 / 160
.25 / .15 / .359 / 120
.50 / .29 / .455 / 80
.75 / .38 / .490 / 40
Total / .20 / .402 / 400
Total / .00 / .16 / .364 / 160
.25 / .18 / .386 / 160
.50 / .35 / .478 / 160
.75 / .35 / .478 / 160
1.00 / .57 / .497 / 160
Total / .32 / .467 / 800
B)A simple evolutionary model of toxin evolution
Stability of automimicry in binary trait systems has been demonstrated by Broom et al. (2005) and Speed et al. (2006). SvennungsenHolen (2007) report a detailed analytical approach to automimicry in continuous traits.
Given this prior work, we do not therefore propose to formally demonstrate stability here. Rather we limit our analysis to a brief demonstration of the characteristics consistent with evolutionary stability in the simple model in the main paper, to demonstrate that the interpretation of stability in Figure 3 (main paper), is sustained when the model is expanded to enable evolution.
To enable an evolutionary model we assume an infinite population size. We determine the fitness of defended and nondefended prey as in the model in the main paper. We then calculate the frequency of the defended form to the next generation as (fitness of defended form / sum of fitness of both forms) or
Fo.SoT=1..t / [ (Fo.SoT=1..t) +Fi.SiT=1..t) ] (S1)
and the frequency of the defended form is 1-Fo.SoT=1..t.
We can now demonstrate that defended prey will evolve to stable intermediate frequencies by iterative use of equation S1 (and of course the rest of the model outlined in the introduction.
We present a limited illustration here, but we provide the R code below, so that readers may explore more widely if they wish.
First, for the example of t=1000, and Fco=0.9, we introduce defended prey from rarity. The defended prey increase, stablising in frequency at 0.78 (Figure S1a). If we consider the converse situation in which edible prey are introduced to a population of defended prey, then the frequency of defended prey reduces from 1 down to the stable level at 0.78 (Figure S1b).
We can introduce some stochastic challenge to the equilibria by modifying the frequency of the defended prey (S1) in each generation by adding a value randomly drawn from a gaussian distribution of mean 0 and SD=1 (so that positive and negative values created symmetrically around the value generated in S1). To prevent stochastic extinction, frequencies that become greater than one are set down to 0.99, frequencies less than zero are set upward to 0.01.
Whether the defended prey start off rare (Figure S1c) or near to fixation (Figure S1d), the frequency of defended prey evolves toward the equilibrium state (0.78), varying around it because of the effects of random perturbations to phenotype frequencies between generations.
Note though that these are still infinite populations being measured, and hence underestimate the effect of random drift when the net benefits of individual toxicity are very small at any frequency.
Figure 1. Frequency of defended prey across generations
- evolving from rarity, no stochastic effect
b. evolving from near fixation, no stochastic effects
c. evolving from rarity, stochastic effects
- evolving from near fixation, stochastic effects
Supplementary Material Reference list
Broom, M., Speed, M. P., & Ruxton, G. D. (2005). Evolutionarily stable investment in secondary defences.Functional Ecology Funct Ecology, 19, 836–843.
Speed, M. P., Ruxton, G. D., & Broom, M. (2006).Automimicry and the evolution of discrete prey defences. Biological Journal Of the Linnean Society, 87, 393–402.
Svennungsen, T. O., & Holen, O. H. (2007).The evolutionary stability of automimicry.Proceedings Of the Royal Society B: Biological Sciences, 274, 2055–2063.
R Code (please contact M Speed at )
# frequency of mimic - mf
# probably of being noticed - p1
# probability of being attacked - function to match observed data - p2
# probability of survival - 99 or 33 - p3
# notional cost - c
# t=time for season
# survivial = (1-p1p2p3)^t
# fecundity = 1-c^y
# y= 0.5, 1.5,1
# plot fitness mimic vs model; find equality
rm(list=ls())
xnewFo<-0
xnewFi<-0
mf<-0.99
mof<-1-mf
p3a=0.38
p3b=0.89
t<-1000
fco<-0.9
fci<-1
p1=0.001
for (z in 1:1000)
{
p2<-(0.13+(0.5*mf)^1.4)
smo<-(1-p1*p2*p3a)^t
smi<-(1-p1*p2*p3b)^t
Fo<-(fco*smo)
Fi<-(fci*smi)
newFo <- mof*Fo
newFi <- mf*Fi
allF<- newFo+newFi
mf<-(newFi/allF)+rnorm(1, mean = 0, sd = 01)/100
if (mf <0) mf<-0.01
if (mf>1) mf=0.99
mof<-1-mf
xnewFo[z]<-1-mf
}
plot(xnewFo, xlab="Generations", ylab="Frequency of defended prey", xlim=c(0, 1000), ylim=c(0, 1) )