Supplementary Material for “Conspecific density modulates the effect of predation on dispersal rates.”

EddHammill1,2,,Richard G. Fitzjohn3, and Diane S. Srivastava2

1School of the Environment, University of Technology, Sydney

Harris and Broadway, Ultimo, Sydney, Australia, NSW 2007

2Department of Zoology and Biodiversity Research Centre, University of British Columbia, 6270 University Blvd.Vancouver, B.C., Canada V6T 1Z4

3Department of Biological Sciences, Macquarie University, Sydney, NSW 2109, Australia

Appendix A1 – Analytical Solutions of the Model
Appendix A2 – R-code for fitting model to data

Appendix A3 – Data used in the MS

Appendix A1: Analytical Solutions of the Model

In order to ascertain whether a switch in the number of dispersers in no predator and predator treatments was possible, we asked whether the equations describing the number of dispersers in the two treatments could be equal:

where the left hand side of the equation represents dispersal number expected in the absence of predation, and the right hand side represents dispersal number expected in the presence of predation. Simplifying S1 results in:

We then solved this equation for different parameter values of SB and GB to determine if switch points were possible at positive prey densities. Results are given in Appendix table 1.

A1 table 1

Parameters / Figure / Intersection at N > 0?
sB = 0, gB = 0 / Figure 1a / No, parallel lines with predators higher by c individuals
sB = 0, gB > 0 / Figure 1b / Yes, at
sB = 0, gB < 0 / Figure 1c / No, only at negative densities
sB < 0, gB = 0 / Figure 1d / Yes at
sB < 0, gB > 0 / Figure 1e / Yes at
sB < 0, gB < 0 / Figure 1f / Yes at
sB > 0, gB = 0 / Figure 1g / No, dispersal rates with predators always higher
sB0, gB> 0 / Figure 1h / If
sB0, gB< 0 / Figure 1i / No, dispersal rates with predators always higher

A1 table 1. Analytical solutions of equation S1 for various values of SB and GB. If the function between dispersal number and prey density in the absence of predators intersected with the analogous function in the presence at predators, at positive prey densities, we concluded that there was a density-dependent switch in dispersal strategy in response to predation.

Appendix A2 – R-code used to fit the model to empirical data

all.data<-read.csv("~/dispersal/2nd expt, gradients/2nd expt results.csv",header=TRUE)

pred.data<-all.data[(all.data$pred=="yes"),]

cont.data<-all.data[(all.data$pred=="no"),]

## Doing it the new way, going back to first principles...

## Logic, there is an inherent risk of starvation (s), which is

## proportional to density. This will affect an individuals'

## desire to move, the actual number will be that desire multiplied

## by the number of individuals, e.g. (s*N)*N, or sN^2

## Also, there is a (g), for the number moved due to density independent factors (c in the MS), this must

## be multiplied by density, therefore gN

## The other way to think of it is the decision of the individual,

## multiplied by the number of individuals N(sN+g)

## IN the predator treatment there's another term, based on risk

## of predation, with a term to convert risk into desire to move.

## risk is generally 1/N, multiplied by p = p/N. this is the

## INDIVIDUAL's desire, to convert to how many individuals will

## move, need to multiply by N, there fore, (p/N)*N = p

## Combine this with the other terms to get the actual model...

## Again, other way to think of it is the decision of the individual, multiplied by number of individuals..

## N(sN+g+p/N)

### Trying the final glm thing ### Think that this works

all.data$p.factor<-ifelse(all.data$pred=="no",0,1)

all.data$dens2<-all.data$start^2

all.mod1<-glm(number~start+p.factor+dens2+start*p.factor+dens2*p.factor-1, data=all.data)

summary(all.mod1)

anova(all.mod1,test="F")

### AIC = 199.35, all vary

## Bits for AIC

## 197.35, s doesn't vary

all.mod1.5<-glm(number~start+p.factor+dens2+dens2*p.factor-1, data=all.data)

summary(all.mod1.5)

AIC(all.mod1.5)

## 203.72, g doesn't vary

all.mod1.3<-glm(number~start+p.factor+dens2-1, data=all.data)

summary(all.mod1.3)

AIC(all.mod1.3)

## 246.90

all.mod1.4<-glm(number~start+dens2-1, data=all.data)

summary(all.mod1.4)

AIC(all.mod1.4)

## 245.12

all.mod2<-glm(number~start+p.factor+dens2+start*p.factor+dens2*p.factor-1,quasipoisson, data=all.data)

summary(all.mod2)

AIC(all.mod2)

all.mod3<-glm(number~start+p.factor+dens2+start*p.factor+dens2*p.factor-1,poisson, data=all.data)

summary(all.mod3)

#### Figures

par(mfrow=c(1,2))

###### ********* WORKS!!!!********#####

## Numbers

plot(pred.data$number~pred.data$start,xlab="Density",ylab="Number dispersed",lty=2,cex=1.5,lwd=2,pch=19,ylim=c(-5,55))

legend(500,50,c("Control","Predator cue"),bty="n",lty=c(3,1),lwd=2,cex=1.5)

points(cont.data$number~cont.data$start,lwd=1.5,cex=1.5, pch=1)

### values for drawing lines obtained from summary(all.mod1)

x.vals<-c(250:4000)

cont.ests<-0.000001737*(x.vals^2)+0.004983*x.vals

lines(cont.ests~x.vals,lwd=2,lty=3)

pred.ests<-(0.000001737+0.0000003073)*(x.vals^2)+(0.004983-0.005499)*x.vals+5.658

lines(pred.ests~x.vals,lwd=2,lty=1)

## Proportions

plot(pred.data$percent*100~pred.data$start,xlab="Density",ylab="Percent dispersed",lty=2,cex=1.5,lwd=2,pch=19)

legend(1500,2.2,c("Control","Predator cue"),bty="n",lty=c(3,1),lwd=2,cex=1.5)

points(cont.data$percent*100~cont.data$start,lwd=1.5,cex=1.5)

cont.ests<-((0.000001737*(x.vals^2)+0.004983*x.vals)/x.vals)*100

lines(cont.ests~x.vals,lwd=2,lty=3)

pred.ests<-(((0.000001737+0.0000003073)*(x.vals^2)+(0.004983-0.005499)*x.vals+5.658)/x.vals)*100

lines(pred.ests~x.vals,lwd=2,lty=1)

### AIC weights.

tots<-exp(-(199.35-197.35)/2)+exp(-(203.72-197.35)/2)+exp(-(246.90-197.35)/2)+exp(-(197.35-197.35)/2)

exp(-(199.35-197.35)/2)/tots

exp(-(203.72-197.35)/2)/tots

exp(-(246.90-197.35)/2)/tots

Appendix A3 – Data used in the MS

“predator cue” denotes whether freeze killed predators were present in the trial. “start” denotes the number of individuals introduced to the entry bottle at the beginning of the experiment. “number” states the number of dispersing individuals.

predator cue / start / number
no / 250 / 0
no / 250 / 1
no / 250 / 2
no / 400 / 2
no / 400 / 3
no / 400 / 4
no / 500 / 2
no / 500 / 4
no / 500 / 4
no / 650 / 2
no / 650 / 3
no / 650 / 4
no / 1250 / 6
no / 1250 / 6
no / 1250 / 11
no / 2500 / 21
no / 2500 / 25
no / 2500 / 28
no / 4000 / 43
no / 4000 / 49
no / 4000 / 50
yes / 250 / 3
yes / 250 / 4
yes / 250 / 7
yes / 400 / 5
yes / 400 / 5
yes / 400 / 7
yes / 500 / 6
yes / 500 / 6
yes / 500 / 12
yes / 650 / 3
yes / 650 / 9
yes / 1250 / 4
yes / 1250 / 6
yes / 1250 / 8
yes / 2500 / 13
yes / 2500 / 18
yes / 2500 / 18
yes / 4000 / 27
yes / 4000 / 34
yes / 4000 / 34