Snails Home Supplementary Electronic Material

Transcription

1 Snails Home Supplementary Electronic Material D.J. Dunstan and D.J. Hodgson This consists of the 2001 and 2002 data file (Excel spreadsheet supplied separately), and also the text below, explaining the Monte Carlo programme and the maximum likelihood code. Monte Carlo: Initialisation Set parameter values nres=200; Number of resident snails, with homing instinct nvis=500; Number of non-resident snails, without homing instinct probout=0.75; Relative probability for snails to leave the garden probvisin=0.025; Relative probability for non-residents to enter the garden probresin=0.25; Relative probability for residents to enter the garden probbirth=0.3; Probability of a birth per unit time (per capture) probdeath=0; Probability of a death per unit time (per capture) probbres=nres/(nres+nvis); Probability of a birth being a resident mixing=30 Mixing rate factor Note the relative probabilities are set to give a quarter of the residents and a thirtieth of the nonresidentsto be in the garden at equilibrium, about 70 snails Create empty lists for record-keeping res={}; vis={}; garden={}; out={}; waste={}; live={}; born={}; recruited to the trial) died={}; the trial) List of resident snails List of non-resident snails List of snails in the garden List of snails outside the garden List of snails in the landing-zone List of snails that are currently alive List of snails that have been born (or List of snails that have died (or left

2 No21={}; the garden of No. 21 List of snails taken to Create table for storing results results=table[0,{i,20},{j,1950}]; Table of lists suitable for plotting as Fig3 Note 20 is more than the expected maximum number of spots; 1950 is the number of captures in the simulations Create the list of 700 snails with three properties snails=table[{0,0,0},{sn,nres+nvis}]; Do[snail[[sn,1]]=1; Make the first batch residents by setting the first property to 1 res=join[res,{sn}],{sn,nres}]; and put them in the list of residents ( isn t used in Mathematica but is included here for clarity) Do[vis=Join[vis,{sn+nres}], {sn,nvis}]; and put the rest in the list of non-residents live=join[res,vis]; Put them all in the list of live snails Note that the first property snail[[sn,1]] indicates non-resident (0) or non-resident (1). The second property indicates where it is: out (0), garden (1), No.21 (3), The third property is the number of spots. The lists res and vis are updated by joining the snail number sn as a list {sn} Allocate residents and non-residents to the garden or out Do[sn=res[[i]]; We do the list of residents prob=probresin/probout If[RandomReal[]<prob, using the random number generator snail[[sn,2]]=1; to allocate at random to the garden (1) garden=join[garden,{sn}], and update the list garden ( isn t used in Mathematica) snail[[sn,2]]=0; or else to allocate to outside the garden (0) out=join[out,{sn}]], and update the list out ( isn t used in Mathematica) {i,length[res]}]; iterating i over the whole list res Note it is not necessary to use the list res, but this code illustrates its use. The variable sn is the snail number. RandomReal[] is a random number equally distributed over the range 0 to 1 Do[prob=probvissin/probout; If[RandomReal[]<prob, We do the list of non-residents

3 snail[[i,2]]=1; Here we take the i th snail directly garden=join[garden,{i}], and put it in the garden... snail[[i,2]]=0;... or out, according to the random number out=join[out,{i}]], {i,nres+1,nres+nvis}]; and iterate over the known non-resident snails Note this code is more compact and faster, but is less easy to read and debug than the code used above for the residents. It does exactly the same thing. Do a single capture to start the results table off i=randominteger[{1,length[garden]}];pick a snail in the garden at random sn=garden[[i]]; use the garden list to get its number sn snail[[sn,2]]=0; give it the property of being out (0) snail[[sn,3]]=1; and give it one spot (third property) garden=drop[garden,{i}]; updating the garden list out=join[out,{sn}]; and the out list results[[1,1]]=1; and make the first entry in the results table Now we iterate over 1949 more captures n=2; done While[n<1951, until n=1950 Capture ng=length[garden]; If[ng>0, empty k=randominteger[{1,ng}]; random sn=garden[[k]]; snail[[s,3]]++; a Tippex spot on it) If[n<1386 Or EvenQ[sn], protocol changed out=join[out,{sn}]; snail[[sn,2]]=0, First capture (n=1) has already been The following code will be iterated Get number ng in the garden Capture can t be done if garden is Pick a snail from the garden at sn is its number increment its spot property (i.e. make n>1385 is the year 2002 when the put it in the out list update its site property

4 No21=Join[No21,{sn}]; put it in the No21 list (2002 and odd-numbered snail snail[[sn,2]]=3]; update its site property (3 for No. 21) garden=drop[garden,{k}]; update the garden list Do[ Update the results list (i.e. fill in the n th column) If[j==snail[[sn,3]], If j is the number of spots the snail now has results[[j,n]]=results[[j,n-1]]+1, increment the value in the previous column results[[j,n]]=results[[j,n-1]]], otherwise copy the previous value {j,length[results]}]]; iterating over the 20 columns n++]; Now increment n as we have made a capture Births If[(n<750 Or n>1385) summer And RandomReal[]<probbirth, capture) < 1 snail=join[snail,{{0,0,0}}]; snail sn=length[snail]; of the list If[RandomReal[]<probbres; or outside garden=join[garden,{sn}]; res=join[res,{sn}]; snail[[sn,1]]=1; snail[[sn,2]]=1; out=join[out,{sn}]; vis=join[vis,{sn}]; snail[[sn,1]]=0; snail[[sn,2]]=0; Births only in the first part of the with a probability (births per Join a new snail to the end of the list its number sn is then the new length Make a random allocation to garden update garden update res set property of being a resident and in garden update out update vis set property of being a non-resident and being out of garden

5 born=join[born,{sn}]]; update list born live=join[live,{sn}]]; update list live Note that we make the rate of births of residents and non-residents proportional to the initial populations and the place of birth is the garden and outside respectively. Deaths These happen at random throughout, and also while overwintering If[RandomReal[]<probdeath, nlive=length[live]; nlive is the number of live snails, in garden and out k=randominteger[{1,nlive}]; so we pick one at random sn=live[[k]]; find its number live=drop[live,{k}]; delete it from the live list died=join[died,{sn}]; add it to the died list If[snail[[sn,2]]0, If the snail in the out list pos=position[out,sn][[1]]; find position in out list out=drop[out,pos]], and delete it If[snail[[sn,2]]1, If the snail is in the garden list pos=position[garden,sn][[1]]; find position in garden list garden=drop[garden,pos]], and delete it If[snail[[sn,2]], If the snail is in the No21 list pos=position[no21,sn][[1]]; find position in No21 list No21=Drop[No21,pos]]; and delete it snail[[sn,2]]=4; set its site property to 4 (dead) pos=position[garden,sn]; Note all snails in garden and outside have the same risk of dying Overwintering deaths If[n==1386, We have just finished 2001 and the next capture will be in 2002 kill=length[born]-length[died]; The number we need to kill to restore the initial population If[kill>0, fkill=kill/length[live]]; The fraction to die is fkill Do[sn=live[[k]]; pick each snail in live in turn If[RandomReal[]<fkill, if it draws the short straw died=join[died,{sn}]; add to died list live=drop[live,{k}]; remove from live list If[snail[[sn,2]]==0, If it is in the out list pos=position[out,sn][[1]]; find its position out=drop[out,pos], and remove it

6 garden list pos=position[garden,sn][[1]];find its position in the garden=drop[garden,pos]];and remove it snail[[sn,2]]=4], set property 2 (place) to 4 (graveyard) {k,length[live],1,-1}]]; iterate over all live snails Mixing Residents go out, and non-residents come visiting ngo=length[garden] probout/mixing;number of snails expected to leave in unit time ngo=integerpart[ngo]+if[random[]<fractionalpart[ngo],1,0]; Make ngo integer Do[k=RandomInteger[{1,Length[garden]}]; sn=garden[[k]]; Pick a snail at random snail[[sn,2]]=0; Set its site to out out=join[out,{sn}]; Add it to the out list garden=drop[garden,{k}], Remove it from the garden list {k,ngo}] Iterate until ngo snails have left Do[sn=out[[j]]; Now look for snails to enter If[snail[[sn,1]]==1, If a resident, prob=probresin, it has the probability probresin to enter prob=probvisin, it has the probability probvisin to enter If[RandomReal[]<prob, garden=join[garden,{sn}]; Add it to the garden list snail[[sn,2]]=1; Set its site to garden out=drop[out,{j}]], Remove it from the out list {j,length[out],1,-mixing}]]]; Repeat until Length[out]/mixing have been looked at End While This is the end of the iteration over 1949 captures

7 Now we plot the results plotresults=table[results[[i]],{i,length[results]}]; fig6a=show[ ListPlot[plotresults, PlotRange->All, Joined->True, PlotStyle->{Blue,Red,Green,Black}], pdata, Graphics[Line[{{1385,600},{1385,0}}]]]

8 SUPPLEMENTARY INFORMATION 2 R Code to re-create the maximum likelihood analyses in R ###Requires library Rcapture library(rcapture) #enter simplified histogram of frequency of number of recaptures observed<-c(136,61,41,56,37,22,24,15,3,3,4,4,2) #enter sequence of number of spots nbcap<-seq(1,13) #create simple recapture matrix snail.cmr<-cbind(nbcap,observed) #fit the homogeneous (Poisson) model snail.closed.mhpoisson<-closedpci.0(snail.cmr, dfreq=true, dtype="nbcap", t=62, t0=13,m="m0", h="poisson") #fit the heterogeneous (Chao) model snail.closed.mhchao<-closedpci.0(snail.cmr, dfreq=true, dtype="nbcap", t=62, t0=13,m="mh", h="chao") #display results, giving estimated population sizes, confidence intervals and AIC snail.closed.mhpoisson snail.closed.mhchao #plot the residuals, allowing inference of surfeit/deficit of snails with s spots plot(snail.closed.mhpoisson) plot(snail.closed.mhchao)