Lecture 7 review Foraging arena theory examines the implications for foraging and predation mortality of spatially restricted foraging behavior that evolves in response to predation risk Spatially restricted foraging intensifies intra-specific competition, and in conjunction with selection to maintain growth rate can lead to Beverton-Holt stock recruitment relationships Estimating the mean stock-recruitment relationship has two main problems: errors-in-variables and time series bias. High recruitment variation is NOT a serious cause of poor estimation performance.

Lecture 8: Mark-recapture methods for abundance and survival Most important application and very broad need is to provide short-term estimate of exploitation rate U, to allow use of N=C/U population estimates, manage U change Mark-recapture data generally analyzed using binomial or Poisson likelihoods Multiple marking and recapture sessions over time can give estimates of survival and recruitment rate along with population size

Mark-recapture experiments Mark M animals, recover n total animals of which r are marked ones Pcap estimate is then r/M, and total population estimate is N=n/Pcap = nM/r, i.e. you assume that n is the proportion Pcap of total N Critical rules for mark-recapture methods: NEVER use same method for both marking and recapture (marking always changes behavior) Try to insure same probability of capture and recapture for all individuals in N (spread marking and recapture effort out over population) Watch out for tag loss/tag induced mortality especially with spagetti tags (use PIT or CWT when possible, or GENETAG)

How uncertain is the estimate of Pcap (U) from simple experiments? Suppose M animals have been marked, and r of these have been recaptured Log Binomial probability for this outcome is lnL(r|M)=r ln(Pcap) + (M-r) ln(1-Pcap) Evaluate uncertainty in Pcap estimate by either profiling likelihood or looking at frequency of Pcap estimates over many simulated experiments; get same answer, as in this example with M=50, r=10:

It takes really big increases in number of fish tagged to improve Pcap estimates The variance of the Pcap estimate is given by σ2pcap=(Pcap)(1-Pcap)/M, where M is number of fish marked. The standard deviation of Pcap estimates depends on Pcap and number marked:

Estimates of N=C/Pcap are quite uncertain for low M, eg 50 fish This would be Lauretta’s luck, getting only 4 recaps when the average is 10 (0.2 x 50 marked fish) Generated using Excel’s data analysis option, random number generation, type binomial with p=0.2 and “number of trials”=50

How uncertain is the estimate of N from simple mark-recapture experiments? Suppose M animals have been marked, and r of these have been recaptured along with u unmarked animals Log Binomial likelihood for this outcome given any N is lnL(r,u|N)=r ln(Pmarked)+u ln(Punmarked)= r ln(M/N) + u ln((N-M)/N) Can also assume Poisson sampling of the two populations M and M-N Pcap=(r+u)/N; predr=pcap*M, predu=pcap*(M-N) lnL=-predr+r ln(predr) – predu + u ln(predu) Evaluate uncertainty in N estimate by profiling likelihood (show how lnL varies with N), as in this example with M=50, r=10, u=100:

Open population mark-recapture experiments (Jolly-Seber models) Mark Mi animals at several occasions i, assuming number alive will decrease as Mit=MiSt where St is survival rate to the tth recapture occasion. Recover rit animals from marking occasion i at each later t. Estimate total marked animals at risk to capture at occasion i as TMit=Σi-1Mit to give Pcapi estimate Pcapi=Σi-1rit/TMit. Total population estimate Ni at occasion i is then just Ni=TNit/Pcapi, where TNi is total catch at i. Estimate recruitment as Ri=Ni-SNi-1 or other more elaborate assumption

Two Pcap estimators for Jolly-Seber experiments Unbiased but inefficient: (capture of marked fish known to be alive) (number of fish known to be alive) (here “known” means were seen later) Efficient but possibly biased: (capture of marked fish) (model predicted number at risk)

Structure of Jolly-Seber experiments Make up a table to show mark cohorts and recapture pattern of these: Predict the number of captures for each table cell Rij=MiS(j-i)Pcapj (or Ni,j-1-rij-1)S if removed) Use Poisson approximation for lnL lnL=Σij[–Rij+rij ln(Rij)] evaluated at conditional ml estimate of Pcapi=Σirij/ΣiMiS(j-i) (only i’s present at sample time j)

Just remember these five steps Array your observed capture, recapture catches in any convenient form, Cij For each distinct tag (and untagged) group i of fish, predict the numbers Nij at risk to capture on occasions j, using survival equation (and recruitments for unmarked N’s) For each recapture occasion, calculate Pcapj as Pcapj=(total catch in j)/(total N at risk in j) For each capture,recapture observation, calculate the predicted number as =pcapjNij Calculate likelihood of the data as Σij(- +Cijln( ))

Don’t make stupid mistakes like this one Buzby and Deegan (2004 CJFAS 61:1954) analyzed PIT tag data from grayling in the Kuparuk River, AK; concluded there had been decrease in Pcap and increase in annual survival rate S over years, tried various models and presented lots of AIC values to justify the estimates below. In fact, (1) high Pcaps in early years are symptomatic of not covering the whole river in m-r efforts; (2) Pcap and S are partially confounded (can increase S and lower Pcap or vice versa, still fit the data).

Using annual tagging to track catchability changes Assume Ut=1-exp(-qtEt), i.e. exploitation rate is a saturating function of effort Et Mark M fish at start of fishing, estimate Ut=(catch of marked fish)/M One-year estimate of qt is then qt=-log(1-Ut)/Et Issue: this estimate of qt is noisy; is there a better estimator based on weighted averaging of qt with predicted value based on past years data (Kalman “filtering” estimate): qtt=Wqt+(1-W)qtt-1 where W=(var of true qt given true qt-1) (total var of qt estimate)