Lecture 7 review

1. 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.

2. 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

3. 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)

4. 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:

5. 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:

6. 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

7. 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:

8. 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

9. 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)

10. 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 lnLlnL=Σij[–Rij+rij ln(Rij)] evaluated at conditional ml estimate of Pcapi=Σirij/ΣiMiS(j-i) (only i’s present at sample time j)

11. 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( ))

12. 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).

13. 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-1where W=(var of true qt given true qt-1) (total var of qt estimate)