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Lecture 8 review

- Options for estimating population size
- Direct census (visual, acoustic, etc.)
- Density expansion (time, area)
- Change in index methods (depletion, ratio)
- C/U methods (Gulland’s old trick)
- Pcap methods using marked animals
- Bt/Bo methods using stock assessment models that estimate Bo as leading parameter

- Designing sampling programs for density
- Define the sampling universe carefully
- Use systematic sampling whenever possible

Lecture 9: 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)

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,Pcap)=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 TMi=Σi-1Mi, to give Pcapi estimate Σi-1rit/TMi.
- Total population estimate Ni at occasion i is then just Ni=TNi/Pcapi, where TNi is total catch at i.
- Estimate recruitment as Ri=Ni-SNi-1 or other more elaborate assumption

Structure of Jolly-Seber experiments models)

- 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)

Just remember these five steps models)

- 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 models)

- 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).

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