Mark-RecaptureMark-RecaptureModern use dates from work by C. G. J. Petersen (Danish fisheries biologist, 1896) and F. C. Lincoln (U. S. Fish and Wildlife Service, 1930)Useful in estimating:a. Size of populationb. Rate of exploitationc. Survival rated. Rate of recruitmentLincoln-Petersen EstimateApplicable to closed populationsA sample is taken, marked and released back into the population = MA second sample is taken of n individuals of which m have marksLincoln-Petersen EstimateProportion in sample marked = m/nThis proportion should be equal to the proportion of marked population (M) to total population (N)= M/Ne.g. M/N = m/nLincoln-Petersen EstimateAssuming second sample is an unbiased sample of population, thenNest = Mn/mThis basic model is the point of departure for multitude of more sophisticated (complicated) models.Chapman’s modification (M+1) (n+1)Nest= ------------------ (m+1) N2 (n-m)Var(N)=-------------- (n+1)(m+2)Rate of ExploitationWhere the second sample is taken in course of harvesting the population (much of fisheries, waterfowl, control efforts for pests)Rate of exploitation (u)uest = m/MAssumptions:1. Population is closed (no births or deaths or movements out or in)2. All animals have same probability of being caught in the first sample3. Marking does not affect catchability of an animal4. Second sample is a random sampleAssumptions:5. No loss of tags between samples6. All tags are reported in the second sampleMultiple Mark-RecapturesAssuming a closed population Schnabel (1938) and Schumacher & Eschmeyers (1943) developed E ni MiNk = ------------------- (E mi ) + 1Note: All sums from i=2 to kMultiple Mark-RecaptureA great advantage of multiple mark-recapture studies is that we can evaluate some of the critical assumptionsand apply more complicated models where the simple assumptions are not appropriate.Probability of Capture“Probability of capture is equal and constant for each animal at each trapping occasion.”Problems:Day to day variation (weather) = timeBehavioral effects (trap happy/shy)Individual differences (heterogeneity)Capture ProgramModels developed to handle these problems based on maximum likelihood (ML) in 50’s, 60’s, 70’s.Not applied until 1980’s because of difficulty of calculations.Otis et al (1978: Wildlife Monograph No.62) developed program CAPTURE to do calculations.CAPTURE Program: ModelsMo Constant capture probabilitiesMt Variation by time =SchnabelMb Behavioral response to trappingMbh Behavior and heterogeneityMth , Mtb , MtbhCAPTURE ProgramKey requirement is to mark animals individually so that their full capture history can be recorded.Numbers vs. densityBoundary problemsTrapping WebStandard approach is to lay out traps in a rectangular grid (See CAPTURE concentric rows of traps)Record location of initial capture of each animal.Density of captures in centermost circles estimates density using variable circular plot approach.Open PopulationLimitation of previous methods is assumption of closure (no births, deaths, immigration or emigration).Can we estimate for open populations?What problem does mortality cause?Marked population is unknown because some of these have died.Open Populationt1t2t3t4t5Jolly (1965) - English statistician andSeber (1965) - New Zeland statisticianindependently developed solution for multiple mark-recapture study based on earlier work by:Darroch (1959) another English statistician and Cormack (1964) a Scottish statistician.Darroch, Cormack, Jolly & Seber’s Ideat1t2t3t4t5NiMinimiJolly-Seber Model (Often called Cormack-Jolly-Seber)Use results of sampling at later time periods to estimate how many of the marked animals were present at an earlier time period.To do this we must give each animal an individual mark so that its entire capture history can be recorded.Capture Recapture TotalsTime Captured Recaptures Released i ni mi Ri 1 54 0 54 2 146 10 143 3 169 37 164 4 209 56 202 5 220 53 214 6 209 77 207Recapture Matrix Time of Capture Time of 1 2 3 4 5 6 last Capture 1 10 3 5 2 2 2 34 18 8 4 3 33 13 8 4 30 20 5 43JS Population Estimate ni MiNi = ---------- miIf we don’t actually know Mi we can use an estimate of M i.JS Survival Estimate Mi+1 si = ----------- Mi + ni - mi JS Estimate of BirthsNi+1 = Additions + Survivors from Ni = Bi + Ni si rearranging this for birthsBi = Ni+1 - Ni si JS Marked PopulationHow do we estimate the size of the marked population?We have to do a mark-recapture estimate of that.Use the animals not seen at the first recapture sample but seen later on as our recaptures.We do a mark-recapture within a mark-recapture estimate.JS Marked PopulationMi actually unknown because of mortality of released animalsWhat is largest known group at i that is a subset of Mi ?mi is known so must estimate the restMi - mi are the restJS Marked PopulationWhich of the rest are known to be alive?Some of the rest are caught after sample iCall these zi zi = Animals marked previous to sample i, not caught at i, but caught later.JS Marked Populationni is largest group of individuals known to be alive at sample i and it is comparable to (Mi - mi ), the “rest”Denote by ri the number of ni observed after sample i.ri is some fraction of ni JS Marked Populationri / ni should be comparable tozi /( Mi - mi )Setting these equal to each other and solving for Mi zi ni Mi = --------- + mi riRecapture Matrix Time of Capture Time of 1 2 3 4 5 6 last Capture 1 10 3 5 2 2 2 Z3 = 34 18 8 4 3 r3= 33 13 8 4