Bayesian Analysis of Extinction Risk in Atlantic Salmon A.J.F. Gibson1, P.G. Amiro1 and 2 R.A. Myers 1Department of Fisheries and Oceans, Dartmouth, NS 2Dalhousie University, Halifax, NS Overview • Endangered Species • Approach to Assessment • Questions - Needs • Inner Bay of Fundy Salmon • Index-based model of population size and trajectory • Summary: • Bayesian considerations for endangered species • Future work Species-at-Risk vs. Fisheries Modeling • The range of actions or activities to be evaluated are typically wider • habitat protection, creation or restoration, supportive rearing, bycatch reduction • Abundance is at levels where dynamics are less certain • potential Allee effects, demographic vs. environmental stochasticity • Populations are often data poor • Higher level of precision is sometimes required • unachievable • Dynamics have likely changed • implications for selection of priors Endangered Species: Challenges for Modelling • Challenges: • often data poor • life history often not well understood • Recovery planning has very specific requirements: • assessment of recovery potential, critical habitat designation, recovery targets • may require greater knowledge and model resolution than for exploited species • What is the role of Bayesian methods in this process? Species at Risk: Three Key Questions 1. What is the population’s current size and trajectory? • Risk, timeline for recovery 2. What are the differences in the past and the present population dynamics? • Which life history parameters changed, when and by how much? • Are there correlates that may explain the pattern? 3. Is the population tracking a declining carrying capacity? • importance of density dependence • key question for recovery planning Inner Bay of Fundy Atlantic Salmon • Designated as endangered by COSEWIC and listed by GoC on Schedule 1 (SARA prohibitions in effect) • IBoF Salmon are different and distinct from other salmon populations • life history: • very high incidence of fish that mature after 1 winter at sea • high incidence of repeat spawning • genetic: • unique mitochrondrial DNA haplotype that has not been found in salmon outside the inner Bay of Fundy IBoF Atlantic Salmon: Area of Occupancy Big Salmon River E E Stewiacke River 0 25 50 kilometers Stewiacke River, NS: Salmon Data Year 1960 1970 1980 1990 2000 Rec. Catch (adults) Adult Counts (fence) Adult Counts (boat electrofishing) Adult M-R estimates Adult Age Comp. Electrofishing Big Salmon River, NB: Salmon Data Year 1960 1970 1980 1990 2000 Rec. Catch (adults) Adult Counts (fence) Adult Counts (snorkel) Adult Age Comp. Redd Counts Electrofishing Smolt Counts (Fence) Smolt Counts (M-R ests.) Smolt Age Comp. The Problem • We want to estimate the population size and trajectory from the available data • when did the decline occur? • is the population still declining? • when is extinction expected to occur? • From a Bayesian perspective can we avoid subjectivity (prior belief) when building the model? Two Model Components: 1. Annual abundance estimated by modeling catch-effort data and any auxiliary data that can be interpreted as an abundance index (Rago 2001) • Parameter estimates are obtained using ML • Parameter uncertainty is assessed using MCMC to derive posteriors 2. Forward projections (PVA) using Monte Carlo simulations using the estimated change in population size from one year to the next (Dennis et al. 1991) • Projections using the posteriors for abundance, and the mean and variance of log(lambda) • Advantages: uncertainty in parameter estimates is included in the PVA; parameter covariance is preserved The Model (catch equations) • We begin with the standard catch equation for a Type I fishery: −Ft ,s Ct,s = Nt,s (1− e ) • We assume Ft,s is proportional to the fishing effort in year t: F = q E t,s s t The Model (harvesting) • Calculated the proportion of the catch that was harvested • 0.824 for small salmon (1983 to 1990) • 0.758 for large salmon in 1983. • used as constants in analysis • could lead to overestimate of escapement in early years Esct,s = Nt,s − Ht,s The Model (fence counts) • Counts assumed complete in 1994 and 1995, but not in 1992 and 1993. • The 1992 and 1993 counts were adjusted upwards using the mark and recapture estimates. • The fence counts then equal to the number of fish returning to the river in each size category and each year: Fencet,s = Nt,s The Model (adult electrofishing) • electrofishing occurs after fishing and is an index of escapement • reported for size categories combined • modeled use the catch equation: qboat Eboat ,t Cboat,t = (1− e )∑ Esct,s s The Model (egg deposition) • mean fecundity of Stewiacke River salmon was calculated using data of Amiro (1990) • weighted by number at age and sex ratio • 2,364 eggs per small salmon • 7,545 eggs per large salmon • used as constants in this analysis Eggst = ∑ Esct,s fecs s The Model (juvenile electrofishing) • we use Beverton-Holt model to describe the relationship between juvenile abundance and egg deposition in the previous years: • age-0 density in year t is a index of egg deposition in year t-1 • age-1 density in year t is a index of egg deposition in year t-2 • age-2 density in year t is a index of egg deposition in year t-3 • two parameters estimated for each size class α Eggs P = a t−a−1 t,a α Eggs 1+ a t−a−1 R0a The Model: estimated parameters • We set up the model to estimate: • the log of the total escapement in each year (37 parameters) • the average proportion of the population that are small salmon (1 parameter) • the catchability coefficients for the recreational fisheries and boat electrofishing (3 parameters) • the slope at the origin and asymptotic level for the 3 ages of fish in the electrofishing data (6 parameters) • Total of 47 parameters Model Fitting: • Parameter estimates were obtained using maximum likelihood • Lognormal errors were assumed for all data except the adult mark-recapture experiments (hyper-geometric structure assumed) • σ could not be estimated for all data sets simultaneously • fixed σ at 0.33 for the electrofishing data and estimated σ for the other model components • Parameter estimates were obtained by minimizing the value of an objective function (O.B.V.) that is the sum of the negative log likelihoods (no weights): O.F.V. = −( fence + catch + electrofishing + boat + mr ) The Model (Bayesian component) • We assumed uniform bounded priors for all model parameters • Bounds were wide enough not to influence the fit. • We used 2,000,000 iterations after a burn in of 200,000 iterations. • We sampled every 2,000th iteration to derive the posterior distribution. • This level of thinning was sufficient to ensure that autocorrelation in the chain was not problematic. The Model (diagnostics) • We ran many iterations of the model using several starting values to ensure convergence to a global maximum • model is robust in this respect • Convergence of the Markov chain was inferred informally • by comparison of the posterior densities based on the first 1,000,000 iterations with those from the second 1,000,000 iterations • by comparison of the posterior densities from several chains Results (recreational fishing) Recreational Fishing Effort 10000 • effort increased through 8000 6000 4000 the '70's and early 80's Days Rod 2000 and then decreased 0 1970 1980 1990 2000 • catches are variable but Year increase in the 1980's Large Salmon Catch 600 400 • fitted catch tracks the 200 observed catch of Fish Number 0 reasonably well 1970 1980 1990 2000 • a near perfect fit is Year Salmon Salmon Catch obtained if the 2000 1500 proportion in each size 1000 500 category is estimated of Fish Number 0 for each year 1970 1980 1990 2000 Year Results (recreational fishing) • MLE's for the log of catchability coefficients are: - 9.522 (small) and –10.214 (large) • smaller fish are easier to catch Given an effort of 3000 rod days: the MLE and 80% B.C.I. for the catch 1.0 0.8 rates are: Small Salmon Large Salmon 0.8 0.6 Large salmon: 10.4% 0.6 0.4 (6.5 to 20.1%) 0.4 0.2 0.2 Probability Density Probability Small salmon: 19.7% Density Probability 0.0 0.0 (13.0 to 34.1%) -14 -12 -10 -9 -8 -14 -12 -10 -9 -8 log(q) log(q) Results (recreational harvest rates) • Shown MLE's and 95 % CI's for the annual harvest rates Recreational Fishery Exploitation Rate: Small Salmon 0.8 MLE's for harvest rates 0.6 increased to a maximum of 0.4 0.2 40.5 % (small salmon) and Rate Exploitation 0.0 21.8 % (large salmon) in 1970 1980 1990 2000 1983 and decreased Year thereafter Recreational Fishery Exploitation Rate: Large Salmon 0.8 Rates would be 0.6 underestimated if catch and 0.4 0.2 release increased during Rate Exploitation 0.0 time 1970 1980 1990 2000 Year Results (boat electrofishing) • Catches with the electrofishing boat ranged between 58 salmon (1991) and 0 (1997) • Effort ranged between 31.8 and 123 km • At an effort of 40 km, the expected catch is 1.4% Electrofishing 0.6 Boat of the population 0.4 Boat El ect r of ishi ng: l ar ge and small sal mon combi ned 60 0.2 40 Probability Density Probability 20 0.0 Number of Fish 0 -9 -8 -7 -6 1985 1990 1995 2000 log(q) Year Results (juvenile electrofishing) 200 • Between 27 and 44 150 age-0 sites electrofished 100 annually since 1984 50 0 • Consistent downward 1985 1990 1995 2000 trends are evident for 200 150 age-1 all three age classes 100 50 • no wild fry captured 0 1985 1990 1995 2000 since 1999 Density (number/100m^2) • slight increase in age-1 30 age-2 20 density in 2002 10 probably due to LGB 0 1985 1990 1995 2000 releases Year Results