Depensation and extinction risk II

References Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. Canadian Journal of Fisheries and Aquatic Sciences 54: Liermann M & Hilborn R (2001) Depensation: evidence, models and implications. Fish and Fisheries 2:33-58 Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:

Review

Depensation due to mating success Depensation: mating Replace spawners S in stock-recruit with p mated × S Number of spawners at which 50% successfully mate A Beverton-Holt curve Proportion mated 12 Depensation and extinction I.xlsx

Low densities (summary) Increased risk of extinction – All births one gender – Random events – Predation – Difficult to find mates – Other (inbreeding, lost group benefits, etc.) The net effect is depensation: lower rate of increase at low densities

Myers analysis Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269: RAM Myers Nick Barrowman Jeff HutchingsAndy Rosenberg

Myers analysis Model 1: δ = 1 (find MLE, likelihood L 1 ) Model 2: δ free (find MLE, likelihood L 2 ) Nested model Likelihood ratio test: R = 2ln(L 2 /L 1 ) is a chi-square distribution with degrees of freedom 1 Detecting depensation

delta=1delta free alpha K delta11.78 sigma NLL nparams34 Likelihood ratio19.35 Degrees of freedom1 Chi-squared prob1.1E-05 Compare model 1 and model 2 Myers analysis Recruitment Recruitment (log-scale) Spawning biomass 13 Depensation and extinction II.xlsx

Myers results Explored 128 data sets Only 3 significant cases of depensation Fewer than expected by chance Of these data sets about 27 had high power Myers analysis Myers RA, Barrowman NJ, Hutchings JA & Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269:

Problems with Myers method Parameterization has no biological interpretation except δ > 1 implies depensation Used p values to test for significant depensation, ignores biological significance Confounding of environmental change (regime shifts) with depensation Myers analysis

Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54: Myers method: two curves, same δ Myers analysis Recruits Spawners

Hilborn depensation method Hilborn method 13 Depensation and extinction II.xlsx Spawning level at which depensation reduces recruitment by 50% Beverton- Holt curve Recruitment Recruitment (log) Spawning biomass

Likelihood profile Hilborn method 13 Depensation and extinction II.xlsx

Liermann & Hilborn (1997) Same data used in Myers et al. New depensation model with parameter q = depensatory recruitment divided by Beverton- Holt recruitment, both at 10% of unfished biomass Calculated Bayesian probabilities of different values of the q parameter Liermann & Hilborn Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54:

Both curves go through the same points at (R*, S*) and (zR*,0.5S*) q = n/m is the ratio of recruitment at 0.1S*. When q 1 there is hyper- compensation. Spawners Recruits Liermann & Hilborn z is analogous to steepness but at 0.5 of max. S max observed spawner level Parameters: q, S* and z Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54:

Liermann & Hilborn depensationhyper- compensation depensationhyper- compensation Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54:

A lot of uncertainty about the underlying distribution Some probability for depensation (q 1) Liermann & Hilborn (1997) Liermann & Hilborn depensationhyper- compensation Liermann M & Hilborn R (1997) Depensation in fish stocks: a hierarchic Bayesian meta-analysis. CJFAS 54:

Population Viability Analysis (PVA), a.k.a. extinction risk

Extinction risk For any model with process error we can calculate the probability of going extinct or, rather, falling below a “quasiextinction threshold” Quasiextinction is a population size that is so low it is likely to become extinction Adding depensation increases the probability of falling below this limit Quasiextinction

random.walk <- function(N, b, d, nyears=100, quasiextinction=10) { N.vector <- vector(length=nyears) if (N < quasiextinction) { #no point in going through all the years N.vector[]<-0 #set all N's to zero } else { N.vector[1] <- N #first year for (yr in 2:nyears) { probs <- runif(n=N.vector[yr-1]) #vector probabilities between 0 and 1 births <- sum(probs<b) #number of cases < b, births deaths =b & (probs < b+d)) #if between b and b+d then death N.vector[yr] <- N.vector[yr-1]+births-deaths if (N.vector[yr] < quasiextinction) { N.vector[yr]<-0 } invisible(N.vector) } R code

Random walk: quasiextinction Quasiextinction b = d = 0.2 (birth probability = death probability) Quasiextinction 13 Random walk quasi.r 13 Depensation and extinction II.xlsx

Random walk: quasiextinction Quasiextinction 13 Random walk quasi.r 13 Depensation and extinction II.xlsx

Dennis model: simple analytic model (diffusion approximation method) Dennis model Dennis B et al. (1991) Estimation of growth and extinction parameters for endangered species. Ecological Monographs 61: Increases when λ > 1 Examine trends in ln(N) Starting population size Next year changes by µ After many years of µ increases Variance grows over time Random process error assumed normally distributed for lnN t

µ = -0.03, σ = 0.15, X 0 = ln(500) Trends in abundance Dennis model 13 Dennis method NAtl right whales.xlsx

µ = -0.03, σ = 0.15, X 0 = ln(500) Trends in log space Dennis model 13 Dennis method NAtl right whales.xlsx

Relation between λ and µ Dennis model

Estimating  and  2 from counts Choose pairs of N i and N j in adjacent years t i and t j e.g. N 1980 =6, N 1981 =8 calculate transformed variables If data each year, X = 1 If data each year, denominator is 1 Dennis model

Steps Do a regression of Y values against X values, forcing the regression through the origin Slope is µ Mean squared residual is σ 2 “residual” is difference between observed and model-predicted values; in this process-error model – observed = lnN t – predicted = lnN t-1 + µ Dennis model

Application: North Atlantic right whales ( ) NA right whales Extremely well studied, abundance Census of annual cow-calf pairs; these counts measure reproductive females. Average inter-calf interval 3-5 years.

Slope of the line is µ = -0.09, while σ = 0.60 NA right whales 13 Dennis method NAtl right whales.xlsx

Probability of falling below 10 individuals (not 10 calves) is 1% after 4 yr 22% after 10 yr 40% after 20 yr 52% after 50 yr N 0 = 400, µ = -0.09, σ = 0.60 NA right whales 13 Dennis method NAtl right whales.xlsx

Caswell H et al. (1999) Declining survival probability threatens the North Atlantic right whale. PNAS 96: NA right whales

Fujiwara M & Caswell H (2001) Demography of the endangered North Atlantic right whale. Nature 414: NA right whales

Kraus SD et al. (2005) North Atlantic right whales in crisis. Science 309: NA right whales

13 Dennis method NAtl right whales.xlsx increasing

Best current estimates Right Whale News December NA right whales

Probability of falling below 10 individuals (not 10 calves) is 7% after 4 yr 26% after 10 yr 37% after 20 yr 43% after 50 yr N 0 = 400, µ = +0.04, σ = 0.82 Mean abundance: 599 after 10 yr, 898 after 20 yr, 3017 after 50 yr. NA right whales Not very different! 13 Dennis method NAtl right whales.xlsx

Disadvantages of the Dennis method The results are highly sensitive to errors in the estimates of  and . The data series is often short which means that  and  may be very imprecise With increasing time, variance increases, predictions range from 0 to very high, and thus extinction risk will always be high in the future (despite increasing trends!) No account is taken of changes in (past or future) management practices and environmental change No allowance for density-dependence The extinction risk can be very sensitive to the initial population age-structure (which is ignored) Dennis model

Sampling stochasticity Abundance estimates are measured with observation error Dennis  based on change in estimated N from year to year High observation error = high  value But actual probability of extinction depends on process error not observation error E.g. perfectly stable population, no process error, high observation error, therefore zero  but high  Dennis method: high estimated extinction risk but in reality a zero extinction risk Herrick GI & Fox GA (2013) Sampling stochasticity leads to overestimation of extinction risk in population viability analysis. Conserv. Lett. doi: /j X x. Dennis model

Calculating extinction risk (any model) Define model and parameters – Exponential, logistic, with or without depensation, Dennis model, etc. Simulate population size into future Generate probability for population size at specified times Define threshold population size – Quasiextinction or critical population sizes Calculate proportion of simulations that fall below critical number

Key lessons Concept of depensation How to add that to models Empirical studies of depensation Quasiextinction criterion Dennis model of stochastic populations leading to extinction Be very wary of predictions!