Monte Carlo methods and extinction risk (Population Viability Analysis)

Modeling extinction risks ProblemModeling solution Decreasing habitatChanging K Environmental changeChanging r or K ExploitationCatch term Demographic stochasticityProcess error Catastrophic eventsExtreme process error DepensationN 50 term Exotic speciesMultispecies model GeneticsNot covered here

Monte Carlo methods Unrealistic to project into the future with no uncertainty Projections are stochastic (include randomness) from two causes – Sample from uncertain current state of nature (e.g. different values of N t, r, K in logistic), using bootstrapping or Bayesian methods – Sample from future possible environmental events, using Monte Carlo methods

Needed for Monte-Carlo simulation Current starting state variables and parameters A model Random process errors for future Rules about future human impacts (e.g. harvest) Rules about other environmental change

Stochastic exponential model Multiplicative error, serially uncorrelated Rate of change Process error in year t Normal distribution with SD = σ w Lognormal error Lognormal correction so that mean of exp term is 1 14 Monte Carlo Methods.xlsx, sheet “exponential”

Demo: 14 Monte Carlo methods.xlsx sheet Exponential

Exponential model lessons Multiplicative error – Most trajectories decline – A few increase (but they increase a lot) – On average, final abundance is the same No density dependence

Often easy to obtain random numbers from a normal distribution with mean 0 and SD 1 This allows conversion to random numbers from a normal distribution with any mean µ and SD σ Trick 1: Random normal numbers

Trick 2: Use same set of w t values when comparing scenarios Generate N sets of random numbers Save them (values only) Compare policies using set 1, then compare using set 2, etc. Same pattern of variability for each scenario In Excel, OFFSET command is useful here In R, use an array and access columns with X[,i] 14 Monte Carlo Methods.xlsx, sheet “randoms”

Demo: 14 Monte Carlo methods.xlsx sheet Randoms

Multiplicative vs. additive error Additive error: – CV = SD/mean = declines with increasing X, e.g. 10% error at low X, 1% error at high X Multiplicative error: – CV = SD/mean = constant with increasing X, e.g. 10% error at all values of X Additive error: amount not dependent on X Multiplicative error: amount proportional to X

Stochastic logistic model + depensation Multiplicative error, serially uncorrelated Variability in process error Multiplicative process error RemovalsDepensationLogistic growth Population at which recruitment is halved through depensation 14 Monte Carlo Methods.xlsx, sheet Logistic dep

Demo: 14 Monte Carlo methods.xlsx sheet Logistic dep

Autocorrelation Environmental conditions are correlated from one year to the next (and from one day to the next too). Autocorrelation (lag 1) = correlation between a column of data minus the first point, and the same column minus the last point Excel Data in A1:A20 =correl(A2:A20, A1:A19) R Data in vector X =cor(X[-1],X[-length(X)]) 14 Monte Carlo Methods.xlsx, sheet Autocorrelation

Weather autocorrelation

Implementing serial autocorrelation “rho”, the autocorrelation parameter Depends on previous year’s process error 14 Monte Carlo Methods.xlsx, sheet Autocorrelation Morris WF & Doak DF (2002) Quantitative conservation biology: theory and practice of population viability analysis. p. 135

Demo: 14 Monte Carlo methods.xlsx sheet Autocorrelation

Adding autocorrelation to logistic 16 Monte Carlo Methods.xlsx, sheet Logistic dep autocorrelation Variability in process error Multiplicative process error RemovalsDepensationLogistic growth Population at which recruitment is halved through depensation Autocorrelation rho parameter

Demo: 14 Monte Carlo methods.xlsx sheet Logistic dep autocorrelation

What does serial autocorrelation do? Reduces year-to-year empirical variance If variance is re-inflated – Makes populations go lower and higher – Extinction is more common – Potential yields are higher but more variable

Jacquet J, Pauly D, Ainley D, Holt S, Dayton P & Jackson J (2010) Seafood stewardship in crisis. Nature 467:28-29

Warning sign: cherry-picking time periods

Eastern Bering Sea pollock (spawning biomass) Assessment of the walleye pollock stock in the Eastern Bering Sea: “64% decline ” Recovery after 2008 Unfished stock Regime shift in environment Full time period Currently 4.7 times higher than it was when fishing started

Practical issues in PVA If all you see is a declining population over time, it could be due to – Declining K (habitat loss) – Negative rates of increase – Bad luck environmental conditions (autocorrelation) – Harvesting These will have very different extinction risks

Don’t rely on numbers over time as the only “information” Declining abundance cannot tell you much about extinction risk What do you know about habitat? What do you know about environmental changes? What do you know about harvest?

Autocorrelation and stationary mean a lognormal “stationary walk” 14 Rand autocorrel logn stationary.r, Slightly modified code from Michael Wilberg original. Wilberg MJ & Miller TJ (2007) Comment on “Impacts of biodiversity loss on ocean ecosystem services”. Science 316: 1285b Random process error CV in N Amount of autocorrelation Stationary mean in N

Autocorrelation and stationary mean properties for large numbers of years

Demo: 14 Rand autocorrel logn stationary.r

“Random” number seeds Computers cannot generate truly random numbers Use a variety of ingenious methods to generate pseudo-random numbers that appear random Each requires a starting point (a random number “seed”) Successive numbers generated from the previous number in the sequence Sequence does not repeat for a very long time In R: set.seed(some.positive.number) picks a sequence Chapter 7 Press et al. (2007) Numerical Recipes. Cambridge University Press. 1235pp.

Year Abundance Seed=5, CV=0.1,0.25,0.4 Year Seed=6, CV=0.1,0.25,0.4 Different seeds, different CVs 14 Rand autocorrel logn stationary.r

Application: assess validity of catch status plots Froese R & Kesner-Reyes K (2002) Impact of fishing on the abundance of marine species. ICES paper CM 2002/L:12: 15pp Pauly D (2008) Global fisheries: a brief review. Journal of Biological Research-Thessaloniki 9: 3-9 Froese R, Zeller D, Kleisner K & Pauly D (2012) What catch data can tell us about the status of global fisheries. Mar. Biol. 159: Pauly D (2013) Does catch reflect abundance? Yes, it is a crucial signal. Nature 494:

Autocorrelated time series, fluctuating around a mean Branch et al. (2011) Conservation Biology 25:

Actual status from biomass Branch, TA unpublished analysis Trends in status Year Percentage of fisheries Collapsed Fully exploited Overexploited Developing Overexploited Fully exploited Catch status method applied to catches Recovering Catch and biomass from fisheries with known status