458 Estimating Extinction Risk (Population Viability Analysis) Fish 458; Lecture 25

458 Identifying Species at Risk of Extinction – (Dennis-type methods) This method estimates the probability of (quasi) extinction within a given time frame based on: a time-series of counts; a rate of change and its coefficient of variation.

458 Some Examples Steller sea lions (pup counts) Marmot Island: Decreasing at 13.4% p.a. (sd 1.0%) Sugerloaf: Decreasing at 5.8% p.a. (sd 0.9%) White Sisters: Increasing at 20% p.a (sd 4.3%) Bowhead whales Increasing at 3.2% (SD 0.76).

458 The Basic Method-I The basic dynamics equation is: Note: the expected population size is given by: Therefore, if  <0, the population will eventually be rendered extinct.

458 Interlude: Modelling This model is (also) the solution of the diffusion equation with a constant diffusion rate and an “absorptive” boundary (at zero). It is not uncommon for the same mathematical formulation to arise from different assumptions. Are there any other cases we have seen when the same model arises from different assumptions?

458 The Basic Method II Probability of extinction. If is the logarithm of the ratio of the current population size to the population size at quasi-extinction, then the probability of extinction is:

458 The Basic Method III Analytic expressions exist for: The distribution of the time to quasi-extinction. The median / mean time to extinction. The variance of the time to extinction. However, we will tend to explore the method using numerical methods as these are more flexible.

458 Computing Extinction Risk Numerically 1. Set the current log-population size, x, to the logarithm of the initial population size. 2. Generate a random variate, , from N(  ;  2 ) and add it to x. 3. Check whether x < the logarithm of the population size that defines quasi- extinction. If so, “extinction” has occurred. 4. Repeat steps 2-3 many times (say 1000 years). 5. Repeat steps 1-4 many times and count the frequency with which extinction occurred.

458 Example: Grizzly Bears-I  =  = Initial population size = 47 Quasi-extinction level = 10 Quasi extinction level

458 Example: Grizzly Bears-II Median time to extinction: 163yrs Mean time to extinction: 218 yrs

458 Analytic vs Numerical Methods Analytic solutions are available for many quantities of interest / problems. However, numerical solutions are more flexible (if rather computationally intensive). For example (Grizzly Bears – 1000 simulations): AnalyticalNumerical Mean time to extinction Median time to extinction152163

458 Extensions Allow the residuals (  ) to be correlated (if suggested by the data). Use integer arithmetic (there were only 47 Grizzlies). Change population size by a randomly selected change from the actual set of changes (non-parametric approach). Allow for multiple populations. Take account of measurement error when computing  and .

458 Sensitivity to  : Bowheads Initial population size = 7800 Quasi-extinction level =10  =0.032 This population is increasing Ignoring measurement error shifts you to the right

458 Computing time to extinction non-parametrically 1. Determine the empirical set of %age changes in abundance. 2. Set the current log-population size, x, to the logarithm of the initial population size. 3. Select a change in abundance at random and add it to x. 4. Check whether x < the logarithm of the population size that defines quasi-extinction. If so, “extinction” has occurred. 5. Repeat steps 2-3 many times (say 1000 years). 6. Repeat steps 1-4 many times and count the frequency with which extinction occurred.

458 Example: Steller sea lions at Sugarloaf-I Rather a question of when rather than whether!

458 Example: Steller sea lions at Sugarloaf-II Mean time to extinction: Normal assumption: 95 years; Non-parametric: 70 years Does this point worry anyone?

458 Multiple Populations If a meta-population consists of n sub-populations (c.f. Steller sea lions). The probably of extinction of the whole meta-population depends of how changes in population size are correlated over space. If the probability of a single sub-population going extinct is p then if all populations are independent, the probability of the whole meta-population going extinct is p n. if the factors impacting the populations are perfectly correlated, the probability of the whole meta-population going extinct is p.

458 Key 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. 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).

458 Explicit Modeling of Extinction Risk (if it is that important…) An alternative to the Dennis-type approach is to develop a specific model(s) of the system under consideration and examine the consequences of future management actions (etc) on extinction risk. The models can include ‘ecological knowledge’. This is, however, highly data intensive (but the consequences of (say) an ESA listing are substantial).

458 Readings Dennis et al. (1991). Holmes (2004). Stobutzki et al. (200)