Population Viability Analysis I: Counts

I.  Population Viability Analysis

A.  Endangered Species and Population Viability
        1.  Desire to bring science to recovery process
        2.  Many attempts to make rigid numbers about targets for recovery, including these early recommendations:
            -- 50/500 rule
            -- 99% chance of persistence for at least 1000 yrs


A.  Goal: To use ecological models to forecast trends in future
        population size

B.  Structure: There is not one standard approach to conduct a PVA
        - let the data dictate the PVA approach
        - keep the model as simple as possible
        - models can easily get very complex
   
C.  Data needs:  depends on model
        1.  For count based data, >10 yrs
        2.  For stochastic demographic model, 4 yrs
        3.  For deterministic demographic model, 2 yrs
        4.  For spatial (metapopulation) PVA, 20 subpopulations


D.  Uses of PVA

        1.  Compare risks in two or more populations
        2.  Analyze or synthesize monitoring data
        3.  Identify key life stages for management
        4.  Determine number of populations to protect

        Problem: Contrast between the dire need for PVA and the scarcity of data in conservation – modeling realistic population changes could involve many factors

E.  PVA emphasizes variability, not just mean!

    1.  Stochasticity, in general, increases extinction risk
       -- geometric v. arithmetic mean







    2.  Variability causes projected population sizes to rapidly diverge
       -- California Clapper Rail example






    3.  Catastrophes and bonanzas in particular are difficult to quantify



    4.  Contrasting effects of negative density dependence

    5.  Positive density-dependence can increase population risk

    6.  Other factors (ie, genetic, spatial) can greatly increase uncertainty in PVA


F. What measure to quantify extinction risk?

    1.  Do not quantify risk of utter extinction - set some higher threshold

    2.  Measures of population size generally not sufficient

    3.  Measures of population growth can be useful

    4.  What measure to use to quantify extinction risk?
         -- the ultimate probability of extinction at any future time - asymptote of Cumulative Distribution Function (CDF)
        -- mean (calculated)
        -- median (50th percentile on CDF)
        -- mode (peak on Probability Distribution Function (PDF))
        -- the probability of extinction at some future time (CDF)






    5.  Probabilities of extinction over certain time horizons are the most useful and robust extinction risk estimates

To get these, need an estimate of probabilities of extinction


II.  PVA with count data

A.  Overview
1.  Start with simplest assumptions
        - Mean and variance of population growth rate remain constant
    - Environmental conditions uncorrelated
    - Environmental variation is small
    - No observer error

2.  Characteristics of density-independence
        - Probability that population will be some size at any future time described by lognormal distribution (thus, log(N) is normally distributed)
        - Over time, mean population size will change in relation to lambda, but variance will grow








        - Characterize population growth by log of the geometric growth rate (mu)
        - Variance parameter for log growth rates - sigma squared







       Note:      Do not quantify risk of utter extinction - set some higher threshold


        - Assuming a diffusion process, can generate PDF (and thus CDF) with an inverse Gaussian distribution that requires parameters μ, σ2, and the difference between population size and the extinction threshold



B. Practical methods for making calculations of mu and sigma square

    1.  Because variance changes over time and time interval may vary, need to normalize with regard to time interval
        - divide log(Nt+1/Nt) by square root of the time interval

    2.  Regress square root time interval against normalized population change (forced through origin)

    3.  Slope is mu and error mean square is the estimate of sigma square



    4.  Confidence intervals for mu calculated with SE and two-tailed Students t

    5.  Confidence intervals for sigma square calculated using chi-square distribution
       
    6.  Create PDF, CDF, and CI using MatLab


C.    Keep in mind assumptions (no temporal autocorrelation,  extreme events, or changes in mu or sigma square over time)


D.  Examples of how PVA can be used









E.  Tips on using PVA:

1. Avoid conducting formal PVA when data are too sparse

2. Present confidence intervals

3. Compare relative rather than absolute changes

4. Do not project population viability too far into the future

5. Start with simple models: let the data choose the model
   
6. Consider PVA to be a work in progress!