Population viability analysis
and sensitivity analysis

What is PVA?

Population Viability Analysis

“The use of quantitative methods to predict the likely future status of a population or collection of populations of conservation concern.”

Morris and Doak (2002)

History

First used by Shaffer (1983) to estimate the minimum viable population size in Yellowstone National Park.

Now routine part of species assessments and recovery plans.

Grizzly recovery in Yellowstone

Uses of PVA

Assessing the extinction risk of a single population.
Comparing relative risks of two or more populations.
Analyzing and synthesizing monitoring data.
Identifying key life stages or demographic processes as management targets (sensitivity analysis).
Determining how large a reserve needs to be to achieve a desired level of protection from extinction.
Determining how many individuals to release to establish a new population.
Setting limits on the harvest or “take” from a population that are compatible with its continued existence.
Determining how many (and which) populations are needed to achieve a desired overall likelihood of species persistence.

Steps of PVA

Develop objectives
Develop a set of competing models
Design a study to collect necessary data
Fit models to data and select the best model(s)
Use model(s) to identify best management options
Implement management option, monitor the consequences, and refine models
Return to step 4

Very similar to adaptive management

Types of PVA

Count-based PVA
- This is the cheapest, but least-informative method

Demographic PVA
- Requires estimates of vital rates
- Useful for identifying key demographic parameters

Metapopulation viability analysis
- Useful in reserve design

Spatially-explicit, individual-based PVA
- Most realistic, but hardest to parameterize

All methods require clear definition of time horizon.

Products of PVA

Central Georgia black bears

Smallest and most isolated bear population in GA.

Little information about the effects of harvest on population viability.

Five years of genetic capture-recapture data.

Hooker et al. (2020, JWM)

Central Georgia black bears

Sensitivity Analysis

Sensitivity analysis

A sensitivity analysis seeks to understand the degree to which \(\lambda\) is sensitive to changes in a vital rate, while holding all other rates constant.

Usually applied to age- or stage-based population models.

The sensitivity of \(\lambda\) to a change in a population parameter \(\theta\) (e.g., survival or fecundity), is (loosely): \[ \text{sensitivity} = {\small \lim\limits_{\Delta \theta \to 0}}\; \frac{\Delta \lambda}{\Delta \theta} \]

Sensitivities allow us to make statements such as:

“Increasing subadult fecundity by a small amount increases \(\lambda\) by 0.01, whereas increasing adult fecundity by the same amount increases \(\lambda\) by 0.02. Therefore, population growth is more sensitive to changes in adult fecundity.”

Sensitivity example

Parameter	\(\Delta \theta\)	\(\Delta \lambda\)	Sensitivity
Fecundity of first age class (\(f_1\))	0.05	0.010	0.20
Fecundity of second age class (\(f_2\))	0.05	0.003	0.06
Survival of first age class (\(s_1\))	0.05	0.040	0.80
Survival of second age class (\(s_2\))	0.05	0.030	0.60

Elasticity

A problem with sensitivities is that it is hard to compare values for parameters on different scales, such as survival and fecundity. For this reason, it can be better to report “elasticities” instead.

Elasticity: the proportional change in \(\lambda\) caused by proportional changes in vital rates.

Easier interpretation because:

Standardized units
Values sum to 1

Examples

“A 1% increase in \(f_1\) increases \(\lambda\) by 0.01%.”
“A 1% increase in \(s_1\) increases \(\lambda\) by 0.05%.”

R code for sensitivity analysis

Suppose we have the following stage-structured projection matrix:

A <- matrix(c(
    0.2, 0.8, 1.0, 0.9,
    0.4, 0.0, 0.0, 0.0,
    0.0, 0.6, 0.0, 0.0,
    0.0, 0.0, 0.8, 0.5), nrow=4, byrow=TRUE)

We can use eigenanalysis to compute \(\lambda\), stable age distribution, reproductive value, sensitivities, and elasticities.

eA <- eigen(A)
lam <- Re(eA$values[1])
lam                              ## Asymtotic growth rate

[1] 1.034908

w <- Re(eA$vectors[,1])
w <- w/sum(w)                    ## Stable age distribution

v <- Re(eigen(t(A))$vectors[,1]) ## Reproductive value

R code for sensitivity analysis

The sensitivities are given by \(z_{ij}=\frac{v_i w_j}{{\bf v}{\bf w}}\)

z <- outer(v,w)/c(v%*%w)
z ## Sensitivites

          [,1]      [,2]       [,3]      [,4]
[1,] 0.3473925 0.1342698 0.07784447 0.1164229
[2,] 0.7251023 0.2802575 0.16248251 0.2430061
[3,] 0.7875009 0.3043751 0.17646493 0.2639179
[4,] 0.5844985 0.2259131 0.13097572 0.1958851

The elasticities are given by \(e_{ij}=\frac{a_{ij}z_{ij}}{\lambda}\)

e <- A*z/lam
e  ## Elasticities

           [,1]      [,2]      [,3]       [,4]
[1,] 0.06713492 0.1037926 0.0752187 0.10124623
[2,] 0.28025755 0.0000000 0.0000000 0.00000000
[3,] 0.00000000 0.1764649 0.0000000 0.00000000
[4,] 0.00000000 0.0000000 0.1012462 0.09463883

Summary

PVA covers almost all methods for predicting how populations will respond to future scenarios.

The most common use is to assess how different conservation strategies will affect extinction risk.

Sensitivity analysis is a component of PVA that can guide conservation planning.

Sensitivity measures the change in \(\lambda\) given an absolute change in a parameter.
Elasticity measures the proportional change in \(\lambda\) given a proportional change in a parameter.

Summary

Problems with PVA

Most PVAs are “essentially games played with guesses.” (Caughley, G. 1994. Conservation Biology).

Many PVA’s don’t include multiple models corresponding to different hypotheses, and population parameters are often not estimated from data.

Some software encourages this.

Solutions

Collect better data! We need more demographic research, conducted in an adaptive management framework.

When data are limited (and they always are), decrease the time horizon and acknowledge uncertainty in decision process.