# Population viability analyses in COSEWIC status reports

Approved by COSEWIC in April 2010

## Introduction

The assessment process for determining a wildlife species status under COSEWIC allows the use of quantitative analyses. This is referred to IUCN Red List criterion E (IUCN Standards and Petitions Working Group 2008). This criterion can be used where a robust estimation of extinction risk can be calculated, for example, a forest dependent endemic species whose habitat is expected to be clear-cut within the next 20 years (Mace et al. 2008). However, the main quantitative analysis available for species status assessment is Population Viability Analysis (PVA). Broadly defined, PVA is the use of quantitative methods to predict the likely future status of a population or collection of populations of conservation concerns (Morris and Doak 2002). Within that definition, "future status" typically refers to whether the population (or total number of individuals across all populations) will be above some minimum size at different future times. Assigning concrete numbers to measures of future status is the aim of PVA.

As of 2009, criterion E has been used in a very small proportion of all COSEWIC assessments. However, because monitoring programs are likely to be put in place for different species at risk in the near future - through the implementation of legally required recovery strategies - PVAs are likely to become more commonly used, especially for reassessment. A study of the use of PVA in recovery plans for endangered species in the US concludes: "Even if insufficient data now exist to use PVA for a particular threatened or endangered species, routine monitoring could make the construction and parameterization of population models feasible in the future. In many cases, population viability analysts may be able to improve the appropriateness of monitoring schemes for modeling purposes without significantly increasing the cost of monitoring" (Morris et al. 2002).

## The Need for Standards

For the results of PVA to inform the (re-) assessment of species, they need to be well communicated to COSEWIC members. The Guidelines (Standards and Petitions Working Group 2006) for using the IUCN Red list categories and criteria (versions 6.2, Dec. 2006) contains a clear and well-documented set of guidelines to apply criterion E. Under the section on "Documentation requirements", the Guidelines state:

- Any [...] assessment that relies on Criterion E should include:
- a document that describes the quantitative methods used, and
- all the data files that were used in the analysis.

The document and accompanying information should include enough detail to allow an evaluator to reconstruct the methods used and the results obtained.

- The documentation should include:
- a list of assumptions of the analysis
- explanations and justifications for these assumptions.
- documentation of the uncertainties in the data

All data used in estimation should be either referenced to a publication that is available in the public domain, or else be included with the listing documentation.

- Methods used in estimating model parameters and in incorporating uncertainties should be described in detail. Time units used for different model parameters and components should be consistent; the periods over which parameters are estimated should be specified.

## Overview of PVA models

One of the challenge for reporting results of PVAs so that they can inform species status assessment is that the types of quantitative methods that can be called PVA are varied, both in terms of the complexity of the underlying models and the quantity of data needed to parameterize them (Burgman et al. 1993, Morris et al. 1999, Beissinger and McCullough 2002). Recent texts describe four types of PVA (Morris and Doak 2002; Lande et al. 2003): unstructured population models, structured population models, metapopulation models, and spatially-explicit population models.

### 1) Unstructured population models

The simplest class of PVA uses time series data on population size to parameterize a basic stochastic exponential growth model with year-to-year variability in the growth rate, and no density-dependence. This type of PVA requires data on both current population size and trends in population size over time, but does not require age- or stage-structured data or spatially explicit information. Note that any age or stage-structured model will converge to the stochastic exponential model, which is why that model is so fundamental.

One property of the basic density-independent growth model is that it can be approximated by a diffusion equation (Dennis et al. 1991). This approximation opens a toolbox of parameterization methods for linear models with normal error. It also provides analytical estimates of passage probabilities, i.e., the probability of crossing a particular threshold within a given time frame, for example quasi-extinction estimates to some critical population size >1 (Holmes 2004).

The diffusion model approximates many types of stochastic age structured population processes, as seen both from simulated and real data (Lande and Orzack 1988, Dennis et al. 1991, Holmes and Fagan 2002; Holmes 2004). There are, however, well-known cases where the diffusion approximation performs poorly, such as when year-to-year variability is high and populations are small (Ludwig 1999), when demographic stochasticity introduces strong nonlinearities (Wilcox and Possingham 2002), and when density dependence is extreme (Sabo et al. 2004).

There are at least four different maximum likelihood methods for estimating the parameters of the diffusion approximation (DA) model: the regression methods (Dennis et al. 1991), the runsum-slope methods (Holmes and Fagan 2002, Holmes 2001), the Kalman filters (Lindley 2003) and the Restricted Maximum Likelihood method (Staples et al. 2004).

The regression method assumes that all the observed year-to-year variability in the population growth rate is due to environmental variation. The other methods assume that some of the variability comes from other sources, such as observation errors, error due to age-structure cycling, etc. By estimating the parameters of a state-space model in which the true size of the population remains unknown, the other methods aim at estimating the variance in population growth due solely to environmental variation, which is critical in PVA (Holmes 2004).

One of the strengths of DA methods is that the statistical distributions of the estimated parameters are known. As a result, the uncertainty in the estimated risks can be calculated. This is often not the case for other PVA approaches, such as matrix models or individual-based simulations, where uncertainty in the estimated model parameters is often poorly known (Holmes 2004).

Variants of the basic stochastic growth model can incorporate density-dependence, autocorrelation in the environment, and large variation in population growth rates (see Morris and Doak 2002). With these more complex unstructured models, viability metrics are obtained by simulating a large number (1000 to 10,000) of population trajectories.

Unstructured stochastic growth models can use the same data used for assessing species under criteria A (declining populations). However, they have been designed to work with count data, so when the population time series are based on an abundance index (ex. catch per unit effort), then either a biologically significant population threshold appropriate for that index must be specified, or count data must be corrected for a unique catch effort.

Examples of PVAs using the DA methods are; for bird species: Dennis et al. (1991), Stacey and Taper (1992), Middleton and Nisbet (1997), Engen and Saether (2000); for mammal species: Dennis et al. (1991), Gerber et al. (1999), Oli et al. (2001); for reptile species: Snover and Heppel (2009); for insect species: Schultz and Hammond (2003).

### 2) Structured population models

Structured population models typically use life tables or projection matrix models, which track changes in the numbers of individuals in different stages (e.g., age or size categories) in a population (Morris et al. 2002). Stochastic population projections with matrices use a given description of the stochastic environment, such as independent and identically distributed sequences. These sequences can be used to select whole matrices, matrix elements, or components of matrix elements from either finite sets of observed values or from parametric distributions (Caswell 2001).

Matrix models can incorporate any of demographic stochasticity, catastrophes or bonanzas, density-dependence, spatial structure, and can be used to simulate interventions such as harvesting or population augmentation (reintroduction, head-starting, etc.). They can also allow more detailed analysis of critical life stages or demographic processes that are potential targets for management (Caswell 2001). They require data on stage-specific fecundity and mortality rates, and the current stage structure of the population to be used to predict viability.

Examples of PVAs using projection matrices are; for bird species: Lande (1988), Beissinger (1995); for reptile species: Crouse et al. (1987); for plant species : Menges (1990), Nantel et al. (1996), Menges and Dolan (1998), Gross et al. (1998), Bell et al. (2003), Garcia (2003); for fish species : Kareiva et al. (2000), Legault (2005).

Stochastic population projections can also be done using individual-based simulations. These can allow the inclusion of genetic information (such as pedigree) and be used also to model the effect of different reproduction systems. Examples of PVAs using individual-based simulations, for plant species, are Schwartz et al. (2000), Kirchner et al. (2006). The software VORTEX is a popular PVA modelling tool that uses individual-based simulations (Miller and Lacy 2005).

### 3) Metapopulation models

"Metapopulation" PVAs follows the fates of multiple subpopulations and attempt to determine whether the rate of establishment of new subpopulations through colonization is sufficiently high to counter the extinction of subpopulations, thus allowing the entire metapopulation to persist. Such PVAs require information on the number of subpopulations, trends in the number of subpopulations or the rate of subpopulation extinction, and the colonization rate, typically as reflected in patterns of dispersal. An example of a metapopulation PVA for a butterfly species is Hanski et al. (1996).

### 4) Spatially explicit population models

"Spatially explicit" PVAs is the most complex and data intensive type. It typically involves simulating the behaviour of individual organisms on detailed landscapes upon which the sizes and locations of suitable habitat patches are mapped. In addition to requiring information about birth and death rates of individuals within each patch, and their movement patterns, this type of PVA also requires data on the degree of isolation and fragmentation of suitable habitat patches. Examples of spatial PVAs that use RAMAS-GIS as the modeling software, for various taxa, can be found in Akçakaya et al. (2004). Note that the software VORTEX can also be used for spatially explicit population viability analysis (Miller and Lacy 2005).

### Sensitivity Analyses

Sensitivity analyses are processes to measure how sensitive or changeable is an indicator of population fate, such as growth rate or extinction risk, to particular changes in the model's input parameters, such as individual vital rates. Sensitivity analyses are an important component of PVAs because they help, among other things, to identify input parameters that are more influential (Burgman et al. 1993, McCarthy et al. 1995, Caswell 2001). This is critical as uncertainties associated with the more influential parameters can contribute to a large portion of the uncertainty of the model's predictions.

The documentation requirements related to Criterion E in the IUCN guidelines do not mention sensitivity analyses. However, the results of a PVA will be more transparent, likely easier to interpret, and therefore more useful for status assessment when presented to COSEWIC with a sensitivity analysis that covers, at least, the most uncertain input parameters.

A large array of mathematical methods, either analytical or statistical, can be used for measuring parameter influence on the predictions of deterministic and stochastic population models. These include elasticity analysis, conventional and relative sensitivity analyses, life table response experiments, life table simulation analysis, retrospective simulation to evaluate responses to alternative future scenarios, and the use of standardized regression coefficients or other statistics to measure relative influence (Burgman et al. 1993, de Kroon et al. 2000, Cross and Beissinger 2001, Rustigian et al. 2003, Fieberg and Jenkins 2005, Curtis and Naujokaitis-Lewis 2008). Among those methods, the logistic regression approach is well established as an acceptable methodology for sensitivity analysis of PVA models (Curtis and Naujokaitis-Lewis 2008).

### Check-list for reporting PVA and population projections in status reports.

In view of the diversity of models used in PVA, report writers who want to present PVA in status report should consider the following check-list of input and output. Note that the input parameters and details of the model structure can be summarized in a set of tables that could be appended to report.

**Input:**

- Nature of the data collected and how they were collected and analysed.
- Type of model used: non-structured or structured by age, stage, sex, subpopulations (patches); individual-based or matrix-based; model of density-dependence and how it has been parameterized.
- If non-structured: time span and frequency of counts (weighted or partial); method used for estimating parameters of the diffusion approximation or other models (ex. Kalman filter).
- If structured: Random matrices (RM) or vital rate simulations (VRS). If VRS, method(s) for estimating (and correcting) vital rates' means and variances. Table of vital rates estimates (mean and variance); probability distribution(s) used to generate random vital rates in stochastic projections. Method used to simulate within-yr and between-yrs correlation (if any). Definition of projection matrix (for vital rate simulations). For both RM and VRS: method (Monte Carlo) used for simulating demographic stochasticity (if simulated).
- Starting number of individuals (per class for structured models)
- Extinction threshold (with weights per class for structured models)
- Time span of projection into the future
^{1} - Number of runs
- Method used to simulate perturbations (population augmentation or harvesting), if any.
- Software package(s) used to build and run models.

**Output:**

- Stochastic population growth rate (with confidence intervals)
- Graph of the quasi-Extinction time cumulative distribution function (CDF)
^{2}, with confidence intervals (if possible) - Sensitivity analysis

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(1) The time span must cover those used in criterion E: 20 yrs or 5 generations (whichever is longer, up to a maximum of 100 yrs) for Endangered; 100 yrs for Threatened.

(2) Quasi-extinction CDF gives the probability that the population will decline below a given threshold (> 1 individual) at or before a given future time; this is considered the single most useful way to present extinction risk information (Morris and Doak 2002).