Robustness and uncertainty in estimates of butterfly abundance from transect counts

Abstract

Many butterfly populations are monitored by counting the number of butterflies observed while walking transects during the butterfly’s flight season. Methods for estimating population abundance from these transect counts are appealing because they allow rare populations to be monitored without capture–recapture studies that could harm fragile individuals. An increasingly popular method for estimating abundance from transect counts relies on strong assumptions about the counting process and the processes that govern butterfly population dynamics. Here, we study the statistical performance of this method when underlying model assumptions are violated. We find that estimates of population size are robust to departures from underlying model assumptions, but that the uncertainty in these estimates (i.e., confidence intervals) is substantially underestimated. Alternative bootstrap and Bayesian methods provide better measures of the uncertainty in estimated population size, but are conditional upon knowledge of butterfly detectability. Because of these requirements, a mixed approach that combines data from small capture–recapture studies with transect counts strikes the best balance between accurate monitoring and minimal injury to individuals. Our study is motivated by monitoring studies for St. Francis satyr (Neonympha mitchelli francisci), a rare and relatively immobile butterfly occurring only in the sandhills region of south-central North Carolina, USA.

Appendix: technical description of Bayesian model

Let d = 1,..., D be a daily time index, where days 1 and D are the days of the first and last transect counts, respectively. Let X(d) be the number of butterflies that emerge on day d, let Y(d) be the number of butterflies alive on day d, and let Z(d) (the data) be the number of butterflies counted on day d if a transect count was conducted. To make the notation more compact, we use the vector notation X = [X(1),..., X(D)], Y = [Y(1),..., Y(D)], and Z = [Z(t ₁),..., Z(t _n )], where t ₁,..., t _n are the days of the n transect counts. In keeping with standard Bayesian practice, we parameterize the variance of the emergence distribution by its inverse τ =  σ ⁻². Also, let ξ be a shape parameter that indicates whether the emergence distribution is normal (ξ = 0) or logistic (ξ = 1). Let ${\varvec{\theta} = (N, \mu, \tau, \varphi, \xi)}$ be a vector that contains the unknown parameters.

Let π(·) denote a probability distribution. Our goal is to estimate ${\pi({\varvec{\theta}} | {\bf Z})},$ the posterior distribution of the unknown parameters given the data. This marginal posterior distribution can be found by integrating X and Y out of the full posterior ${\pi({\varvec{\theta}}, {\bf X}, {\bf Y} | {\bf Z})}.$ The full posterior is proportional to the product of likelihood of the data and the prior distribution

$$ \pi {\left({{\varvec{\theta}},{\mathbf{X}},{\mathbf{Y}}|{\mathbf{Z}}} \right)} \propto \pi {\left({{\mathbf{Z}}|{\varvec{\theta}},{\mathbf{X}},{\mathbf{Y}}} \right)}\pi {\left({{\varvec{\theta}},{\mathbf{X}},{\mathbf{Y}}} \right)}.$$

(2)

The terms on the right-hand side of Eq. 2 can be simplified by observing that the distribution of the data Z depends only on Y, and by factoring the prior:

$$ \pi {\left({{\varvec{\theta}},{\mathbf{X}},{\mathbf{Y}}|{\mathbf{Z}}} \right)} \propto \pi {\left({{\mathbf{Z}}|{\mathbf{Y}}} \right)}\pi {\left({{\mathbf{X}},{\mathbf{Y}}|{\varvec{\theta}}} \right)}\pi {\left({\varvec{\theta}} \right)}.$$

(3)

The first term on the right of Eq. 3 is the probability distribution of the data given the actual (unobserved) dynamics, and can be written as

$$ \pi {\left({{\mathbf{Z}}|{\mathbf{Y}}} \right)} = {\prod\limits_{i = 1}^n {\pi {\left[{Z(t_{i})|Y(t_{i})} \right]}}} $$

(4)

where π[Z(t _i )| Y(t _i )] is the appropriate probability model for the transect counts. For SFS, we use a beta-binomial distribution with a fixed overdispersion parameter η.

The second term on the right of Eq. 3 is the probability distribution of the actual butterfly dynamics given unobserved parameters, and comes directly from a modified version of Zonneveld’s model. First, the vector of emergence times X has a multinomial distribution with parameters N and p ₁,..., p _D , where p _d is the probability of emergence on day d. The emergence probabilities p ₁,..., p _D are determined by discretizing the underlying continuous distribution for emergence times,

$$ p_{d} = \frac{{{\int_{d - 1}^d {f{\left({t; \mu, \tau} \right)}\hbox{d}t}}}}{{{\int_0^D {f{\left({t; \mu, \tau} \right)}\hbox{d}t}}}}, \quad d = 1,...,D.$$

(5)

Second, the number of butterflies flying on day d, Y(d), is the sum of the butterflies that emerged on day d, X(d), plus those butterflies that were flying on day d−1 and survived. In equations,

$$ \begin{aligned} Y(1) &= X(1), \\ Y(d) &= X(d) + {\hbox{binomial }}[X(d - 1),\varphi] \quad d = 2,...,D. \\ \end{aligned}$$

(6)

For each SFS time series, we used the following proper noninformative priors for μ, τ, and N

$$ \begin{aligned} \mu &\sim {\hbox{Unif }}(1,D), \\ \tau &\sim {\hbox{Exp }}(1), \\ N &\sim {\hbox{Unif }}(1,N_{0}) \\ \end{aligned} $$

where N ₀ is a relatively large value (5,000). ξ had equal prior probability of indicating either a logistic or normal distribution. The prior for the daily survival probability ϕ was based on capture–recapture studies for the same subpopulations in 2002 and 2003

$$ \pi {\left(\varphi \right)} \propto N(\mu = 0.669,\sigma = 0.109) \times I_{{(0,1)}}. $$

We approximated the posterior distribution numerically using the Markov chain Monte Carlo (MCMC) algorithm described below. We used separate Metropolis–Hastings updating steps for the parameters μ, τ, and ξ, as well as the parameter blocks (ϕ, N, X, Y), (N, X, Y), (X, Y), and Y. N, X, and Y were updated as blocks because of the conditional dependence structure of the model, while ϕ was updated in a block because of its strong posterior correlation with N. Updates of the single parameters used standard proposal distributions (folded normals for μ and τ; because ξ is a binary variable, its proposal was whatever the current value was not), but block updates used non-standard proposals, and so we describe these here. In what follows, we use primes (′) to denote proposed parameter values. In all cases, substantial cancellation follows from the model factorization (Eq. 3), which eases the calculation of the acceptance ratio considerably.

To update the block (N, X, Y), we first flip a fair coin to decide whether to add (N′ = N + 1) or subtract (N′ = N − 1) an individual, and then roll a fair D-sided die to determine the day on which an individual is added or subtracted. We add or subtract the individual from both X and Y. To keep the proposal density symmetric, moves to regions of zero prior probability [e.g., X′(d) = −1] are allowed, but are automatically rejected because the acceptance ratio is zero. To update the block (ϕ, N, X,Y), we propose a value ϕ′ separately, but evaluate the entire proposed move to ϕ′, N′, X′, and Y′ at once.

We update the block (X, Y) by moving a randomly selected individual’s emergence time forward or backward by a day. To do so, we first choose the day from which an individual is moved, d ^*, with probability P(d ^* = d) = X(d)/N. If 2 ≤ d ^* ≤ N − 1, we flip a fair coin to determine whether to move that individual’s emergence time forward or backward by a day. If d ^* = 1 or d ^* = N, we move the individual’s emergence time ahead or back by a day, respectively. For example, if we choose to move an individual’s emergence time one day later, then X′(d ^*) = X(d ^*) − 1, X′(d ^* + 1) = X(d ^* + 1) + 1, Y′(d ^*) = Y(d ^*) − 1, and Y′(d ^* + 1) = Y(d ^* + 1) + 1. Not all moves of this type result in a legal X and Y; such moves are considered but are automatically rejected for lack of prior support. The proposal distribution is not symmetric, and must be factored into the acceptance ratio.

To update Y, d ^* is selected at random from 2,..., N (each with equal probability), and a fair coin is flipped to determine whether to increase or decrease Y(d ^*) by 1. Again, not all possible moves result in legal combinations of Y and X, and illegal moves are considered but are automatically rejected for lack of prior support. The proposal distribution is symmetric.

Ten separate chains were run from different starting values, and convergence was monitored by Gelman and Rubin’s (1992) potential scale reduction factor. For the fits to SFS data, Monte Carlo standard errors of posterior medians were less than 2.5% of posterior standard errors in all cases.