Monitoring abundance and phenology in (multivoltine) butterfly species: a novel mixture model

Abstract

Summary1. Data from 'citizen science' surveys are increasingly valuable in identifying declines in widespread species, but require special attention in the case of invertebrates, with considerable variation in number, seasonal flight patterns and, potentially, voltinism. There is a need for reliable and more informative methods of inference in such cases. 2. We focus on data consisting of sample counts of individuals that are not uniquely identifiable, collected at one or more sites. Arrival or emergence and departure or death of individuals take place during the study. We introduce a new modelling approach, which borrows ideas from the 'stopover' capture-recapture literature, that permits the estimation of parameters of interest, such as mean arrival times and relative abundance, or in some cases, absolute abundance, and the comparison of these between sites. 3. The model is evaluated using an extensive simulation study which demonstrates that the estimates for the parameters of interest obtained by the model are reliable, even when the data sets are sparse, as is often the case in reality. 4. When applied to data for the common blue butterfly Polyommatus icarus at a large number of sites, the results suggest that mean emergence times, as well as the relative sizes of the broods, are linked to site northing, and confirm field experience that the species is bivoltine in the south of the UK but practically univoltine in the north. 5. Synthesis and applications. Our proposed 'stopover' model is parameterized with biologically informative constituents: times of emergence, survival rate and relative brood sizes. Estimates of absolute or relative abundance that can be obtained alongside these underlying variables are robust to the presence of missing observations and can be compared in a statistically rigorous framework. These estimates are direct indices of abundance, rather than 'sightings', implicitly adjusted for the possible presence of repeat sightings during a season. At the same time, they provide indices of change in demographic and phenological parameters that may be of use in identifying the factors underlying population change. The model is widely applicable and this will increase the utility of already valuable and influential long-standing surveys in monitoring the effects of environmental change on phenology or abundance.

Dynamic Models for Longitudinal Butterfly Data

“…Typically, the resulting counts, graphed over time, reflect successive waves of birds arriving, staying and then leaving. Matechou et al (2014) observe that this is the same pattern seen when adult butterflies are counted within a season, with for example bivoltine species analogous to two waves of birds observed at a stopover site.…”

“…This development is an extension of that in the original specification of Matechou et al (2014). For brood b, the parameters…”

“…This is a simple stopover model, proposed for butterfly data by Matechou et al (2014); see also Dennis et al (2014). Stopover models are used in describing data on migrating birds, which rest and feed during their journey at particular stopover sites where observations take place.…”

“…In Matechou et al (2014), the above model is extended to account for multivoltine data, and the expression of Eq. (6) then becomes a mixture of terms, each of which is an area under an appropriate probability density function.…”

“…Soulsby and Thomas (2012) developed a mathematical model for this variation, but only allowed for discrete, non-overlapping generations. Other models have been proposed to describe the within-year variation, both non-parametrically using generalised additive models (GAMs, Rothery and Roy 2001;Dennis et al 2013) and via stochastic mixture models (Matechou et al 2014;Dennis et al 2014). Counts adjusted for seasonal fluctuations can then be used to produce longer-term trends, but existing methods do not impose any relationship between counts from one year to the next, which is the topic of this paper.…”