The N-mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication (Royle, 2004, Biometrics 60, 105–115). We explain and exploit the equivalence of N-mixture and multivariate Poisson and negative-binomial models, which provides powerful new approaches for fitting these models. We show that particularly when detection probability and the number of sampling occasions are small, infinite estimates of abundance can arise. We propose a sample covariance as a diagnostic for this event, and demonstrate its good performance in the Poisson case. Infinite estimates may be missed in practice, due to numerical optimization procedures terminating at arbitrarily large values. It is shown that the use of a bound, K, for an infinite summation in the N-mixture likelihood can result in underestimation of abundance, so that default values of K in computer packages should be avoided. Instead we propose a simple automatic way to choose K. The methods are illustrated by analysis of data on Hermann's tortoise Testudo hermanni.

Summary1. Volunteer-based 'citizen science' schemes now play a valuable role in deriving biodiversity indicators, both aiding the development of conservation policies and measuring the success of management. We provide a new method for analysing such data based on counts of invertebrate species characterised by highly variable numbers within a season combined with a substantial proportion of proposed survey visits not made. 2. Using the UK Butterfly Monitoring Scheme (UKBMS) for illustration, we propose a two-stage model that makes more efficient use of the data than previous analyses, whilst accounting for missing values. Firstly, generalised additive models were applied separately to data from each year to estimate the annual seasonal flight patterns. The estimated daily values were then normalised to estimate a seasonal pattern that is the same across sites but differs between years. A model was then fitted to the full set of annual counts, with seasonal values as an offset, to estimate annual changes in abundance accounting for the varying seasonality.3. The method was tested and compared against the current approach and a simple linear interpolation using simulated data, parameterised with values estimated from UKBMS data for three example species. The simulation study demonstrated accurate estimation of linear time trends and improved power for detecting trends compared with the current model. 4. Comparison of indices for species covered by the UKBMS under the various model approaches showed similar predicted trends over time, but confidence intervals were generally narrower for the two-stage model. 5. In addition to creating more robust trend estimates, the new method allows all volunteer records to contribute to the indices and thus incorporates data from more populations within the geographical range of a species. On average, the current model only enables data from 60% of 10 km 2 grid squares with monitored sites to be included, whereas the two-stage model uses all available data and hence provides full coverage at least of the monitored area. As many invertebrate species exhibit similar patterns of emergence or voltinism, our two-stage method could be applied to other taxa.

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.

Citizen scientists are increasingly engaged in gathering biodiversity information, but trade-offsare often required between public engagement goals and reliable data collection. We compared population estimates for 18 widespread butterfly species derived from the first 4 years (2011)(2012)(2013)(2014) (2011 -2014) de un proyecto de ciencia ciudadana de corta duración (Gran Conteo de Mariposas [GCM]) con aquellos estimados de datos de largo plazo y monitoreo estandarizado recolectados por observadores experimentados (Esquema de Monitoreo de Mariposas del Reino Unido [EMMRU]). Los datos del GCM son recopilados durante un periodo anual de tres semanas, mientras que los muestreos del EMMRU se realizan durante seis meses cada año. Una comparación inicial con los datos del EMMRU restringida al periodo de tres semanas del GCM reveló que los cambios en la población de las especies