## Introduction

There is a need to obtain accurate measurements of the abundance of multiple species of insects across regions and ranges, for example to describe patterns of change, to identify and understand the drivers of species’ population dynamics (Nowicki *et al.* 2008; Thomas, Simcox & Hovestadt 2011) or to monitor the attainment of national and international conservation targets (Nayar 2010). Whereas freshwater systems are well monitored by professional entomologists in many developed countries, extensive monitoring of terrestrial insects depends largely on schemes that utilise enthusiastic, skilled amateur recorders. In practice, this often restricts wide-scale, long-term monitoring to butterflies; although in the UK, a national scheme exists for moths and is proposed for dragonflies and bumblebees (Thomas 2005; Conrad, Fox & Woiwod 2007).

The transect walk is the most widely used scientific method of monitoring butterfly numbers at a site (Pollard *et al.* 1975). A fixed route through the site is walked once a week under prescribed weather conditions, from 1st April to 29th September inclusive in the UK, by a recorder who counts the number of each species seen within a constant distance (usually a moving 5 × 5 × 5 m box). The UK Butterfly Monitoring Scheme (UKBMS) started in 1976 and currently samples *c.* 900 sites annually: results, analysis and interpretation of transect walks over the first 15 years were presented by Pollard & Yates (1993). The method is increasingly applied across Europe, where comprehensive schemes have been established in 14 nations, as well as in China and, more locally, in the US and elsewhere (Van Swaay *et al.* 2008).

From species data in standard field guidebooks, about 80% of the resident British butterfly species, up to 87% of European species, and a similar proportion in other temperate latitudes exhibit discrete flight periods for each generation, with daily counts of the number of adults in a sufficiently large population forming a well-defined and relatively smooth curve vs. time (the population curve). Moreover, the *shape* of this curve is rather similar for a wide variety of cases: for different species, generations, sites, years, and population sizes (Pollard & Yates 1993; Rothery & Roy 2001). The timing, duration and peak number vary with species, site and year (e.g. according to annual climatic conditions) but can be treated as parameters in a single general-purpose shape of curve.

In current butterfly and moth monitoring schemes, the results are expressed as relative changes in abundance obtained by summing the weekly mean count for each species, providing an Index of Abundance for every generation. The advantages of this simple method were considered to outweigh its disadvantages, such as the uncertainties generated by missing counts. Pollard & Yates (1993) advocated devising a generic mathematical formula for the time-variation in counts, which could be fitted to data with missing weeks so that the Index of Abundance could still be calculated. Such a formula would also be valuable for exploring the reliability of population indexes based on smaller numbers (Zonneveld 1991; Rothery & Roy 2001; Gross *et al.* 2007; Nowicki *et al.* 2008) and for calculating key demographic parameter values used in pure and applied ecology.

A model of this type was presented by Manly (1974; subsequently M74), to represent the population curves of individuals at successive life stages of insects. He assumed that the time at which insects enter a life stage follows a normal distribution, and that the survival rate per day after entry to the stage follows an exponential decay with time, that is, a constant age-specific death rate. He obtained a mathematical solution for the generalised form of the population curve expressed as an integral (also see Rothery & Roy 2001). Zonneveld (1991; subsequently Z91), who devised another model of this class, also assumed a constant death rate but chose a logistic distribution for the rate at which adult butterflies emerge (eclose). He assumed that the stochastic variation in individual counts is Poisson-distributed and used a maximum-likelihood method to fit the model to field data. Use of the Z91 model was greatly eased by its incorporation into free software, the Insect Count Analyzer (INCA, 2002).

General additive models (Rothery & Roy 2001) serve a similar purpose of smoothing and interpolating transect data but do not yield a ‘universal’ mathematical formula for the population curve nor allow quantities such as total population size and life span to be estimated.

Applications of the M74 and Z91/INCA models fall into two types:

- 1 To fit curves to field data in order to derive total (transect) population, life span and eclosion parameters (e.g.) to grasshoppers and locusts (M74), and various butterflies (Z91; Mattoni
*et al.*2001; Gross*et al.*2007; Haddad*et al.*2008; Marschalek & Deutschman 2008); - 2 To synthesise idealised data sets for statistical studies, such as testing analysis techniques for transect data (Rothery & Roy 2001) and the design of monitoring schemes (Zonneveld, Longcore & Mulder 2003).

However, in a review of butterfly monitoring methods, Nowicki *et al.* (2008) commented that the Z91 model remains difficult to apply and consequently rarely used. It has yet to be adopted by major schemes, perhaps because, while providing a good fit to the population curves of several example species, it fails to generate estimates when numbers are low or counts are irregular or infrequent, a common occurrence in transect data sets. One practical and conceptual disadvantage of the M74 and Z91 models, which may affect their adoption, is that their population-curve equations were presented as integrals that can only be evaluated numerically.

Here, building on the M74 and Z91 models and in the hope of promoting wider use of such models, we present a new algebraic equation to describe the shape of the population curve (POPFIT). It contains four readily interpretable parameters that can be fitted to transect data from individual species, sites and years. As with earlier models, our mathematical representation is not applicable to the 13–20% of temperate butterfly species that are continuously brooded, are predominantly migratory, have overlapping broods, or over-winter as adults, nor to sites with very small observable populations (peak daily count <15). The model is equally applicable to other taxa, including UK moths (Conrad, Fox & Woiwod 2007), whose discrete populations are regularly sampled.