## Introduction

Eradication of invasive species requires the removal of every individual of a species from a target area – for plants this entails the removal of both adults and seeds. There have been many successful eradications of both animals and plants (Mack & Lonsdale 2002; Simberloff 2002, 2003). A major challenge facing eradication managers is deciding when a programme can be declared successful (Morrison *et al*. 2007). Survey techniques are imperfect, and thus, the failure to detect a species does not necessarily mean it is absent. An invasive species can re-emerge if eradication is declared prematurely, and monitoring stopped, with resulting ecological impacts and costs of further management. Despite this risk, eradication is still declared on an *ad hoc* basis (Regan *et al*. 2006), for example, after 3 years without detection (Rejmanek & Pitcairn 2002).

Regan *et al*. (2006) took an economic approach to the question of when to declare eradication, using decision theory. They found the stopping time (based on the number of previous consecutive surveys in which the species is not found, hereafter referred to as absent surveys) that minimizes the net expected cost. This is essentially a trade-off between the cost of continued surveying and the cost if eradication is declared when the species is still present.

Although this work represents a new way of thinking about how we approach setting guidelines for invasive species eradication, its practicality is reduced by the data requirements of the model. To calculate the probability that an invasive species is still present after a number of absent surveys, Regan *et al*. (2006) used probabilities of persistence and detection. These parameters are difficult to estimate for many invasive species. For example, in the field of population viability analysis, uncertainty around the estimates of probabilities of persistence can span zero and one (Ludwig 1999; McCarthy, Burgman & Ferson 1996). Similarly, methods for estimating detection probabilities usually require labour-intensive data (MacKenzie *et al*. 2002; Tyre *et al*. 2003; Wintle *et al*. 2004), and detection probabilities for newly invading species are likely to be very uncertain. Instead of estimating these parameters, we can use the presence–absence sighting record of the species.

There are several methods documented in conservation literature that use a species’ sighting record to infer persistence. Solow (1993a) used a presence–absence sighting record, and assumed a constant pre-extinction sighting rate. This essentially assumes that the species’ population level remains constant prior to extinction. Solow (1993b) described a variation of the equation for use in declining populations where the pre-extinction sighting rate declines. Solow & Roberts (2003) described a nonparametric test, based on the two most recent sightings of the species. These methods and variations are summarized by Solow (2005).

In addition to these, several papers have focused on using sighting records and collection data to identify declining or threatened species. Burgman, Grimson & Ferson (1995) extended the equation from Solow (1993a) to account for multiple sightings within one time period, and explored methods that are sensitive to patterns in sighting data. McCarthy (1998) used five different methods of identifying declining species from museum records, including methods that account for variable collection effort. McInerny *et al*. (2006) modified Solow's equation to remove the influence of the length of the initial sighting period. Of these methods that use sighting data to infer extinction and decline, Solow (1993a) is the only one to include a Bayesian formulation of the probability of presence.

Two other studies tested these statistical methods. Burgman *et al*. (2000) calculated the power of Solow's original equation and a runs test (Grimson, Aldrich & Wanzer Drane 1992) by applying them to data generated from a scenario where the ‘true’ rate of population decline was known. They found both equations had a type I error rate (probability of detecting a decline when there is none) of less than or equal to the conventional limit of 0·05. Robbirt, Roberts & Hawkins (2006) used herbarium data for endemic Ecuadorian species of *Guzmania* (Bromeliaceae) to compare the results from five different statistical methods with the IUCN (International Union for the Conservation of Nature) listing of each species. The correlation results were not significant in the traditional sense (*P* value < 0·05), but were close to significant (*P* value < 0·1).

We use the Bayesian formulation of the equation from Solow (1993a) to calculate the probability that a weed is still extant, given a presence–absence sighting record. This equation assumes a constant pre-extinction sighting rate, meaning the invasive species population level is constant until eradication. We incorporate this into the decision-making framework of Regan *et al*. (2006) to determine the optimal number of absent surveys after which eradication should be declared. First, we examine the analytical solution of the new equation, and find a simple approximation for when to declare eradication. Then, we use stochastic dynamic programming to find an exact optimal solution that incorporates the possibility that the weed may be seen in future surveys. In addition, we include a solution with a declining pre-extinction sighting rate, meaning the invasive species population declines prior to eradication. We apply these methods to the example of bitterweed *Helenium amarum*, the same case study used by Regan *et al*. (2006).