## Introduction

The apparent paradigm shift from the theory of island biogeography to the theory of metapopulation dynamics has created a demand for quantitative metapopulation models for application to conservation (McCullough 1996; Hanski & Simberloff 1997). The construction of a practical metapopulation model begins with the choice of the factors presumed to affect metapopulation dynamics. Here the basic variables of classic metapopulation dynamics, habitat patch area and isolation (following MacArthur & Wilson 1967), have a prominent place, but additional factors may be included as found appropriate. After choosing the variables, a modelling approach and a particular model structure are chosen. Modelling of local dynamics, dispersal, and the effects of habitat patch area and isolation on local extinction and colonization are among the components that are typically included into metapopulation models (e.g. Hanski 1994a).

The next step is model parameterization using empirical data. Here, it is important to consider the quality and quantity of data needed for reliable parameter estimation, as gathering data from metapopulations generally requires substantial resources. Stochastic patch occupancy models (SPOM; see Moilanen 1999 and references therein) ignore local dynamics and only model the presence and absence of the species within discrete habitat patches. One practical reason for using SPOMs is the relative simplicity of these models; because data on local dynamics are not required, the amount of work needed for parameter estimation is significantly reduced. SPOMs that have been used for prediction include the incidence function model (IFM; Hanski 1994a) and the logistic regression model (Sjögren Gulve & Ray 1996; ter Braak, Hanski & Verboom 1998).

This study is concerned with the parameterization of SPOMs and especially with a particular assumption relevant to parameter estimation, namely whether the metapopulation is assumed to be at a stochastic quasi-equilibrium or not. (By quasi-equilibrium I mean the ‘typical’ dynamic state of the metapopulation before the inevitable eventual extinction, characterized by a stationary distribution of the number of turnover events per time unit and a characteristic pattern of patch occupancy reflecting the long-term probabilities of different spatial patterns of occupancy.) This assumption has generally been coupled with the choice of the metapopulation model: for example, the IFM in its original form assumes that the metapopulation is at a stochastic quasi-equilibrium (Hanski 1994a), whereas the logistic regression model (Sjögren Gulve & Ray 1996) does not assume stability.

Here, I demonstrate, how, in the case only turnover data is used in model parameterization and metapopulation stability is thus implicitly not assumed, estimation from empirical data is liable to produce parameter values that predict a spurious trend in metapopulation size. The implicit estimation of a trend occurs because empirical data are liable to show a spurious trend during a study period of only a few years. Such an apparent trend will almost certainly be present in empirical data even if the metapopulation truly is at a stochastic quasi-equilibrium. This is because extinction-colonization stochasticity, possibly amplified by regional stochasticity (spatially-correlated environmental stochasticity, Hanski 1991), is likely to cause the numbers of extinction and colonization events to be unequal during a short study period. As an extreme case one may envision a situation, where by chance only extinctions or colonizations are observed during a 2-year study period.

In this study, a number of simulated patch occupancy data sets were generated, and subsequently parameters relevant for the logistic regression model and the IFM were estimated from these simulated data sets. For the IFM, parameters were estimated using a new method for SPOMs, which is based on Monte Carlo inference for statistically implicit models (Moilanen 1999). This method computes maximum likelihood estimates from an observed sequence of patch occupancy patterns, and it allows one to make an explicit choice of whether metapopulation stability is assumed or not. Finally, metapopulation dynamics were predicted using the different parameterized models and the models/estimation methods were compared in their susceptibility to the implicit estimation of a trend.