## Introduction

Species distribution models are static, probabilistic models of the relationship between the distribution or abundance of species and environmental variables (Guisan & Zimmerman 2000). Often, they predict the probability of presence of a species from presence/absence data using logistic regression. Such models rely on the Hutchinson niche concept (Guisan & Thuiller 2005), in which niche space is defined as a real space whose axes are environmental variables. The Hutchinson fundamental niche is the set of points in this niche space at which the species is able to persist indefinitely, and is bounded by the set of points on which the growth rate is zero (Hutchinson 1957). Any point in physical space maps to a point in niche space (Colwell & Rangel 2009), although a point in niche space may correspond to one, more than one, or no point in physical space. If the boundary of the Hutchinson niche is determined by experimental or observational measurements of population growth rate (e.g. Birch 1953; Doak & Morris 2010), then the set of points in physical space that map to points inside the Hutchinson niche can be determined, and may represent a likely distribution for the species.

More often, presence/absence and environmental data are recorded from points in physical space, and used to build a species distribution model which predicts the probability of presence at points in niche space. It is generally assumed that points at which a species occurs are contained in a subset of the Hutchinson fundamental niche known as the realized niche (Guisan & Thuiller 2005), with caveats discussed below. The resulting predictions can then be mapped back to points in physical space. The output of such a model is probabilistic for several reasons. First, a species may be absent from suitable sites due to chance events or dispersal limitation. Second, a typical model is based on the projection of niche space onto a relatively small number of environmental variables, so that an apparently suitable site may have an unsuitable value of an unmeasured variable. Third, a species may be present in unsuitable sites as a result of dispersal from nearby suitable sites.

The output of a species distribution model is static, in the sense that it has no explicit temporal interpretation. A predicted probability of presence can be interpreted as the probability that a species will be present at a randomly-chosen site having specified environmental conditions, but not as the expected proportion of time for which the species will be found at a single site. In addition, such models cannot make conditional predictions, such as the probability that a species will be present at a site in the future, given that it is present now. Thus, although ideas about both spatial distributions and population dynamics underpin the Hutchinson niche concept, species distribution models have retained only the spatial component of this concept.

One reason for the relative neglect of population dynamics in this context is that it is very difficult to build a plausible model for environmental effects on population dynamics. A plausible model needs to account for both density dependence and stochasticity, and how they are affected by the environment. Developing and fitting such a model requires a great deal of biological knowledge, and large amounts of data on how abundances change over time under a range of environmental conditions. These data are rarely available. However, a new class of density-structured models makes it easier to include population dynamics in species distribution models. Density-structured models are structured population models in which the population is structured by density, rather than by age or life stage (Taylor & Hastings 2004; Freckleton *et al*. 2011), in the same way that multivariate abundance data are sometimes discretized into community states (e.g. Johnson 2005). Such models can be based on categorical estimates of abundance (e.g. ‘not seen’, ‘rare’, ‘occasional’, …), which are faster and cheaper to obtain than precise estimates of abundance. This allows larger numbers of sites to be surveyed for the same effort. For example, Queenborough *et al*. (2010) estimated the probabilities of transitions between ordered categorical abundance classes for arable weeds from surveys of 500 agricultural fields over 3 years. Density dependence is built into these models in a crude way: the probability of a transition into a given destination class can be different for each source class, and can therefore approximate arbitrary relationships between abundance and population growth (Freckleton *et al*. 2011). Such models are also stochastic: both environmental and demographic stochasticity may contribute to the estimated transition probabilities. Lowe *et al*. (2011) described an extension of these models in which transition probabilities are simple parametric functions of environmental variables. Since only categorical abundance data are needed, it becomes possible to survey large numbers of sites with a range of environmental conditions, and thus to study how these conditions affect population dynamics. Density-structured models with environmental explanatory variables can be used to construct dynamic species distribution models, whose predictions are conditional probabilities of future abundance categories, given current abundance categories. There are very close parallels between these models and dynamic occupancy models, used to describe changes in the occupancy of sites over time (e.g. Erwin *et al*. 1998; Royle & Kéry 2007; MacKenzie *et al*. 2003, 2006, 2009). In fact, it is only terminological differences that divide the occupancy literature from the species distribution model literature.

The stochasticity in dynamic species distribution models has an explicit temporal interpretation. In particular, the stationary probability of presence under given environmental conditions can be obtained. This stationary probability can be interpreted both as the probability that the species will be present at a randomly-chosen site with the specified conditions, and as the expected proportion of time for which the species will be present at a particular site with these conditions. The dynamic nature of these models means that they can be used to investigate temporal features of population dynamics. For example, there are plausible mechanisms which could lead to either higher or lower temporal variability close to the limits of a species' range (e.g. Williams, Ives & Applegate 2003). From a density structured model, it is possible to calculate the normalized entropy (Hill, Witman & Caswell 2004), which measures uncertainty in abundance category one time step in the future for a randomly-chosen site with given environmental conditions. This can be used to examine the relationship between unpredictability of abundance (on a categorical scale) and the suitability of a site. In addition, it is likely that populations at some sites will persist for a long time, while others will quickly go extinct. The expected persistence time at a site can be determined from a density-structured model.

Here, we develop density-structured population models for two intertidal invertebrates, the gastropods *Phorcus lineatus* and *Gibbula umbilicalis*, based on large amounts of categorical abundance data from the same sites in consecutive years. We use these models to obtain the predicted probability of presence for each species under a range of environmental conditions (mean winter sea surface temperature and wave fetch), and to examine how normalized entropy and expected persistence time change with environmental conditions. We explain how these models can be viewed as dynamic species distribution models.