### Abstract

- Top of page
- Abstract
- Introduction
- The Model
- Model Validation
- Results
- Discussion
- Acknowledgments
- Conflict of Interest
- References
- Supporting Information

Frequently, vital rates are driven by directional, long-term environmental changes. Many of these are of great importance, such as land degradation, climate change, and succession. Traditional demographic methods assume a constant or stationary environment, and thus are inappropriate to analyze populations subject to these changes. They also require repeat surveys of the individuals as change unfolds. Methods for reconstructing such lengthy processes are needed. We present a model that, based on a time series of population size structures and densities, reconstructs the impact of directional environmental changes on vital rates. The model uses integral projection models and maximum likelihood to identify the rates that best reconstructs the time series. The procedure was validated with artificial and real data. The former involved simulated species with widely different demographic behaviors. The latter used a chronosequence of populations of an endangered cactus subject to increasing anthropogenic disturbance. In our simulations, the vital rates and their change were always reconstructed accurately. Nevertheless, the model frequently produced alternative results. The use of coarse knowledge of the species' biology (whether vital rates increase or decrease with size or their plausible values) allowed the correct rates to be identified with a 90% success rate. With real data, the model correctly reconstructed the effects of disturbance on vital rates. These effects were previously known from two populations for which demographic data were available. Our procedure seems robust, as the data violated several of the model's assumptions. Thus, time series of size structures and densities contain the necessary information to reconstruct changing vital rates. However, additional biological knowledge may be required to provide reliable results. Because time series of size structures and densities are available for many species or can be rapidly generated, our model can contribute to understand populations that face highly pressing environmental problems.

### Introduction

- Top of page
- Abstract
- Introduction
- The Model
- Model Validation
- Results
- Discussion
- Acknowledgments
- Conflict of Interest
- References
- Supporting Information

Understanding the effects of the environment on populations is central to ecology (Heller and Zavaleta 2009; Pereira et al. 2010; Crone et al. 2011). However, many environmental drivers of population change, such as land degradation, climate change, pollutant buildup, ocean acidification, and succession, operate on a long-term, directional basis (Singh 1998; Parr et al. 2003; Kroeker et al. 2010; Wake 2012). The timescales involved make the study of the impact of environmental change on vital rates (survival, growth, and reproduction) impracticable. The correct identification of such impact will allow conservation efforts to be directed more appropriately, to better understand the basis of population change, or even to track the evolutionary changes in life-history traits through time. This calls for specific methods that tackle this problem (Doak and Morris 1999; Pereira et al. 2010; Crone et al. 2011).

Traditional demographic modeling does not provide a solution, as it often assumes that environment change does not occur in a directional fashion (Caswell 2001; Ellner and Rees 2007). Nevertheless, if the environmental driver we are studying changes directionally, population would never reach stability, which is usually the focus of traditional models. Assuming stability in a changing population leads to biased conclusions (Koons et al. 2005). Furthermore, traditional models use repeat surveys of the individuals as input (Caswell 2001). Doing this for the decades or centuries required for environmental change to unfold is impracticable. A substitute, but also costly, approach would be to survey over a representative time period the individuals of a series of populations at different stages of environmental change (Dahlgren and Ehrlén 2011). However, if we are to accomplish global goals such as the assessment of the conservation status and long-term threats for all plant species by 2020 (COP 10 2010), a faster and cheaper alternative to such traditional demographic methods becomes imperative.

A viable approach would be to use time series of static, population-level data, such as population densities and structures, to reconstruct the species vital rates and their change through time as the environment changes. Such datasets have been recorded over several years for different species in the context of forestry, hunting, fisheries, and long-term ecological research (Waters 1999; Hobbie et al. 2003; Parr et al. 2003; Clucas 2011). Also, this kind of data can be rapidly collected for several populations that represent different stages of environmental change, and integrated into a chronosequence (Matthews and Whittaker 1987; Mori et al. 2007). This reconstruction of vital rates from static data has been successfully applied in the context of fisheries stock assessment (e.g., Fournier et al. 1998; Quinn 2003; Maury et al. 2005; Hilborn 2012). However, the translation of these models into an ecological context is not straightforward, as the amount of information and biological knowledge available in fisheries rarely exists for noncommercial species (Quinn 2003). For instance, the available data for most species will usually be sparsely distributed in time, and not surveyed annually as in fisheries. Also, the demographic behavior of the species that ecologists study can be quite complex, as in many species the vital rates depend on size, rather than on age. In plants, for example, organisms having originally different sizes may end up having the same size after 1 year, due to growth, shrinkage, or stasis (Caswell 2001), thus complicating the relationship between size structure and vital rates. Therefore, a model is needed that accommodates these complexities as well as a wide variety of life cycles.

As a time series of static, population-level data does not inform on the fate of individuals, more than one combination of vital rates would be expected to lead to the same series. For instance, a high proportion of seedlings in a population may result from a large fecundity, a low seedling mortality, or impediments to seedling growth. In a model that uses population-level data, Ghosh et al. (2012) envisage this problem. However, as their aim is to forecast population structures, they circumvent the problem by making assumptions on the vital rates that simplify their model but that do not reflect their behavior at the individual level (Ellner 2012). However, if we are interested in correctly reconstructing vital rates, we cannot make such assumptions.

In this article, we develop a model that, based on a time series of population size structures and densities, reconstructs the shifts in vital rates caused by a directionally changing environmental driver. The model was validated with artificially generated data and with data from a threatened cactus subject to long-term human disturbance. We show that, although more than one scenario may be obtained, the correct solution is always provided by the model, and that basic information on the biology of the species is frequently enough to discard alternative solutions.

### The Model

- Top of page
- Abstract
- Introduction
- The Model
- Model Validation
- Results
- Discussion
- Acknowledgments
- Conflict of Interest
- References
- Supporting Information

Our model attempts to reconstruct the vital rates (survival, growth, and reproduction) and their change over time based on a time series of size structures and densities. If these rates change as the environmental driver shifts, the structure and density of the population would be expected to evolve accordingly. The model explores a variety of vital rates, seeking the ones that produce the size structures and densities that best resemble the observed time series. A succinct description of the model is presented below. Please refer to Appendix S1 for the full details.

The vital rates of the size-structured population were modeled by means of an integral projection model (IPM; Easterling et al. 2000). An IPM integrates the vital rates into a function *k* known as the kernel. This function establishes the log-sizes *y* that individuals of log-size *x* may reach from time *t* to *t *+* *1, as well as the number and sizes of their descendants. The IPM is expressed through the equation

- (1)

where *n* is the size structure of the population. Note that, in our model, *k* is a function of time because the vital rates are driven by environmental change.

The kernel comprises the functions associated to the survival probability, *s*, growth, *g*, number of newborns, *f*_{1}, and the sizes of these, *f*_{2}, which relate as

- (2)

We used the following simple functions to determine these vital rates:

- (3)

As can be seen from these equations, the vital rates and their change through time are determined by 16 parameters.

To assess whether any given set of 16 parameter values is able to reproduce the observed time series, we first calculated the vital rates for every year in the period over which environmental change takes place by substituting the parameter values in equations (3). We then calculated the time series of size structures through the iteration of equation (1). To do so, an initial size structure, *n*(*x*,*t*_{0}), is required. If no environmental change had occurred before the initial time (i.e., if the environment had remained constant), it would be safe to assume that the population was in its stable state (Caswell 2001). Therefore, in the first iteration of equation (1), we used the stable (asymptotic) size structure associated with the vital rates at the initial observed time. The time series of densities was obtained by integrating the size structures. Finally, we compared these two time series with the observed ones through the composite log-likelihood:

- (4)

where *l*_{n} and *l*_{d} are the log-likelihoods associated with the size structures and with the densities, respectively, and *w* is a weighting factor of the relative importance of the fit of the observed size structures versus that of the observed population densities. Because at each observed point in time there is only one datum for density, but several for size structure, not using a weighting factor could belittle the contribution of density to *l*. The right value for *w* was determined experimentally (see below). The values of the 16 parameters that resulted in the highest *l*-value determined the kernel that best resembled the observed data.