Uncertainty in spatially explicit population models

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Abstract

Spatially explicit population models (SEPMs) are often used in conservation planning. However, confidence intervals around predictions of spatially explicit population models can greatly underestimate model uncertainty. This is partly because some sources of uncertainty are not amenable to the classic methods of uncertainty analysis. Here, we present a method that can be used to include multiple sources of uncertainty into more realistic confidence intervals. To illustrate our approach, we use a case study of the wood thrush (Hylocichla mustelina) in the fragmented forest of the North Carolina Piedmont. We examine 6 important sources of uncertainty in our spatially explicit population model: (1) the habitat map, (2) the dispersal algorithm, (3) clutch size, (4) edge effects, (5) dispersal distance, and (6) the intrinsic variability in our model. We found that uncertainty in the habitat map had the largest effect on model output, but each of the six factors had a significant effect and most had significant interactions with the other factors as well. We also found that our method of incorporating multiple sources of uncertainty created much larger confidence intervals than the projections that incorporated only sources of uncertainty included in most spatially explicit population model predictions.

Introduction

Conservation scientists and practitioners often rely on spatially explicit population models (SEPMs) to predict the response of species to management schemes, assist with reserve site selection, or guide reintroduction efforts (Liu et al., 1995, Gerber and VanBlaricom, 2001, Carroll et al., 2003, Kramer-Schadt et al., 2005, Pearson and Dawson, 2005, Rushton et al., 2006, Schiegg et al., 2006, Vandel et al., 2006). These complex models typically use patches or a lattice to represent the landscape, identify the location of every object of interest, and simulate birth, mortality, and dispersal at the individual or population level (Dunning et al., 1995). In an attempt to improve the realism of model output, modelers sometimes fall prey to a natural inclination to increase the number of explanatory variables and the complexity of these models (Gardner and Urban, 2003). However, as models grow in complexity, it becomes increasingly difficult to quantify the various sources of uncertainty, which can cloud the interpretation of model results. Even more importantly, model output that does not include an estimate of uncertainty may invoke a false sense of confidence, resulting in uninformed conservation decisions with potentially serious consequences.

There is a large literature on uncertainty analysis based in a regression framework, in which parameter values are sampled from their distributions, (e.g., Gardner et al., 1981, Gardner, 1984, Haefner, 1996, Crosetto et al., 2000; reviewed by Gardner and Urban, 2003), but all components of spatially explicit population models are not equally amenable to this procedure. Traditionally, the standard errors of the estimates of model coefficients are used to constrain Monte Carlo methods for assessing uncertainty (Gardner and Urban, 2003). In this, several replicate simulations are conducted, and in each simulation a different set of model parameters is selected randomly from the joint distribution of parameters. The model is then run with each parameter set, and model output (some dependent variable selected for its diagnostic value) is regressed on the input parameters. Sensitivity analysis is often conducted in the same framework, except the range of variation for each parameter is constrained to be some fraction of its nominal value (e.g., 10%) and Monte Carlo simulations are run for a large number of slightly perturbed parameter sets. By convention, a parameter’s uncertainty is indexed by the amount of variation that its estimation error induces in the output, while its sensitivity is indexed as its partial regression slope (i.e., a sensitive parameter is one for which a small change in the parameter elicits a large change in model output). There are many examples of this type of analysis in the literature (Crosetto and Tarantola, 2001, Cox et al., 2003, Harmon et al., 2004), but this approach can be difficult to apply to several sources of data error that are common in spatially explicit population models (e.g., habitat maps that have different numbers of patches, or alternative algorithms for animal dispersal. As a result, uncertainty in less traditional (although increasingly common) model inputs, such as GIS habitat maps, have been largely ignored in terms of effect on spatially explicit population model output. What is needed is a way to recast these and other, non-traditional, sources of uncertainty so that they can fit into the Monte Carlo regression framework with which most modelers are familiar.

In this paper, we demonstrate a way to extend the general approach described above to a wide variety of sources of model error by simply relaxing the mechanics of the approach. The “parameter sets” can then be drawn from a collection of predefined alternatives including input maps, boundary conditions, alternative model algorithms, and conventional parameter values. To illustrate our approach, we use a case study of the wood thrush (Hylocichla mustelina) in the fragmented forests of the North Carolina Piedmont. We examine what we consider to be the 6 most likely sources of uncertainty in our spatially explicit population model: (1) the habitat map, (2) the dispersal algorithm, (3) clutch size, (4) edge effects, (5) dispersal distance, and (6) the intrinsic variability in our model. With little prior knowledge of their relative importance, we selected these factors because they were often associated with great uncertainty in the literature and they covered a range of sources to illustrate our approach. The results highlight model components that need to be more accurately calibrated to improve the utility of our spatially explicit population model, give a general indication of the reliability of the model predictions, and identify trends that may be meaningful to users of other models.

Section snippets

Methods

Our goal in this paper was to establish the importance of incorporating diverse sources of uncertainty into model results and to illustrate an easy approach for doing so. To accomplish this as simply as possible, we have selected six key model factors and represented the uncertainty in each one with just two alternative values or possibilities. Five of these sources are model inputs representing aspects of avian biology, and the sixth is the intrinsic model variability that can affect any given

Results

Because the model was initialized with the population at 50% of carrying capacity, the number of birds on the landscape had the potential to either increase or decrease as the simulation proceeds, depending on the set of input factors. To illustrate a typical model run, we show the mean output from one particular scenario as a time-series graph with 95% confidence intervals (Fig. 6). The confidence intervals are produced from the intrinsic model variability alone, by running the model 100 times

Discussion

Every factor that we vary in our model has a significant effect on the output. In addition, most interaction terms are also significant. The range of output of the model is quite large when taking all these uncertainties into account. While this may be disconcerting, our results show that there are some steps that can be taken to reduce model error as much as possible. Although every factor is significant, the majority of overall model uncertainty is due to just a few factors (habitat map and

Acknowledgements

This research was initiated as a working group project in Duke’s Center on Global Change. J. Clark and M. Levine provided valuable feedback on early drafts of the manuscript. We would also like to thank John Dunning and the anonymous reviewers whose comments helped to improve the manuscript. The project was supported in part by NSF Grant SBR-9817755 to D.L.U.

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