Embedding Population Dynamics Models in Inference
📝 Abstract
Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ``building block’’ approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population.
💡 Analysis
Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the effects of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ``building block’’ approach to formulating discrete-time models. We show how to estimate the parameters of such models from time series of data, and how to quantify uncertainty in those estimates and in numbers of individuals of different types in populations, using computer-intensive Bayesian methods. We also discuss advantages and pitfalls of the approach, and give an example using the British grey seal population.
📄 Content
At the 2002 World Summit on Sustainable Development in Johannesburg, political leaders agreed to strive for “a significant reduction in the current rate of loss of biological diversity” by the year 2010. Initial steps to achieve this include developing and implementing monitoring programs and data collection procedures that quantify the rate of loss of biodiversity (Buckland,Magurran,Green and Fewster,?yearBMGF05), allowing assessment of the success of management actions. However, monitoring is a blunt instrument for management, because of the long lag between action and observed effects. Explanatory mathematical models provide “what if” tools for predicting the impact of different management actions on populations and biodiversity before a course of action is chosen. Both elements are needed: models, to guide and increase the effectiveness of management action; and monitoring, to provide a retrospective measure of whether the predicted effects have been achieved. Indeed, the two approaches should be fully integrated, by using the data from monitoring programs to update and finetune the mathematical models. One way to integrate the two elements is through the use of adaptive management techniques (Walters, 2002).
The process of creating explanatory mathematical models involves several steps, including formulating one or more models, fitting the models to data, and, in situations of multiple models, selecting or averaging models that will be used for prediction (Burnham and Anderson, 2002). Realistic models that meet the needs of policymakers and biodiversity managers often have a high degree of complexity. If these people are not to be misled, it is essential that uncertainty in these models is quantified (Harwood and Stokes, 2003;Clark and Bjørnstad, 2004). The uncertainty in modeling population dynamics that arises includes four main sources: process variation (demographic and environmental stochasticity), observation error, parameter uncertainty and model uncertainty.
A popular and powerful approach to formulating models for plant and animal population dynamics has been the use of matrix models (Caswell, 2001). Such models explicitly account for the various processes that affect the dynamics of populations, for example, survival, birth, movement. However, fitting such matrix models to data, for example, estimating survival and fecundity rates, has often been carried out in a somewhat piecemeal fashion, and uncertainty about model parameters has not always been incorporated into model projections, let alone uncertainty associated with the initial choice of the particular matrix model.
In contrast, many statistical models for time series of population data can readily be fit in an integrated manner and various forms of uncertainty simultaneously accounted for. However, model formulation is often empirical, with no attempt to incorporate the processes underlying population dynamics explicitly, for example, AR(p) and ARIMA models.
In this paper we review recent developments in modeling population dynamics and describe an integrated approach to formulating, fitting and selecting realistic models for population dynamics that builds upon the matrix model framework. The resulting models are not necessarily matrix models, and are in fact more flexible. The approach is embedded within a Bayesian inferential framework, so that the main sources of uncertainty are accommodated. Two model fitting approaches are discussed, Markov chain Monte Carlo (MCMC) and sequential importance sampling (SIS), and an example of the approach is given for the British grey seal population.
Matrix population models have long been used for describing population dynamics. Following the pioneering work of Leslie (1945Leslie ( , 1948)), such models were usually deterministic; if observational data were used at all for fitting models, it was for ad hoc estimation of parameters of the population model, to allow deterministic projection of the population of interest. Caswell (2001) gives a comprehensive account of the mathematical development of the topic, which includes stochastic extensions (demographic and environmental variation), and asymptotic analyses of growth rates and age and stage class distributions for deterministic and stochastic matrix models. However, in many cases an integration of field or laboratory experiment data with an underlying population dynamics model is lacking. Quoting Tuljapurkar (1997) following an exposition of analysis of stochastic matrix models: “A useful application of stochastic models that we have not discussed here is the development of methods for estimating the vital rates of structured populations from data. Such estimation methods . . . are desirable in ecology, although rarely used.”
An integration of data with population dynamics models, and an embedding of such models in statistical inference, is needed to allow for simultaneous accounting of uncertainties about model parameter values, observa
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