These traditional approaches to incorporating stochastic variation into the model for- mulating process will often fail to adequately account for significant sources of uncer- tainty. Modelling approaches which incorporate both process error and measurement error have been developed for animal population dynamics. The fisheries industry is an area that experienced significant development in the formulation of new modelling tech- niques (Schnute, 1994; Newman, 1998; Meyer and Millar, 1999; Millar and Meyer, 2000a). Schnute (1994) introduces a general approach to developing sequential models for the
dynamics of fisheries. The traditional approaches to constructing fisheries models can be separated into two main classes; stock recruitment models (Millar and Meyer, 2000b) or catch-at-age models (Millar and Meyer, 2000a). The stock recruitment approach mod- els the current level of fish stock recruitment as a function of the sizes of fish stocks at previous time periods. The catch-at-age models necessitate the derivation of a function that describes the linkage between the number of fish of a particular age category (say, A) caught at the current time period with the number of fish in the previous age category (A-1), caught in the previous time period.
Schnute (1994) describes three main approaches to formulating sequential dynamics models. The first approach incorporates error only from the observations; the parame- ters of the process model completely determine the dynamics of the fish stocks. For these types of models Schnute (1994) describes the linkage between theoretical and observed age structures and how the parameters of the deterministic process model can be adjusted to obtain an optimal fit to the data. The second approach allows the sequential population process equations to include error but regards the observations as being known exactly. Schnute (1994) raises the issue that, for models created under this paradigm, the output can be highly sensitive to the manner in which the process error is introduced to the mod- elling framework. The third approach describes techniques for simultaneously allowing both measurement and process error to be incorporated into the models; an example of this approach is described by Millar and Meyer (2000b).
The models Schnute (1994) develop are sequential in form whereby the characteristics of interest for the population being modelled, and their linkage with the recorded ob- servations, are defined at a sequence of distinct time points. These models allow for the accommodation of non-stationary behaviour providing that the requisite process equations
describing the evolution of the dynamics can be specified for each distinct time period. To make the calculations described by Schnute (1994) tractable a number of simplifying assumptions are required. The distributions of the stochastic elements of process and observation are assumed to be normal (Gaussian) and the form taken by the process and observation equations is assumed to be linear. Models of this set are typically referred to as normal dynamic linear models and the necessary calculations for estimating the parameters can be performed using the Kalman filter (Besbeas et al., 2002, 2003, 2005), with specific application to a fisheries model presented in Newman (1998).
Although the normal dynamic linear models can incorporate both process and ob- servation error simultaneously the required assumptions of normality and linearity are fairly restrictive. By sacrificing reality for computational expediency these models can still often fail to capture adequately the true stochastic nature inherent in the underlying population dynamics. A relaxation of the assumption of linearity allows the use of the extended Kalman filter which uses linear approximations to fit non-linear process and observation equations to the observed data. When the stochastic elements of the model are Gaussian, the extended Kalman filter is optimal; however when the dynamic system is non-Gaussian the extended Kalman filter is only the best linear predictor (Carlinet al., 1992). There exists the potential for the difference between the best linear predictor and the optimal forecast to be substantial.
When attempting to transform realistic population dynamics models into a framework of normal dynamic linear models the resulting transformed models often convey behaviour that is significantly altered and no longer realistic. Realistic population dynamics models that are non-linear and non-Gaussian would often be far less tractable than their normal and linear counterparts. Recent developments in computer intensive simulation methods
have allowed complicated statistical estimation procedures to be applied to these prob- lems. Sequential Monte Carlo Methods (Doucetet al., 2001; Liu, 2001) and Markov Chain Monte Carlo (Gilkset al., 1996), hereafter MCMC, represent extremely powerful tools for fitting more flexible, and potentially more realistic, animal population dynamics models to observed data. These statistical estimation procedures have been applied to a vari- ety of animal population dynamics models which incorporate non-Gaussian process and measurement errors and have non-linear form. Recent applications include models for red deer (Cervus elaphus) (Trenkelet al., 2000), South Atlantic albacore (Thunnus alalunga) (Millar and Meyer, 2000b), and grey seals (Halichoerus grypus) (Thomas et al., 2005).
In many of these recent applications the modelling approaches have shared certain common features. Although the complete specification of the models differed between the studies, there were several components of the models that were common to each of the projects. This motivated the development of a unified framework that allows the joint definition of both population dynamics models and the measurements taken on a population (Buckland et al., 2004; Thomaset al., 2005; Newman et al., 2006).
In the following section a generalised approach to formulating a framework that allows stochastic population dynamics models to be embedded into statistical inference is de- scribed. Although the modelling approach can be applied to a wide variety of situations, the following section will describe the structure of these models only in the context of modelling animal population dynamics.