## Introduction

There is a pressing need for models that can reliably predict the consequences of human-modified habitats, climates and ecosystem contexts for animal and plant populations (Norris 2004; Coreau *et al.* 2009). Theoretical population ecology has developed increasingly sophisticated models, but these models can prove inadequate for diagnosing the causes of increasing or decreasing populations (Caughley 1994). According to Getz (1998), ‘the art of modelling in population ecology may have more to do with fixing what is most glaringly wrong with models than in finding the right model’. Models encapsulate, in heuristically simplified form, what we currently know and understand about the system being modelled. Yet, a gulf still exists between elegant theoretical models and the real-world biological processes that they supposedly represent. Nevertheless, judging models merely by their fit to data is misdirected; simplicity and generality are equally important (Ginzburg & Jensen 2004). Clearly, some reconciliation is needed between the theoretical models that have become standard in the literature and the biological and environmental processes that generate changes in populations (Owen-Smith 2010a). The manifold influences of climatic variation on the population dynamics of large mammalian herbivores are especially well documented (Owen-Smith 2010b). In this article, I describe how various aspects of environmental variation can be accommodated within population models in ways that have the potential to be heuristically productive, illustrated using working examples.

Models of population dynamics have taken various forms. Single-species population models such as logistic growth equations and variants thereof represent density-dependent feedbacks regulating populations around some environmental carrying capacity, but not what determines this capacity. Time series elaborations allow for the perturbing effects of stochastic environmental variation on the population density level manifested, but still assume some constant ‘attractor’ about which population abundance varies (Royama 1992; Denis & Taper 1994; Bjornstad & Grenfell 2001). Assumptions of a stable attractor become increasingly untenable the longer populations are observed (Pimm & Redfearn 1988; Owen-Smith & Marshal 2010), and spatial variation modifies the population abundance manifested (Brown, Mehlman & Stevens 1995; Hobbs & Gordon 2010). Moreover, the discrete, generally annual time steps usually adopted in these models obscure the seasonal variation in resources and conditions that contributes to the population dynamics generated (Owen-Smith 2002a). Models of interacting populations based on the Lotka–Volterra equations emphasize the propensity to oscillations of predators and their prey, or more generally consumers and resources, but do not represent the environmental contexts that may promote or suppress oscillatory dynamics (Turchin 2003; Owen-Smith 2002b; Gross, Gordon & Owen-Smith 2010). Furthermore, for most consumers, the resource base is actually constituted by multiple populations with distinct dynamics and other properties affecting choices among them. Demographically structured models highlight how distinctions in rates of survival and reproduction among age or stage classes modify the dynamics generated, but only superficially represent the environmental influences affecting these vital rates (Caswell 2001; Lande, Engen & Saether 2003). Individual-based models incorporate unlimited contextual detail in principle, but in practice pose considerable challenges for parameterization as well as conceptual interpretation of the sources of the dynamics generated (Grimm & Railsback 2005).

In this ‘how to’ article, I describe how these standard models can be modified to represent intrinsic and extrinsic processes generating population dynamics in ways that are more conducive to diagnosing the causes of population change. This approach draws on metaphysiological modelling concepts developed originally by Getz (1991, 1993) to embed interacting populations within food web contexts and expanded by me to link population dynamics to the behavioural ecology of resource selection in variable environments (Owen-Smith 2002a). Parallels exist with the dynamic energy balance models formulated by Kooijman *et al.* (2008); Kooijman, van der Hoeven & van der Werf (1989); Nisbet *et al.* (2000) and De Roos & Persson (2001), but these authors place their emphasis on intrinsic physiological mechanisms rather than on the extrinsic environmental influences. Moreover, material resources needed for constructing biomass can restrict population performance as well as the supply of energizing substrates.

The structure of this article will be as follows. First, I will describe how the Lotka–Volterra equations, and modifications thereof, can be elaborated to accommodate environmental influences potentially affecting population dynamics in a general conceptual way. Second, I will show how this approach can be used to explain spatial variation in habitat capacity, as reflected by the population abundance supported, using real-world data. Third, I will illustrate how population structure can be incorporated into these models, using a classical case study as an example. Last, I will suggest how simple single-species models could be modified to reconcile them with the metaphysiological perspective through a general framework for biomass loss accounting. To assist readers in developing and applying this approach, I have supplied the computational code that was developed for these specific examples in Appendices S1–S5, together with listings and some explanation of the parameter values used, provided in these appendices. This approach has proved its heuristic value in my postgraduate teaching (Owen-Smith 2007).