Spatial scale has been shown to be of fundamental importance in basic ecological studies (Wiens 1989; Levin 1992; Chase & Leibold 2002; Willis & Whittaker 2002) as well as in static, species-environment modelling (Orrock et al. 2000; Guisan & Thuiller 2005; Heikkinen et al. 2007). Here, we have shown that scale critically matters also for the application and outcomes of dynamical modelling of species’ range expansions. Dynamic models offer a promising approach for moving beyond envelope models, but it is imperative to acknowledge the potential biases in their projections that result from choices relating to the spatial resolution at which key patterns and processes are represented.
Upscaling landscape maps is inevitable when moving from local to larger scale predictions, especially as range dynamics involve large spatial extents (Wiens 1989), whereas computational constrains and data limitations place constraints on the resolution of simulations (Urban 2005; Govindarajan et al. 2007; O’Sullivan & Perry 2009; Slone 2011). Using a spatially explicit individual-based model for species’ range expansions, we have illustrated that a ‘naïve’ upscaling of landscape maps and/or of ecological processes, or a noninformed choice of spatial resolution, can yield major biases in the form of overestimating species’ range shifting, expansion capacities, landscape connectivity and population sizes. These biases lead therefore to over-optimistic conclusions about the potential for species to respond effectively to the synergistic drivers of habitat loss, fragmentation and climate change. In the following paragraphs, we elaborate on the mechanisms that led to the biases before discussing currently available solutions and highlighting key remaining challenges.
Discerning the Sources of Biases
The first source of bias is the landscape homogenisation owing to upscaling, which results in loss of information about landscape structure (Turner et al. 1989; Wiens 1989; Heikkinen et al. 2007). Aggregating cells lead to homogenisation of the landscape composition, loss of spatial structure and an increase in the proportion of cells offering suitable habitats, even when the total carrying capacity and habitat cover remains equal. In this study, this homogenisation resulted in an overall increase in dispersal success, as all sections within a cell (formerly, potentially separated habitat patches) became equally reachable. This bias was amplified when habitat cover was low, and patches were highly fragmented, namely exactly in those spatial configurations that are of utmost conservation interest and where dispersal limitations because of patch isolation are known to play a key role in species’ dynamics (King & With 2002).
The modification of realised dispersal kernels also occurred owing to the discretisation of space (i.e. moving from continuous to discrete, grid-based space). This modelling artefact has been described already (Holland, Aegerter & Smith 2007b; O’Sullivan & Perry 2009; Chipperfield et al. 2011; Slone 2011); yet, it is often overlooked when modelling dispersal and estimating range expansion. The effect is highly scale dependent: the coarser the spatial resolution, the higher the overestimation of individuals’ displacements, especially when the species’ expected dispersal distance is small relative to the cell size (Chipperfield et al. 2011).
The third source of bias originated from aggregating population dynamics. By upscaling the landscape, all individuals in a larger area are assumed to interact homogenously with each other and to function as a single population (mean-field assumption or mean-field approximation; Murrell, Dieckmann & Law 2004; Bergström, Englund & Leonardsson 2006; Englund & Leonardsson 2008; Morozov & Poggiale 2012). This averages out, and in fact ignores, the spatial structure of the local populations at the finer scale, making the system spatially implicit. However, it is well known, especially when dealing with nonlinear dynamics such as density-dependent population growth, that the aggregation of fine-scale components does not behave as the components themselves (‘fallacy of the averages’; Rastetter et al. 1992; Chesson 1998; Melbourne & Chesson 2005), the error being bigger the greater the variability between the fine-scale elements. By applying the same function at coarser scale without any correction, the increase in carrying capacity, and hence in population size per cell, results in more stable dynamics with reduced demographic stochasticity, and hence reduces the risk of local extinction (Holland, Aegerter & Smith 2007b). This yields a larger number of individuals at coarser resolutions, and boosts the potential for range expansion (Neubert & Caswell 2000; Söndgerath & Schröder 2002; With 2002; Hastings et al. 2005).
Importantly, it was the combined effect of upscaling both population dynamics and dispersal that resulted in the greatest overestimation in projecting range expansion at coarser resolutions. The relative importance of the different sources of bias was mediated by the landscape structure and by the dispersal abilities of the species. In highly fragmented landscapes, the upscaling of dispersal dominates the upscaling of population dynamics as a source of bias, regardless of the dispersal capacities of the species. This is because in such landscapes, the main limiting factor for range expansion is low dispersal success, in our case because of a high probability of arriving in unsuitable cells. As habitat cover increases and fragmentation decreases, the effect of population dynamics becomes dominant as neighbouring cells become unified because of upscaling. In these situations, the dispersal capacity of the species determines the relative influence of different processes as sources of biases.
For species with short expected dispersal distance, dispersal represents the major limitation to range expansion: by upscaling dispersal, individuals are assumed to move longer distances and with lower dispersal mortality. On the other hand, upscaling only the population dynamics does not eliminate the main dispersal limitation. The expansion of species with higher dispersal capacities is limited primarily by dispersal mortality, the number of dispersers and stochastic extinction events [as well as Allee effects (Keitt, Lewis & Holt 2001; Tobin et al. 2009), although these do not act in our simulated system]. These are especially important at those newly occupied sites at the expanding front that are critical in driving the range shifting dynamic. Hence, upscaling dispersal alone does not lead to much overestimation, whereas it can even lead to underestimation if the original landscape was highly autocorrelated. In this case, individuals have a higher chance to be lost in the matrix, thus slowing the expansion rate.
These results underline the extreme importance of considering the interaction between the dispersal capacities of the species and the landscape structure when evaluating the biases as a result of model upscaling. Interestingly, this has emerged even in our model where individuals did not actively move through the landscape. In more realistic dispersal models, where movement processes are mechanistically described and individuals respond to the landscape according to a given set of movement rules, the loss of landscape structures because of upscaling is very likely to yield an even greater bias in the projected range dynamics and population persistence. The same is true for more realistic models of population dynamics.
Where Do We Go from Here?
In accordance with the three main sources of bias discussed earlier, we see the need for potential solutions, involving better methods for upscaling (i) landscape structures, (ii) dispersal and (iii) population dynamics. We strongly advocate that the best way to do so would be to derive upscaled models by starting from the mechanisms and processes working at the fine, individual scale and by understanding which are the relevant details and realistic behaviours on which larger scale models need to be built (Levin 1992; Rastetter et al. 1992, 2003; Urban 2005; Pe’er & Kramer-Schadt 2008; Schick et al. 2008).
Concerning the modelling of dispersal, different promising methods are starting to be developed to resolve scale-related biases. These include, for example, the approximation of ecological diffusion models to larger scales with habitat-dependent diffusion coefficients parameterised at small scales (Garlick et al. 2011) and the approximation of continuous dispersal kernels (Chesson & Lee 2005; Govindarajan et al. 2007; Chipperfield et al. 2011; Slone 2011) or random walk models (O’Sullivan & Perry 2009) in discrete space, correcting, to a certain extent, for scale-related bias. Another possible solution is the identification of characteristic spatial and temporal scales of movements allowing the essential characteristics of animal movements at different scales to be captured (Gurarie & Ovaskainen 2011).
Similarly, the scale at which population dynamics are modelled cannot be arbitrarily chosen (Holland, Aegerter & Smith 2007b) but should emerge from the scale relevant to individual interactions, the scale at which density dependence is acting (Chesson 1998; Bergström, Englund & Leonardsson 2006; Englund & Leonardsson 2008). In the last two decades, much effort has been expended in moving beyond mean-field models, and different methods have been developed for building coarse-scale models from fine-scale relationship and reducing the aggregation error (Rastetter et al. 1992; Morozov & Poggiale 2012). A promising approach derives from the application of the scale transition theory proposed by Chesson (Chesson 1996, 1998, 2012; Chesson et al. 2005; Melbourne & Chesson 2005, 2006; Bergström, Englund & Leonardsson 2006; Englund & Leonardsson 2008) who uses a second-order moment approximation to derive changes in population dynamics with increasing spatial scale; these changes are explained in terms of interactions between nonlinearities and spatial heterogeneities. A second main approach, and to some extent related to the first, is represented by moment closure methods, which aim to capture the effect of spatial heterogeneity and stochastic fluctuations on population dynamics by approximating underlying individual-based stochastic models, where the interactions between individuals are described with distance-dependent kernels in continuous space (Murrell, Dieckmann & Law 2004; Ovaskainen & Cornell 2006; Raghib, Hill & Dieckmann 2011). Moreover, in addition to these analytical methods, approaches that aim to infer large-scale dynamics from very detailed fine-scale IBMs have also been proposed (e.g. Vance, Steele & Forrester 2010).
Another possibility for upscaling population dynamics could be inspired from recent developments of the metapopulation theory and in particular from the ‘effective metapopulation’ approach (Ovaskainen 2002; Hanski & Ovaskainen 2003). This approach approximates a stochastic patch occupancy model (SPOM) for a heterogeneous patch network with a corresponding homogeneous model that behaves in same way as the first. In the new model, the interaction between population dynamics and spatial heterogeneity is expressed by three parameters: the effective patch number, the effective colonisation rate and the effective extinction rate.
When dynamic models are built specifically for inferring spatial processes from field data, a more sophisticated yet more computational intensive approach has been proposed, which allows mapping field observations onto discrete space models accounting for the uncertainty caused by arbitrary discretisations (Baird & Santos 2010; Estoup et al. 2010). The grid characteristics, such as origin, orientation, cell size and cell carrying capacity are treated as parameters themselves together with species’ demographic parameters and the best parameters combination is inferred by integrating across different alternatives through Bayesian averaging.
There remains the issue of a better upscaling of the landscape structure. Interestingly, if methods for upscaling the fundamental ecological processes of dispersal and population dynamics are developed in such a way that the coarse model parameters emerge from the interaction between these processes and the fine-scale spatial configuration, the landscape structure becomes automatically upscaled in a species-specific way. Where this would not be possible, approaches that establish correlations between ecological processes and landscape indices could be explored (e.g. Vos et al. 2001).
As discussed earlier, various approaches for dealing with scaling problems are currently being developed, but still they remain quite sparse in the literature. Moreover, they are not fully integrated in a clear framework for dealing with projections of range dynamics under environmental changes. Indeed, most of the current models do not deal with the problem at all, typically modelling all the processes at somewhat arbitrary resolutions. Thus, we need to initially make better use of existing approaches as well as beginning to develop novel solutions to the particular problem of expansion dynamics across heterogeneous landscapes.
The strongest outcome of this work is the critical point that a potentially ‘smarter’ upscaling of any of the three components – landscapes, dispersal and population dynamics – will remain insufficient if it focuses on each of them separately instead of considering their combined effects. Thus, the most promising solution is likely to lie in methods for parameterising and calibrating large-scale model rules and parameters directly from fine scale process-based models (Urban 2005), so that larger scale, simplified models successfully retain those details that really matter. Increased computer performance and the increasing application of advanced programming techniques in ecological modelling will facilitate rapid developments in scaling methodologies (van de Koppel, Gupta & Vuik 2011), and the results of this study highlight the critical need for such future advances.