1. Dynamic simulation models are a promising tool for assessing how species respond to habitat fragmentation and climate change. However, sensitivity of their outputs to impacts of spatial resolution is insufficiently known.
2. Using an individual-based dynamic model for species’ range expansion, we demonstrate an inherent risk of substantial biases resulting from choices relating to the resolution at which key patterns and processes are modelled.
3. Increasing cell size leads to overestimating dispersal distances, the extent of the range shift and population size. Overestimation accelerates with cell size for species with short dispersal capacity and is particularly severe in highly fragmented landscapes.
4. The overestimation results from three main interacting sources: homogenisation of spatial information, alteration of dispersal kernels and stabilisation/aggregation of population dynamics.
5. We urge for caution in selecting the spatial resolution used in dynamic simulations and other predictive models and highlight the urgent need to develop upscaling methods that maintain important patterns and processes at fine scales.
Developing reliable methods that can be used to improve understanding and prediction of species responses to rapid environmental changes represents a key challenge for ecology. Such methods can provide a means for in-silico testing of alternative future management scenarios and thus may play an important role in informing policy makers and conservation organisations as they aim to devise and implement biodiversity management strategies that will be robust to future environmental changes. To date, the majority of studies focussing on predicting the response of species to climate change have utilised static, correlative ‘climate envelope’ or ‘environmental niche’ models (e.g. Araújo et al. 2005; Thuiller et al. 2005), which relate a species’ current distribution to current climate and then project the change in distribution based on the shift in suitable climate for the given species under future climate scenarios. The obvious shortcoming of these methods is the limited consideration of species’ true capacity to track the changing climate, as usually the potential for incorporating actual movements can only be achieved indirectly by assuming either no dispersal or full dispersal capacity (Thomas et al. 2004). Unsurprisingly, comparing the projections based on these two extremes reveals substantial uncertainty that originates from a range of poorly described ecological processes including dispersal, which affects the spreading potential of species, and the absence of population dynamics and interspecific interactions (Pearson & Dawson 2003; Heikkinen et al. 2006; Dormann 2007; Sinclair, White & Newell 2010).
In the last few years, there has been a move from static, statistical modelling approaches towards dynamic modelling where key ecological processes such as reproduction, mortality, dispersal and density dependence are explicitly incorporated (Keith et al. 2008; McRae et al. 2008; Anderson et al. 2009; Engler & Guisan 2009; Willis et al. 2009; Franklin 2010; Huntley et al. 2010; Midgley et al. 2010; Pagel & Schurr 2012). When coupled with niche modelling approaches, these dynamic models can be employed to simulate spatial population dynamics and dispersal processes with respect to future-suitable climate space, accounting for landscape structure (fragmentation), land-use change and potential conservation actions. The integration of both approaches (Dormann et al. 2012) offers the potential means for providing improved understanding and predictions of how biodiversity is affected by both climate change and habitat loss, whose synergistic effect has been clearly suggested by theory (e.g. Travis 2003; McInerny, Travis & Dytham 2007). Furthermore, it would allow much more robust assessment of the uncertainty around the projections to be obtained (Pagel & Schurr 2012) and to inform both further data collection and management strategies for facing conservation challenges that are dynamic in nature.
While this new approach offers considerable opportunities, some important caveats need to be carefully addressed before we begin applying these ‘second generation models’ to a range of important applied questions. Critical questions relate to the impacts of scales: it is vital to determine how sensitive the model outputs are to the spatial resolution at which we represent key patterns and processes (Holland et al. 2007a; Henle et al. 2010). There is a tension between the desire for realism, that is, incorporating details on landscape characteristics and spatial ecological processes at a high resolution and the limitations of feasibility if models are to run at spatial extents that are large enough to generate large-scale predictions. Yet, in the choice of resolution, we first need to understand whether there may be systematic biases in our predictions because of that choice.
In terms of the landscape, we are likely to implement the models on discrete raster landscapes with a resolution that may be determined by map availability or by limitations of feasibility. For example, land-use data for certain European countries (e.g. Finland, Sweden and the UK) are collected at a very fine resolution of 25 m; running a model at this resolution over large spatial extents introduces computational constraints potentially hampering the model implementation. Scaling up the maps may be inevitable, but what are the potential consequences of doing so? By aggregating cells, we lose information and this ultimately may have consequences for our projections. The same is true for the representation of local density dependence when simulating population dynamics: we are likely to model this process at the same resolution as we incorporate the landscape, but this resolution may be quite different to that at which density dependence, environmental or demographic stochasticity occurs in reality (Holland, Aegerter & Smith 2007b). Similarly, dispersal can be modelled in a range of different ways, but the nature of the simulated individual–landscape interactions will strongly depend on the quality of spatial details, especially if individuals are simulated as belonging to a cell rather than to an explicit coordinate in continuous space, as is often the case.
To provide insights into the different sources of potential biases owing to upscaling, we assess how model outcomes of range expansion dynamics are influenced by the resolution at which we represent (i) the landscape, (ii) the dispersal process and (iii) the population dynamics. Thereby, we seek to understand and quantify the effects of upscaling patterns and processes on our projections, to provide the potential means for developing modelling approaches, or upscaling approaches, which are free of inherent biases.
We used an individual-based model to simulate range dynamics on grid-based landscapes. The model mimics a situation where climatic changes have opened the opportunity for species to expand into new areas, so far unoccupied by the species. The species is thus located at the edge of an unoccupied area that is climatically and environmentally suitable. The dynamics of expansion into this region, however, will depend on the landscape structure, the dispersal capacity of the species, and its population dynamics.
To maintain generality of our investigation we chose to run simulations on artificial landscapes and for a range of hypothetical species, differing in dispersal capacities.
We modelled hypothetical species with discrete generations. Each time step t started with reproduction, where recruitment was determined by a stochastic, individual-based formulation of Hassell & Comins (1976) single-species population model. Each individual produced a number of offspring which was randomly drawn from a Poisson distribution with a mean μ given by
where λ is the finite rate of growth, N(x,y,t) is the local population size at time t, and K(x,y) is the cell’s carrying capacity. Density regulation therefore occurred at the cell level. The parameter b determines the type of competition (Travis et al. 2009) which in this study was kept constant and equal to 1·0, hence simulating ‘contest’ competition and compensatory dynamics. After reproduction, all adults died. At the next time step, all offspring had to decide whether to disperse or not and, if dispersing, they moved simultaneously and without affecting each other’s movements. Those landing in suitable habitat cells would then join the ‘residents’ prior to the next reproduction event (i.e. next generation).
Each individual had a constant emigration probability of 0·1. If the individual dispersed, the distance and the movement direction were determined in continuous space; the distance was drawn from a dispersal kernel represented by a negative exponential distribution with a given mean δ, and the direction was selected randomly from a uniform distribution between 0 and 2π. If the arrival point was outside the landscape, distance and direction were redrawn. The starting location for each disperser was determined by selecting random coordinates within the natal cell; the individual was then displaced from this point to an arrival cell, switching back from a continuous space representation of movement to the grid discrete space in which we represent the population dynamics process. If the habitat proportion in the arrival cell was <1%, the individual was assumed to die. Additionally, each dispersing individual had a constant mortality probability of 0·1.
Landscape Generation and Upscaling
We produced discrete fractal landscapes (i.e. neutral landscape models) by applying the midpoint displacement algorithm (Saupe 1988; With 1997) where the landscape structure was characterised by two parameters: the proportion of landscape occupied by suitable habitat (p), and the degree of spatial autocorrelation (Hurst exponent, H) which ranges from 0·0 (low autocorrelation but still not completely spatially independent) to 1·0 (high autocorrelation, i.e. high habitat aggregation). The resulting landscapes (at the finest resolution) were binary in the sense that they divided into suitable versus nonsuitable cells, the former having a habitat cover of 100%. The habitat coverage determined the species’ carrying capacity (K) for the cell.
For this study, we generated maps of 2049 rows and 1025 columns with a resolution of 25 × 25 m (total length = 51·225 Km, total width = 25·625 Km), with four values of p (=0·1, 0·3, 0·5 and 0·7) and three of H (= 0·1, 0·5 and 0·9, i.e. high, medium and low level of fragmentation). For each parameter combination, we produced 50 maps, yielding in total 600 landscapes. We then upscaled these maps to coarser resolutions of 100, 250, 500 and 1000 m, taking the typical upscaling approach where the habitat area in all cells at the finer resolution is summed up and the habitat proportion is recalculated for the cell at the coarser resolution. The carrying capacity of the coarser cell is calculated as the sum of the carrying capacities of the cells at finer resolution.
Simulations and Analyses
On each of the landscapes described above, we ran one simulation for a set of hypothetical species with mean dispersal distances δ of 25, 100, 250, 500 or 1000 m. All species had a fixed growth rate r of 1·2 and carrying capacity K of 160 individuals/ha (i.e. 10 individuals for the finest cell resolution), both values being arbitrarily chosen. At the beginning of every simulation, we populated the first 2 km of the landscape’s length, placing in each suitable cell a number of individuals equal to the cell’s carrying capacity. The simulation was then run for 100 generations to allow range expansion to occur. At the end of each run, we registered the range margin position along the landscape length, the total population size and the proportion of occupied cells.
We considered three potential causes of bias in the predicted range expansion when upscaling landscape maps: (i) the change in landscape structure because of cell aggregation (i.e. homogenisation of spatial information), (ii) the effect on dispersal because of the loss of spatial structure and (iii) the effect on population dynamics because of decreased demographic stochasticity when simulating larger populations per cell. To disentangle the effects of these components, we performed three analyses. First, we assessed how the habitat availability changed with cell size, in terms of proportion of cells that contained any suitable habitat and whether the landscape characteristics, as determined by p or H, affected this change across resolutions. Second, we performed simulations where we upscaled the landscape maps for the dispersal process, but simulated population dynamics over the original map with the finest resolution (25 m). In these simulations, the starting point of each dispersing individual was no longer drawn from its exact location within the fine-resolution map where reproduction occurred, but instead was randomly drawn from any point within an upscaled cell size resolution (100, 250, 500 and 1000 m). The arrival point of individuals, however, was re-assigned to the cell at 25 m resolution correspondent to the continuous coordinates. If landing in a hostile area, the individuals were assigned to a randomly chosen 25 m cell within the correspondent coarser cell. If no suitable cell was available within the coarser resolution cell, the individuals were assumed to die. Thus, the modification of dispersal kernels owing to upscaling was deliberately separated from the effects of increased dispersal success because of the homogenisation of the habitat cells and of upscaled population dynamics.
In a third and complementary analysis, we upscaled the population dynamics but the maps remained at their original resolution of 25 m when simulating the dispersal processes. Here, upscaling meant that the recruitment (i.e. density dependence) was affected by the carrying capacity of the coarser cell (i.e. assuming interactions between all individuals in the cell) with increasing resolution from 25 m to 100, 250, 500 and 1000 m, but the process of dispersal still occurred on the finest 25 m resolution map. To this end, individuals emerging from the reproduction process were assigned to their parent’s habitat cells within the fine-scale map.
Simulated range expansion dynamics were strongly affected by the upscaling process. Figure 1 illustrates exemplary simulations, demonstrating the effect of upscaling for landscapes differing in spatial structure. The increase in cell size resulted in a substantial increase not only in the range expansion relative to the original resolution, but also in the proportion of occupied cells and the overall population size.
The dispersal capacity of species had a substantial effect on the magnitude of the expansion (Fig. 2): not surprisingly, the further the individuals could disperse, the further the species expanded. In terms of the effects of upscaling, while for species with short dispersal distance the overestimation of the range expansion ‘accelerated’ as the cell size increased (Fig. 2a, grey), for species with high dispersal capacity the overestimation decelerated with cell size (asymptotic effect; Fig. 2a, black). Overall, however, the magnitude of the overestimation (expressed as the difference between estimates of range expansion at 1000 and 25 m resolution) increased with the dispersal capacities of the species (Fig. 2b). Yet, it was not only the distance, or the extent of the range expansion that was overestimated, but also the population size and the proportion of the landscape occupied (Fig. 2c), with increasing overestimation as the species’ dispersal capacity increased (Fig. 2d).
The landscape structure had a strong effect in landscapes with low habitat cover (p = 0·1), where greatest overestimation occurred for high level of fragmentation (H = 0·1), particularly for species with high dispersal capacities (Fig. 3, top-right panel). When habitat cover was relatively high (p = 0·5 or 0·7), we found a smaller effect of landscape structure (both p and H) on the extent of the range expansion overestimation (Fig. 3, lower 4 panels).
Effect of Upscaling the Different Processes
Upscaling only dispersal or population dynamics yielded different results in terms of the overestimation of range expansion, depending on the dispersal capacities of the species and on the landscape structure. For species with small dispersal capacity (δ = 25 m), it was the upscaling of the dispersal processes that led to a greater bias (relative to the upscaling of the only population dynamics), while for species with larger dispersal capacity, upscaling the population dynamics had a much greater effect in most cases (Fig. 4, black lines).
The interaction between habitat fragmentation and proportion of available landscape influenced the relative importance of dispersal and population dynamics. In landscapes with low habitat cover and high fragmentation (p = 0.l and H = 0·1, Fig. 4 upper panels), upscaling dispersal explained a larger proportion of the overestimation, regardless of δ. For higher values of p and H (more clustered landscapes; Fig. 4 lower panels and Fig. S1), dispersal was more important for species with small dispersal distances, but for species with larger dispersal capacity upscaling population dynamics yielded greater overestimation. Interestingly, in some cases, upscaling dispersal alone led to underestimating range expansion at coarser resolutions, especially for species with moderate or high dispersal capacities (δ = 250 and 1000 m) in more clustered landscapes (H = 0·9). Overall, regardless of the dispersal characteristics or the landscape structure, the combined effect of upscaling both dispersal and population dynamics yielded the greatest bias, indicating a strong combined effect of the two processes (Fig. 4, solid lines).
Upscaling separately dispersal or population dynamics also led to differences in the total population size and, particularly, upscaling the population dynamics yielded a larger number of individuals (Fig. 4, solid and dotted grey lines). When only the dispersal process was upscaled, the effect of demographic stochasticity became evident (Fig. 4, dashed grey lines): in these cases upscaling always led to a lower number of individuals regardless of the species’ dispersal capacities, because of the combined effect of high local extinction probability and increasingly global dispersal. Interestingly, for species with short dispersal distances, upscaling only dispersal led to higher estimates of range expansion (relative to the upscaling of population dynamics only), even if the total number of individuals was much smaller.
For aiding a better understanding of the results, it is worth considering how the upscaling process influences the realisation of dispersal kernels over a grid-based system. Because of the discretisation of the landscape and the consequent loss of the exact location of individuals within cells, increasing cell size stretches the distribution of movement distances, increasing the probability that individuals move further. This leads to overestimating dispersal distances (Fig. 5), particularly for species with small dispersal capacity in comparison with cell size.
Effect of Upscaling Habitat Patterns
Further understanding of the consequences of upscaling is facilitated by examining the impacts on the landscape characteristics themselves. Aggregating habitat cover caused a substantial increase in the proportion of cells that contained at least some suitable habitat (Fig. 6). This effect was enhanced by the interaction between reducing habitat cover (p) and increasing habitat fragmentation (H), resulting in the aggregation of small and isolated patches.
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).
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.
We acknowledge funding from the EU FP7 project SCALES (‘Securing the Conservation of biodiversity across Administrative Levels and spatial, temporal and Ecological Scales’, project no. 226852; Henle et al. 2010). We are grateful to two anonymous reviewers whose comments helped improve the manuscript.
GB, RH & JT conceived the study, GB did the modelling work, analysed the outputs, and wrote the first draft of the manuscript to which all authors contributed revisions.