Spread rates on fragmented landscapes: the interacting roles of demography, dispersal and habitat availability

Introduction

Understanding and predicting how species will spread across landscapes have become major goals within spatial ecology (Hastings et al., 2005). Substantial effort related to these broad goals focuses both on the range expansion of invasive species (With, 2002; Jongejans et al., 2008; Miller & Tenhumberg, 2010) and on native species shifting their distributions into newly suitable climate space (Dullinger et al., 2012; Pagel & Schurr, 2012; Steenbeek et al., 2016). Classic theory on the rate of spread of expanding populations has illustrated the importance of reproductive rate and dispersal (Skellam, 1951; see Hastings et al., 2005); indeed, these are the two key processes driving the spread rates predicted by many models, including many that are widely used in a predictive context today (e.g. Jongejans et al., 2011; Leroux et al., 2013; Matlaga & Davis, 2013; Zhou & Kot, 2013). Many of these models make the simplifying assumption of a homogenous landscape. Clearly, for most species this is unrealistic, especially in increasingly anthropogenically modified landscapes. There is a general expectation that species will spread less rapidly over landscapes with less habitat cover (Schwartz, 1993; Collingham & Huntley, 2000; With, 2002; Dewhirst & Lutscher, 2009; but see Bocedi et al., 2014a) and that, when the amount of habitat is above the extinction threshold, higher fragmentation could enhance range expansion, especially for species with good dispersal abilities (McInerny et al., 2007; Hodgson et al., 2012). However, we still lack a robust body of theory that makes clear predictions as to how the amount and spatial arrangement of suitable habitat will influence species’ spread rates.

Theory on the spatial spread of populations across homogenous landscapes is well developed, and, in the main, this knowledge has been acquired by analytical modelling approaches. As long ago as the 1950s, Skellam (1951) demonstrated that, assuming indefinite or logistic growth and random displacement, populations would achieve an equilibrium spread rate determined solely by the intrinsic growth rate and the mean square dispersal per generation. Amongst many insights generated by further mathematical modelling of range expansions that have built on Skellam's pioneering work, key ones include the following: the demonstration that fat-tailed dispersal kernels generate faster and potentially ever-accelerating spread rates (Kot et al., 1996); that Allee effects can substantially reduce spread rates (Veit & Lewis, 1996; Keitt et al., 2001; Tobin et al., 2009); that ignoring stage structure, whereby different population stages contribute differently to demographic and dispersal processes, can result in substantial overestimation of spread rates (Neubert & Caswell, 2000; Clark et al., 2001); and, most recently, that intraspecific phenotypic variability, in terms of growth and dispersal rates, can increase spread rates, especially in temporally variable environments (Elliott & Cornell, 2012, 2013; Meeker, 2013).

This theory has provided an improving understanding of how a species’ life history characteristics, including dispersal and interindividual variability, and environmental stochasticity, influence spread rates across homogenous landscapes. However, we currently have far less understanding of how rates of population spread will be impacted by habitat loss and fragmentation and how the extent of the impact will depend upon a species’ demographic and dispersal attributes. Taking a reaction–diffusion approach, Shigeseda et al. (Shigesada et al., 1986; Shigesada & Kawasaki, 1997) explored the spread rate of a species characterized by periodic variation in either diffusivity or growth rate. They established that, at least for this stylized environmental variability, the rate of spread is provided by the geometric mean of the rates of spread in the different environments. Notably, very recent extensions of the integro-difference modelling approach are also showing considerable promise in terms of including spatial heterogeneity (Gilbert et al., 2014a,b; Harsch et al., 2014; Musgrave & Lutscher, 2014), although here again the environmental variability is highly stylized, most often assumed to be periodic. For instance, Gilbert et al. (2014a) developed second-order approximations to wave speeds for non-homogeneous landscapes; Gilbert et al. (2014b) demonstrated the importance of choosing adequate dispersal kernels to study the relationship between landscape structure and spread rates, and Harsch et al. (2014) noted that in a moving habitat model the speed at which patches move can influence population growth rates and survival, as well as influence the importance of fecundity and the effects of long-distance dispersal. As these analytical approaches develop, we can also make complementary use of spatially explicit simulation modelling to gain additional insights into how habitat loss and fragmentation will influence spread rates.

Simulation models, both patch occupancy (With, 2002; e.g. Travis, 2003; McInerny et al., 2007) and individual-based (Hodgson et al., 2012; Bartoń & Hovestadt, 2013; Watkins et al., 2015), have already been used to investigate the spread of populations across fragmented landscapes. Mostly, this has been for tactical applications (e.g. Higgins et al., 1996; Smith et al., 2002), although there is also an increasing number of examples where simulations have been used strategically to develop theory (e.g. With, 2002; McInerny et al., 2007; Dewhirst & Lutscher, 2009; Pachepsky & Levine, 2011; Hodgson et al., 2012). Tactical models have provided valuable information in the context of management strategies for particular invasive species (Smolik et al., 2010; Travis et al., 2011), understanding how well-connected habitat patches are for threatened species (e.g. Kanagaraj et al., 2013; Aben et al., 2014) and for the development of corridors and stepping stones for species of conservation concern (e.g. Landguth et al., 2012; Watkins et al., 2015). In addition, they have provided insights into the spread of species under climate change (e.g. Willis et al., 2009; Cobben et al., 2012). Strategic modelling has helped advance generic understanding of range expansions, showing, for example, that there is likely to be a nonlinear relationship between spread potential and habitat availability (With, 2002); that spread rates may, for a given amount of habitat available, sometimes be faster when habitat is more fragmented and the average gap size between patches is smaller (McInerny et al., 2007); and that, in fragmented landscapes, carrying capacity as well as growth rate can have an important role (e.g. Bocedi et al., 2014a). While we have increasing understanding of how different factors influence range expansions across landscapes, one aspect that remains quite poorly considered is the relative importance of demographic parameters, dispersal parameters and landscape characteristics on the rate of spread and, additionally, on how these factors interact. Some recent work (Pachepsky & Levine, 2011) has already indicated that effects of intraspecific competition can slow the spread of plant populations through patchy landscapes, by influencing how fast a population produces enough seeds to surpass gaps between habitat patches. This highlights the importance of interactions between demography and landscape heterogeneity in driving spread rates across patchy landscapes, emphasizing the need for greater effort to explore interacting effects.

Here, we use a newly available individual-based modelling platform, RangeShifter (Bocedi et al., 2014b), to examine how the rate of range expansion across patchy landscapes is determined by the amount and spatial pattern of suitable habitat, the reproductive rate and local carrying capacity of the species, and two key characteristics of dispersal, the emigration rate and the mean distance that emigrants travel. We seek general understanding on the relative roles of demographic, dispersal and landscape characteristics on driving the rate of expansion and examine whether there are important interactions between them. Additionally, we seek insight into how predictable are rates of range expansion (RRE), provided that we have good knowledge of a species’ demography, dispersal and habitat availability.

Methods

We used RangeShifter v1.0 (Bocedi et al., 2014b) to conduct a series of individual-based and spatially explicit simulations of a single species spreading across a set of landscapes varying in the amount of available habitat and spatial structure.

Using the landscape generator embedded within RangeShifter v1.0, we created different binary landscapes of 129 × 257 cells at 100 m resolution, across which we applied five levels of the proportion of suitable habitat (p) and five levels of the degree of spatial autocorrelation (each combination was replicated 20 times, totalling 500 landscapes; Table 1). The first level of spatial autocorrelation was random (i.e. no autocorrelation); the remaining four levels were fractal landscapes created using the midpoint displacement algorithm (Saupe, 1988), in which the degree of spatial autocorrelation was determined by the Hurst exponent (H).

The population model was an only-females model with non-overlapping generations. At each generation, individuals reproduced, adults then died and offspring dispersed or remained in their natal cell. The number of offspring M produced by a single individual in cell i at time t was drawn from a stochastic, individual-based formulation of Maynard Smith & Slatkin's (1973) population model:

M∼PoissonRmaxi,t1+(Rmaxi,t−1)×Ni,tKi,tbc(1)

where Rmax is the maximum growth rate, K_i,t is the carrying capacity, N_i,t is the current population size and b_c is the competition coefficient describing the type of density regulation. We assumed a compensatory density regulation (b_c = 1), which leads to the self-regulation of populations around carrying capacity (Münkemüller & Johst, 2007), and varied Rmax and K (Table 1).

Dispersal in RangeShifter v1.0 is explicitly modelled in three phases: emigration (the probability of an individual leaving its natal patch), transfer and settlement. We considered emigration to be passive and density independent, and therefore, the proportion of emigrants, em, remained constant throughout each simulation. The transfer of individual emigrants followed a negative exponential dispersal kernel with a mean dispersal distance, d, from which the distance travelled by each individual was sampled. The movement direction was sampled from a uniform distribution between 0 and 2π. The parameter d was also assumed to be constant during each simulation. However, both em and d were varied between simulations to simulate species with different dispersal traits. Settlement of the individuals in a new cell was determined by its suitability: if the arrival cell was unsuitable the dispersing individual died.

Results

Only a few populations became extinct before the end of the simulation (0.9% of the total number of replicates) and extinctions were more common in landscapes with low p and high H, probably due to inherent stochastic effects arising from the populations not being able to compensate demographically for poorly suitable and highly fragmented landscapes. Extinct populations were omitted from further analyses.

The total variance (adjusted R²) in RRE explained by only the six main effects was 55%, which was slightly increased to 56% by addition of interaction terms within factor groups (landscape, habitat quality, dispersal ability), but substantially more to 66% by further addition of all other between-group two-way interactions (Table 2). However, further addition of all three-way interactions led to a negligible further increase, and even if all possible interaction terms were included, the total variance explained was 67%, that is stochasticity in the replication of randomly generated landscapes together with demographic and dispersal stochasticity accounted for roughly one-third of the variance in RRE. On that basis, interaction terms involving three or more factors were relatively unimportant and are not considered further.

By far, the most important factor in determining RRE was the proportion of suitable habitat in the landscape; in terms of its mean square value, it had almost threefold the effect of the next highest ranked factor (Table 3). The proportion of emigrants, em, and the carrying capacity of the landscape, K, were the next most important factors. The highest ranked interaction terms were those including the highest ranked main effect p, that is p × em, p × K, p × d and p × Rmax, followed by those including the second highest ranked main effect, em, that is em × Rmax, em × d and em × K. Terms incorporating the degree of aggregation in the landscape (H) and interactions between dispersal distance (d) and the two habitat quality variables (Rmax and K) were mostly of relatively low importance; indeed, the effect of the degree of aggregation was essentially limited to slightly higher RRE on the fractal landscapes than on the random landscapes, and H itself had negligible effect (see example in Fig. S2).

Invariably, RRE increased as p increased and the rate at which it did so increased with increasing values of both dispersal parameters, em and d (Fig. 1). When p was low (0.1 or less), RRE was always very low (typically <10 m year⁻¹) and it increased roughly in proportion to the increase in p. Increasing em from 0.1 to 0.2 had a greater effect on the rate of increase of RRE with increasing p than increasing em further from 0.2 to 0.3 (Fig. 1a), whereas constant increases in d resulted in a proportional increase in the response of RRE to p (Fig. 1b). When dispersal capability was at its highest (em = 0.3, d = 400 m) and the proportion of suitable habitat was high (P ≥ 0.4), RRE almost invariably exceeded 50 m year⁻¹, and at the highest value of K = 50, RRE could range up to 100 m year⁻¹ (i.e. one cell per generation) (data not shown).

The effect of habitat quality on the increase in RRE with increasing p was similar whether it was implemented through the demographic parameters Rmax or K (Fig. 2). For the highest value of p, decreasing Rmax by 20% and increasing or decreasing K by 25%, relative to their intermediate values, produced changes in RRE of similar magnitude, whereas increasing Rmax by 20% (relative to its intermediate value) had rather less effect, presumably owing to the effect of density dependence implemented in the fecundity model.

Discussion

Individual-based simulations provide a complementary approach to analytical methods for providing insights into the dynamics of species range expansion. Here, we have taken advantage of a recently released simulation platform (RangeShifter; Bocedi et al., 2014b) to explore the potentially interacting roles played by a species’ demographic characteristics, its dispersal characteristics and the structure of the landscape in determining the rate of range expansion. Our results highlight important interactions, in particular between the proportion of habitat availability and both the demographic and dispersal characteristics, as well as emphasizing that there are inherent limits to predictability.

Our results confirm past studies that demonstrate the effects of increased dispersal ability, increased growth rate, increased carrying capacity and increased amount of habitat on rates of spread. However, they point out important interactions that may have substantial implications for conservation: notably, the interaction between the proportion of available habitat and the probability of emigration. By distinguishing between two types of dispersal parameters (probability of emigration and mean dispersal distance), we were able to disentangle their different effects on the RRE, especially when low proportions of habitat are available.

For instance, at low proportions of suitable habitat (P < 0.2), species range expansion does not seem to be increased with improvement of habitat quality, nor with increased numbers of emigrant individuals. In this case, only increases in the mean dispersal distance produced slightly faster range expansions (considering fairly low fragmentation levels, H = 0.1; Fig. 1b). This is in accordance with the study by Musgrave et al. (2015), who have shown that the percentage of hostile patches at which populations collapse can depend on individuals’ movement behaviour, as introducing hostile patches can increase spread rates if individuals move fast enough.

We also found that for completely passive dispersers, as is the case of wind-dispersed seeds and some mites and spiders, the spatial arrangement of available habitat has a relatively minor effect on the projected rates of spread (at least relative to the amount of suitable habitat, its quality in terms of carrying capacity and growth rates and species’ dispersal characteristics). Furthermore, the degree of habitat fragmentation did not show important interaction effects with either habitat quality parameters or species dispersal ability. For example, the interaction between dispersal parameters (such as the probability of emigrating) and landscape configuration is more important regarding proportion of available habitat than the degree of fragmentation. The effects of habitat fragmentation on plants and plant community dynamics are not always generalizable and may be weak in certain cases, and, in general, habitat aggregation may not fully compensate for low habitat quantity and quality (Hodgson et al., 2011). For instance, habitat fragmentation was predicted to have only a slight effect on the expansion of the invasive biomass crop Miscanthus × giganteus when compared to the amount of suitable habitat and propagule pressure (Muthukrishnan et al., 2015). However, the importance of habitat fragmentation will certainly be higher for species with active dispersal behaviour, whose dispersal kernels likely result from strong interactions between dispersal behaviour and landscape structure (Baguette & Dyck, 2007; Doerr et al., 2010).

There is a lively current debate related to the relative importance of investing conservation resources in improving habitat quality, increasing the extent of suitable habitat or managing the matrix to improve the dispersal of organisms (Hodgson et al., 2009; Mortelliti et al., 2010; Doerr et al., 2011a; Synes et al., 2015). Results from simulations similar to those presented here can be useful in resolving this debate. For instance, although improving habitat quality is thought to lead to improvements in connectivity (Lawson et al., 2012), being thus an important conservation strategy in terms of climate change (Hodgson et al., 2009), we have not observed this phenomenon in our simulations of passively dispersing organisms. Rather, in situations where habitat quality is already above critical thresholds for species persistence, our results suggest that further improvement of habitat quality may not counterbalance the effect of highly fragmented landscapes, while increasing the proportion of available suitable habitat can (see Fig. S2). We note that this outcome can be quite different for active dispersers (see Bocedi et al., 2014a), where interactions between dispersal behaviour and landscape configuration are likely to affect dispersal cost and effective dispersal distances. In addition, active dispersers are likely to be influenced not only by the degree of habitat aggregation but also by the structural connectivity between patches (Hilty et al., 2006; Doerr et al., 2011a,b). This raises challenges for designing conservation strategies focused on promoting the spread of species responding to climate change that work across species with very different behaviours, as the degree of habitat fragmentation and connectivity between patches will have different consequences depending on the dispersal behaviour of the species.

Importantly, the general approach that we have taken here can provide answers to key questions such as how much would the amount of suitable habitat need to be increased to provide the same benefit (in terms of increased RRE) as increasing the quality of existing habitat (in terms of, for example, species growth rate or habitat carrying capacity). As illustration of this, let us take the example presented in Fig. 2(a) and assume that the starting point is 20% habitat available (p), a habitat quality that results in a carrying capacity (K) of 40 individuals and our hypothetical species having a maximum growth rate (Rmax) of 3.0. Here, managing the existing habitat such that it increases Rmax by 20% from 3.0 to 3.6 would yield an increase in RRE from approximately 17–18 m year⁻¹, while managing it to increase K from 40 to 50 would yield an increase to approximately 19 m year⁻¹. To gain an equivalent increase in RRE by increasing the extent of available habitat (assuming the additional habitat provides the same initial Rmax = 3.0 and K = 40) would require approximately 3% more of similarly suitable habitat. If the sole conservation objective was managing to optimize spread rates, the equation is then related to the relative costs and potential of improving the existing habitat compared to purchasing and managing the additional land. Although here we explored a set of hypothetical species, our approach can be used to investigate similar questions in real-world scenarios and help provide guidelines for management (Fraser et al., 2015; Sun et al., 2016), provided that species demographic and dispersal parameters are known, or can be approximated. Alternatively, our approach can provide the range of possible outcomes for species range expansion using a set of demographic, dispersal and/or habitat parameters that vary within a plausible range obtained from existing data (Santini et al., 2016; Sun et al., 2016). Given that we may expect different species groups to respond in different ways, a major future challenge will be extending this general approach for assessing the relative merits of alternative interventions to work for multiple species (Synes et al., 2015).

Notably, we find that around one-third of the overall variation in rates of range of expansion is not explained by the landscape characteristics, species demography, dispersal or interactions between them. Uncertainty in our results arises from differences between the generated artificial landscapes for a given set of landscape parameters and from stochasticity in the modelled demographic and dispersal processes. As in any modelling approach, stochastic processes were included to reflect natural unexplained variance that stems from unmeasured or unknown processes. Thus, we consider that the unexplained variance in our results represents irreducible uncertainty that is invariably encountered in empirical studies of RRE. This uncertainty arises from both the inherent stochasticity of species demography and dispersal, which can be particularly pronounced at expanding margins (Clark et al., 2001; Travis et al., 2011), as well as from idiosyncratic, stochastic variation of natural landscapes. In future work, it will be important to recognize that this inherent stochasticity places limits on predictability. Theory can usefully be developed to provide understanding of the conditions under which there will be greater inherent uncertainty in rates of expansion, while methods for effectively illustrating this inherent uncertainty when presenting results to stakeholders require development; moreover, we need a more active consideration of how this uncertainty influences conservation management planning.

Our study presents a first insight into how a species’ range expansion dynamics are determined by a combination of demographic and dispersal characteristics together with properties of the landscape. In order to be able to disentangle the effects of the six tested variables, we have made a number of simplifying assumptions, such as assuming no stage structure, passive dispersal, no adaptive or evolutionary mechanisms and a static environment. However, we believe that these factors should be explored in future studies to expand on the theoretical clarifications that we provide here. Extending our approach to actively dispersing species will be important, as the effects of landscape configuration may change for these species and even affect their dispersal behaviour. Previous theoretical work has already shown that species settlement strategies will interact with habitat availability (Bocedi et al., 2014a), and empirical studies have shown that movement speed (Barnes et al., 2015) and movement probability (Vasudev & Fletcher, 2015) depend on habitat characteristics. Considering the different dispersal capabilities across life stages and sexes is also likely to be relevant for many species. For instance, seed dispersal may lead to faster niche adaptation than pollen dispersal, when habitat heterogeneity and selection are not too strong (Aguilée et al., 2013). Simulating pollen and seed dispersal separately is undoubtedly an interesting expansion to our study, which allows further decomposing dispersal into its different phases. Female-biased dispersal can also change the speed of range expansion by increasing it, especially if there are mating opportunities before females disperse to patches at leading edges (Miller & Inouye, 2013). Adaptation will also play an important role in species expansions, especially if dispersal capabilities are enhanced at leading edges (Fayard et al., 2009; Travis et al., 2010), and may have critical implications in face of environmental change (Dytham et al., 2014). Although here we assumed that the species’ rate of expansion is not limited by climate, we highlight the importance of developing theory for how ranges expand when a species is shifting its range due to a shift in an environmental gradient across fragmented landscapes. Both individual-based simulations (e.g. Mustin et al., 2009; Dytham et al., 2014) and recent analytical developments (e.g. Potapov & Lewis, 2004; Harsch et al., 2014) have developed theory in this context. These studies have shown that population dynamics can influence range expansions under moving environmental gradients, with rates of colonization increasing the rate of range expansion (Mustin et al., 2009). They have also shown that the speed at which suitable habitat patches become available can decrease the importance of survival characteristics of the species and increase the importance of fecundity and growth (Harsch et al., 2014), while also increasing the critical patch size for species persistence (Potapov & Lewis, 2004).

However simplified our approach may be, existing theory has not considered the range of effects we have explored here. Future work is required to establish whether the relative importance of different effects and their interactions holds for expansions across environmental gradients and more complex population dynamics. In the context of recent calls for the development of approaches to improve our forecasts for species’ distributions under climate change (Urban et al., 2016), it is important that we focus not only on expanding fronts, but also consider the dynamics of range retraction. To date, there has been far less attention paid to the dynamics of retraction both for analytical and individual-based approaches, despite suggestions that negative rates of expansion can occur in populations with Allee effects in fragmented landscapes (Maciel & Lutscher, 2015) and that mismatches between species’ local adaptations and climate optimums can also lead to decelerating rates of expansion (Mustin et al., 2009). Future work should apply both approaches to address some key questions related to range retraction, including asking how the spatial lag of a species behind its ‘climate envelope’ depends upon demographic, dispersal and landscape characteristics. Such work can be very informative in helping to determine species and landscape characteristics for which we are most likely to see extinction debts (i.e. species persist for some period in time in regions where, even in the absence of further environmental change, they are doomed to extinction); this information can be extremely valuable in guiding the allocation of scarce resources for conservation actions.

Acknowledgements

This work was supported by the award of the Natural Environment Research Council grant NE/J008001/1 to J.M.J.T.

Biosketch

Ceres Barros is motivated to explore how different species, communities and ecosystems are capable of adapting to and withstand environmental change. Although her work has spanned from using lichens as biomonitors for atmospheric pollution, to theoretical developments of behavioural models and, lately, understanding ecosystem stability under climate and land use changes from different perspectives, she mostly sees herself as an ecological modeller who is drawn to the study of ecosystem stability, resilience and adaptation to disturbances.

Author contributions: J.M.J.T., C.B. and S.C.F.P. conceived the ideas; C.B. conducted simulations and analysis; G.B. and S.C.F.P. developed the RangeShifter software; C.B. led the writing of the manuscript, to which all authors contributed.