Red noise increases extinction risk during rapid climate change

Abstract

Aim

As the global climate is changing rapidly, there is a need to make conservation decisions to facilitate species' persistence under climate change. Models employed to make predictions regarding the impacts of climate change on species' distributions, and ultimately persistence, typically assume that interannual variability in environmental conditions is independent between years. However, the colour of environmental noise has been shown to affect extinction risk in populations occupying spatially static environments, and should therefore affect persistence during climate change. This study aims to investigate the importance of noise colour for extinction risk during climate-induced range shifts.

Methods

We use a spatially explicit coupled map lattice with a latitudinal gradient in climatic suitability, together with time series of environmental noise, to simulate periods of directional climate change and investigate the effects of noise colour on extinction risk and range size.

Results

Extinction risk increases with reddening of the environmental noise, and this effect is particularly pronounced over short time frames when climate change is rapid.

Main conclusions

Given that management decisions are typically made over such short time frames, and the rapid rates of climate change currently being experienced, we highlight the importance of incorporating realistic time series of environmental noise into models used for conservation planning under climate change.

Introduction

The global climate is changing rapidly, and impacts on species distribution and local persistence are now documented across environments and taxa (Walther et al., 2002; Parmesan & Yohe, 2003; Thomas, 2010). Given the need to make management decisions to facilitate species' persistence under climate change, models are often employed to make predictions regarding the future (e.g. Carey, 1996; Sykes et al., 1996; Hill et al., 1999; Berry et al., 2002; Thuiller, 2003; Wilson et al., 2005; Best et al., 2007; Brooker et al., 2007; Jiguet et al., 2007; Keith et al., 2008; Anderson et al., 2009). However, these models either ignore the interannual variability in environmental conditions (environmental noise) or assume that this variability is independent between years. Theoretical and empirical work has shown that population processes, and thus extinction risk, should be strongly affected by the ‘colour’ of environmental noise (e.g. Greenman & Benton, 2003, 2005a; Benton & Beckerman, 2005; Reuman et al., 2006; Ruokolainen et al., 2009).

By analogy with optics, time series of interannual environmental variation of different frequencies can be described by their colour (Fig. 1). Time series that exhibit no temporal autocorrelation are termed ‘white noise’; time series that are positively autocorrelated, and therefore characterized by low-frequency fluctuations, are referred to as ‘red noise’ (Fig. 1c), and time series that exhibit high-frequency fluctuations show negative autocorrelation and are ‘blue’ (Fig. 1b). Many measured time series of environmental noise are reddened over generational time-scales, more extremely in marine and coastal environments, whereas terrestrial environmental noise tends to fall somewhere between white and red noise (pink noise), and some environmental factors can exhibit extremely low-frequency variations (brown or even black noise) (e.g. Halley, 1996; Vasseur & Yodzis, 2004; Garcia-Carreras & Reuman, 2011).

Figure 1. Schematic to show how the wavelength of light increases as its colour changes from blue to red (a); and generated environmental noise time series, with amplitude σ_e = 0.1, for: (b) ‘blue’ noise (autocorrelation coefficient К = −0.95) and (c) ‘red’ noise (К = 0.95). By analogy with optics, environmental noise is referred to as having a ‘colour’. Where the wavelength is short, and the time series is characterized by high-frequency variation as a result of negative temporal autocorrelation, the noise is termed ‘blue’. Where the wavelength is long, and the time series exhibits low-frequency variation as a result of positive temporal autocorrelation, the noise is termed ‘red’. ‘White’ noise is used to describe environmental time series that are not temporally correlated.

The modern ecological synthesis accepts that population dynamics arise as a combination of density-dependent and density-independent effects (Bjørnstad & Grenfell, 2001). A corollary of this is that all organisms' dynamics are affected by the way that the environment varies. Specific studies on the relationship between noise and dynamics have included a number of taxa including birds, mammals and plants (e.g. Benton et al., 1995; Freckleton & Watkinson, 1998; Engen et al., 2001; Carroll, 2007; Hilderbrand et al., 2007; van de Pol et al., 2011). Given that individual life histories, and therefore population dynamics, integrate over time periods of years or generations, the low-frequency component of environmental variation is likely to be particularly important in extinction dynamics (because a sequence of poor years is likely to have a strong cumulative effect on population size) (Ripa & Lundberg, 1996; Johst & Wissel, 1997; Petchey et al., 1997; Heino, 1998; Greenman & Benton, 2003; Schwager et al., 2006). Depending on the underlying population dynamics, reddening of the environmental noise can increase extinction risk due to long runs of ‘bad’ years or decrease extinction risk due to the relatively lower probability of an extremely bad year in any given time period relative to white noise (Ripa & Lundberg, 1996; Petchey et al., 1997; Ripa & Heino, 1999; Schwager et al., 2006). Given that most species exhibit ‘undercompensatory’ dynamics, as a result of contest competition for resources, extinction risk over a given time period will generally be underestimated if the environmental noise is assumed to be white (Petchey et al., 1997). The effect of reddening of the environment on extinction probability also varies by life history strategy and stage, community abundance rank, the nature of interspecific interactions and the strength of the correlation between the responses of individual species within a community (Heino & Sabadell, 2003; Ruokolainen et al., 2007; Ruokolainen & Fowler, 2008). For example, Heino & Sabadell (2003) found that reddening of environmental noise decreases extinction risk in annually reproducing species, but increases extinction risk for semelparous and iteroparous biennial, and perennial reproducers.

In spatially structured populations, where key demographic rates or life history characteristics vary through space, reddening of the noise generally increases the global extinction risk, even though local extinction risk may decrease, regardless of the pattern of population dynamics (Petchey et al., 1997; Heino, 1998). This is because the spatial heterogeneity in patch quality means that when an unfavourable environmental event occurs, populations in better-quality patches will be more likely to persist than those in lower-quality patches. These patches then act as sources, once conditions improve, to recolonize poorer-quality patches from which the population has gone locally extinct. This is referred to as the ‘rescue effect’ (Brown & Kodric-Brown, 1977). The key point is that conditions will improve more quickly in an environment of white noise, whereas under reddened noise, there are more likely to be long runs of unfavourable conditions, increasing the likelihood of more patches going extinct, decreasing the pool of potential source patches and therefore threatening the viability of the metapopulation. This effect may be amplified by a high degree of spatial environmental correlation and increased noise amplitude (Palmqvist & Lundberg, 1998). Given the apparent importance of noise colour and amplitude in populations occupying spatially static environments, it may be expected that they will have important effects on species persistence and range dynamics under climate change. Much theoretical and conservation interest has been generated in the effects of habitat heterogeneity, spatial demography, spatial population dynamics and life history characteristics on range dynamics, species distribution and persistence under climate change (e.g. Travis, 2003; Opdam & Wascher, 2004; Carroll, 2007; Hilderbrand et al., 2007; Anderson et al., 2009; Mustin et al., 2009; Doxford & Freckleton, 2012; Urban et al., 2012).

Here, we combine a spatially explicit coupled map lattice (CML) model that incorporates a broad-scale latitudinal gradient in climatic suitability with different time series of environmental noise to investigate how the colour of environmental noise affects species' persistence and range size during rapid climate change.

Methods

The model consists of a landscape represented by a cellular lattice of 100 × 600 habitat patches, and all patches are considered to be potentially suitable for occupation. Within-patch dynamics are described by an individual-based formulation of Maynard Smith & Slatkins (1973) single-species population model. Each individual present in patch i at time t produces offspring, the number of which is drawn from a Poisson distribution with mean μ_i,t defined by:

• 

where r describes the maximum growth rate, N_i,t describes the number of individuals in patch i at time t, and K is the carrying capacity. After producing offspring, the individual dies, and a proportion of the offspring disperse. Dispersal is usually to any unoccupied patch from the nearest eight neighbouring patches.

Environmental variation through space and time

Latitudinal variation in mean climate is imposed by defining the position of optimum climatic conditions as a row on the lattice (E_max) and assuming that conditions decline sigmoidally away from this optimum in both directions:

• 

where E_j,t is the mean climate suitability at row j in time t, d_j,t is the distance of row j at time t from E_max, h is the half-saturation constant that defines the distance at which E_j,t = 0.5, and s is a shape parameter that defines the direction (negative values of s give a negative slope and vice versa) and shape of the curve (when s > 1 the curve is sigmoidal). In all simulations presented here, h = 10 and s = −3. If d_j,t is zero, then E_j,t is set as 1. This approach is similar to that used by Mustin et al. (2009), and the model has been extended to incorporate environmental noise around this mean, which varies temporally but not spatially. Time series were produced using r, version 2.10.1 (R Development Core Team). The environmental noise is generated using an autoregressive (AR) method (Ripa & Lundberg, 1996):

• 

where К is the autocorrelation coefficient, and ω_t is a standard normal random component (mean = 0, variance = 1). Previous studies have highlighted that the method used to scale the variance can affect the outcome of modelling exercises (Johst & Wissel, 1997; Heino et al., 2000; Wichmann et al., 2005), and so, the method used here scales the noise to have a variance of w² after Wichmann et al. (2005):

• 

where σ_e defines the standard deviation of the noise process (and can therefore also be used to scale the amplitude of the time series), w is the standard deviation of a white noise time series, and ε' is the unscaled time series produced by the AR process. is the long-term mean of ε'.

The resulting spatio-temporal variation is related to within-patch dynamics by assuming a direct relationship between environmental conditions and intrinsic population growth rate r:

• 

where r_j,t is the growth rate for patches in row j at time t, and r is the expected growth rate in the absence of environmental forcing.

Simulation experiments

The optimum climatic conditions are set to occur at one row on the lattice, and the model was run for 500 time steps. The optimum was then moved unidirectionally for the next 1500 time steps to simulate a period of directional climate change. The model is stochastic, and to obtain a good representation of the system's behaviour, 250 independent realizations were run for each parameter set.

To assess the impacts of different colours of environmental noise on extinction risk under climate change, we consider two conditions. Firstly, we model relatively rapid climate change (0.33 rows^t−1) and assess the extinction risk over time-scales that are relevant to conservation management (30, 50 and 100 years). Secondly, we model slower climate change (0.25 rows^t−1) and assess the extinction risk over long time-scales as a theoretical exercise. All simulations were run for values of κ of 0.99 (extremely reddened), 0.95 (red), 0.9 (red – see Fig. 2b), 0.75 (pink), 0.5 (pink) and 0 (white – see Fig. 2a).

Figure 2. Generated environmental noise time series for (a) autocorrelation coefficient К = 0, noise amplitude σ_e = 0.1; (b) К = 0.95 σ_e = 0.1; (c) К = 0.95 σ^e = 0.05; (d) К = 0.95 σ_e = 0.2.

Sensitivity analyses were conducted to examine the robustness of our findings to:

1. varying amplitude of environmental noise (σ_e of 0.05, 0.1, 0.15 and 0.2); and
2. dispersal neighbourhood (nearest 8, 24 or 48 neighbours and nearest eight neighbours but with 5% of individuals able to disperse globally).

Results

For a given rate of climate change, the extinction risk increases with reddening of the environmental noise (Fig. 3a). However, the average size of extant ranges is larger when the noise is extremely reddened (К = 0.99; Fig. 3b).

Figure 3. (a) Number of extant runs of the simulation against time, and (b) mean size of extant ranges through time where the noise is increasingly reddened [autocorrelation coefficient К of 0.5 (solid black line), 0.75 (dashed line), 0.9 (dotted line) and 0.99 (solid grey line)]. Results are not shown for white noise (К = 0) as no extinctions occurred. Here, the rate of climate change is 0.33 rows ^t−1, noise amplitude σ_e = 0.2, and dispersal is to the nearest eight neighbouring patches only. The vertical dashed line shows the onset of climate change.

When climate change is relatively rapid (0.33 rows^t−1), the rate of extinction (number of simulations that went extinct per time step) increases substantially with noise reddening over management-relevant time-scales (30, 50 and 100 years; Fig. 4a). When climate change is relatively slow (0.25 rows^t−1), the extinction rate is only slightly increased over a long time-scale (1500 years; Fig. 4b).

Figure 4. Extinction rate under: (a) rapid climate change (0.33 rows ^t−1) over 30 (squares), 50 (triangles) and 100 (circles) years; and (b) slow climate change (0.25 rows ^t−1) over 1500 years, where the noise is increasingly reddened (autocorrelation coefficient К of 0.5, 0.75, 0.9 and 0.99). Results are not shown for white noise (К = 0) as no extinctions occurred. In both sets of simulations, noise amplitude σ_e = 0.2, and dispersal is to the nearest eight neighbouring patches only.

Decreasing the amplitude of the environmental noise reduces the extinction risk such that the number of simulations where the range persists through rapid climate change (0.33 rows^t−1) increases from 3 (1.2%) where σ^e = 0.2 to 183 (73.2%) where σ_e = 0.1; however, this sensitivity does not qualitatively change our result, and extinction risk is higher under red than white noise for any given noise amplitude (Fig. 5).

Figure 5. Number of extant runs of the simulation against time under (a) red noise (autocorrelation coefficient К = 0.95) and (b) white noise (К = 0), for noise amplitude σ_e = 0.05 (solid black line), 0.1 (dashed line), 0.15 (dotted line) and 0.2 (solid grey line). Here, climate change is at a rate of 0.33 rows ^t−1, and dispersal is to the nearest eight neighbouring patches only. The vertical dashed line shows the onset of climate change.

Increasing the dispersal neighbourhood, or assuming some long-distance dispersal events to anywhere on the lattice, decreases the extinction risk under red noise such that the number of simulations where the range persists through rapid climate change (0.33 rows^t−1) increases from 183 (73.2%) where dispersal is to the nearest eight neighbouring patches, to 239 (95.6%) where 5% global dispersal occurs (Fig. 6a). This sensitivity does not qualitatively change our result that the extinction risk is higher under red than white noise; however, widening the dispersal neighbourhood to the nearest 24 or 48 patches reduces extinction to zero over the 2000 time steps modelled (Fig. 6a). Furthermore, there is a reduction in range size during climate change under all dispersal scenarios (Fig. 6b).

Figure 6. The effect of varying dispersal strategy, under red noise (autocorrelation coefficient К = 0.95), on: (a) number of extant runs of the simulation against time, with dispersal to the nearest 24 or 48 patches (dashed line), the nearest eight patches (solid line) or with 5% global dispersal with the rest to the nearest eight patches (dotted line); and (b) mean size of extant ranges through time, with dispersal to the nearest 24 patches (dashed line), the nearest 48 patches (solid grey line), the nearest eight patches (solid black line) or with 5% global dispersal with the rest to the nearest eight patches (dotted line). The rate of climate change is 0.33 rows ^t−1, and noise amplitude σ_e = 0.1. The vertical dashed line shows the onset of climate change.

Discussion

For our model of a spatially explicit population, with no age or stage structure, the risk of extinction during a period of directional climate change is increased when the environmental noise is reddened (Fig. 4). This is true for both rapid climate change over short time periods of relevance to management decisions (Fig. 4a), and slow climate change over much longer time periods (Fig. 4b). These effects are probably due to the greater probability of consecutive time steps of poor environmental conditions when noise is reddened relative to uncorrelated time series of white noise, such that the population has no chance to recover. Most environmental noise is reddened (Halley, 1996; Vasseur & Yodzis, 2004), in particular air and sea-surface temperatures, which have been found to have spectral exponents (here equivalent to К) of 0.5–1.5 on average (Vasseur & Yodzis, 2004), and our results therefore have important implications for species persistence under climate change.

The impact of climate change has typically been assumed to relate to the way in which the ‘envelope’ of mean climatic conditions is shifting through time and space. However, there has been a recent resurgence in interest in the impacts of variability in weather relative to these climatic means, and how the frequency or magnitude of extreme weather events might increase as a result of ongoing climate change (Coumou & Rahmstorf, 2012). The variability is expressed as the shape of the distribution of weather around the climatic mean, and empirical data are suggesting that the width of this distribution is moving 2–2.5 times faster than the mean climatic conditions (Hansen et al., 2012); thus, globally, weather is becoming more variable. Given this rapid increase in the frequency and magnitude of extreme events, it is now essential that species distribution modelling takes into account the amplitude and colour of environmental noise to make projections regarding future distribution and persistence under climate change. Specifically, we expect that for many species the predicted threshold rate of climate change for persistence over a given time period will be overestimated if the colour of environmental noise is not considered.

For many biological phenomena, such as growth, as a function of temperature or light, fecundity and population growth, underlying processes are essentially geometric as opposed to arithmetical, and therefore, variance in parameter values has a significant impact on the eventual outcome, especially population persistence. Given that population persistence is typically the goal of conservation management actions, it is therefore essential to account for environmental noise in models that seek to predict future distributions and persistence and that are being used to assess the efficacy of different management options. As a concrete example, the spring of 2012 was characterized as a drought in NW Europe, being amongst the driest on record. This impacted a range of processes across taxa, such as reproduction, seedling emergence and dispersal. The subsequent summer, however, has been amongst the wettest on record, which has impacted juvenile survival, seed set and organismal condition. Overall, however, the total rainfall may emerge as close to average. Modelling this climatic average would mask the effects of the extreme dry and wet periods on population processes and would therefore severely underestimate the effect of the environmental conditions on predictions of range shifting and population persistence. Making predictions for future persistence under predicted climate change, using only mean climatic conditions and ignoring the likely increased variability, will similarly underestimate extinction risk and therefore potentially bias the investment of resources for conservation management.

Interestingly, however, we also find that when the noise is extremely reddened (К = 0.99), the average size of extant ranges is larger than under any other noise conditions (Fig. 3b). This pattern is probably a reflection of the possibility that subpopulations can persist where the average climatic conditions have become unsuitable due to the directional climate change, because consecutive ‘good years’ improve conditions for the species. This potential importance of the ‘trailing edge’ for overall patterns of range change and extinction risk under climate change has also been highlighted elsewhere (Hampe & Petit, 2005). Hampe & Petit (2005) suggest two extremes of ‘behaviour’ at the low-latitude distribution edge: ‘trailing edges’ where populations become extirpated as a result of latitudinal displacement of a species range, and ‘stable rear edges’ where the overall species range expands as a varying fraction of the populations at the rear edge are able to persist. We find both patterns in our results, with the most extremely reddened noise (К = 0.99) producing patterns more akin to ‘stable rear edges’ and less reddened noise (К = 0.5–0.9) producing a pattern more akin to ‘trailing edges’. Further empirical work is required to understand the importance of these rear edge populations, across taxa, under climate change, and currently most evidence comes from studies of perennial plants (Hampe & Petit, 2005 and references therein). In common with previous findings (Johst & Wissel, 1997; Petchey et al., 1997; Heino, 1998; Schwager et al., 2006), we have also shown that when there is no directional climate change and the amplitude of the noise is sufficiently high, extinction is more likely in red than white environments (Fig. 3a).

We have shown that reddening of the environmental noise increases extinction risk in a spatially structured population during a period of climate change. Another important source of structure in populations is the age or stage structure, which refers to the number of individuals of different age classes or stages (i.e. adult versus juvenile) and the probabilities with which they move from one age or stage to another. We use a population model that has no age or stage structure, such that the noise at time t has an impact only on the population growth rate at time t. The focus of this research was the impact of environmental noise colour, during climate change, on range dynamics of a spatially structured population, and hence, we chose to use a simple population model without stage structure. However, in reality, life histories are shaped by environmental conditions throughout life, and maternal effects, and environmental conditions early on in life have been repeatedly shown to produce prolonged effects during organisms' lifetimes. The impact of environmental noise (weather variation around the climatic mean) on dynamics acts through the colour of the resulting population dynamics, which may be linearly related to the environmental noise, as is likely to be the case in models without age or stage structure so that red environmental noise produces red population dynamics. If the dynamics are ‘reddened’, a run of bad years may drive the population extinct. However, in age- or stage-structured models, the colour of the resultant population dynamics can be quite different than the colour of the environmental noise. In part, this results from poor environmental tracking, where demographic rates do not respond linearly to the colour of environmental noise but rather ‘filter’ the noise and change its colour. For example, in stage-structured models, blue environmental noise (negatively temporally autocorrelated) may result in red population dynamics due to the lagged effects inherent in modelling the life history, increasing extinction risk (Greenman & Benton, 2005b). It has been suggested that in such cases the colour of noise will be less important than either the mean environmental change or the extent of the interannual variability (amplitude of the noise) in determining the mean time to extinction (van de Pol et al., 2011). Our results are certainly quantitatively sensitive to the amplitude of environmental noise and rate of climate change (Figs 4 and 5); however, the qualitative effect whereby extinction risk increases with red noise is unchanged and is in common with previous findings (e.g. Ripa & Lundberg, 1996; Petchey et al., 1997; Heino, 1998; Ripa & Heino, 1999; van de Pol et al., 2011). Given that the frequency and magnitude of extreme weather events is predicted to increase under future climate change (IPCC, 2007), our results suggest that in reality extinction risk will increase under future climate change as the amplitude of the environmental noise increases. The utility of our approach is not to say only red noise is important in determining extinction risk under climate change, but more to highlight that the colour of the dynamics (whether driven by coloured noise, or the filtration of noise through the life history) is important to consider. This will particularly be the case for species with more limited dispersal distances and especially those with narrow climatic tolerances such as amphibians, which are more vulnerable to climate change due to an inability to rapidly migrate and keep pace with their necessary, and shifting, climatic conditions (Fig. 6 and Trakhtenbrot et al., 2005; Araújo et al., 2006). However, species with wide dispersal neighbourhoods, or those capable of long-distance dispersal events, such as many bird species, will be more likely to persist (Fig. 6 and Trakhtenbrot et al., 2005).

We suggest four possible extensions to the work presented here. Firstly, there is evidence that the colour of environmental noise may be redder at high and low latitudes compared with temperate latitudes (Vasseur & Yodzis, 2004). This is likely to have important implications as it may lead to, for example for more northerly distributed species, more reddened noise at the leading edge compared with the trailing edge. The greater stochasticity at the trailing edge may allow for persistence over much longer time-scales in environments that are, on average, unsuitable. This would in turn impact on range extent and persistence, and from a conservation perspective, this may also necessitate different management actions in different parts of the species range: for example, assisted colonization (e.g. Hoegh-Guldberg et al., 2008; Willis et al., 2009) at the leading edge versus habitat management or reduction in other threats at the trailing edge. We therefore contend that an interesting extension to the work presented here would be the inclusion of spatial variation in the colour of environmental noise, and if parameterized for a real system, then the effect of different management options in different parts of the range could also be explored in a decision theory framework to find cost-effective management plans. Secondly, many species live in ephemeral habitats, characterized by destruction and regeneration of suitable habitat ‘patches’. For some species, this patch lifespan may be linked to climatic conditions, and for example, increases in the frequency of extreme weather events could reduce patch lifespan. One such species, the grasshopper Bryodema tuberculata, in central Europe survives only on gravel bars along braided rivers in the Northern Alps, a habitat characterized by succession and floods. Stelter et al. (1997) used simulation models to show that persistence time for metapopulations of this species is low if the time between floods is too short (because many subpopulations are washed away at the same time) or too long (because local subpopulations are eliminated by succession). The persistence of species in such dynamic landscapes has received much attention (Fahrig, 1992; Hanski, 1999; Keymer et al., 2000; Johst et al., 2002), and there would be merit in considering a possible interaction between changed frequency and magnitude of extreme climatic events and habitat patch destruction and regeneration. From a conservation perspective, it is possible to envisage two possible routes through which such changes might lead to population declines for species dependent on these ephemeral habitats: patches may have insufficient time to regenerate before the next destructive climate event as a result of increased frequency of such events; or multiple patches could be destroyed simultaneously as a result of increased magnitude of climate events, which could then reduce the probability of patch recolonization from neighbouring patches. Thirdly, previous studies have found that the effect of noise colour on extinction risk varies according to the interspecific interactions between species and structure of the community as a whole (Ruokolainen et al., 2007; Ruokolainen & Fowler, 2008). Here, we have considered a single-species model, and it is certainly reasonable to expect that extinction risk under climate change for any given species will be affected by the range dynamics of competitors, predators, mutualists and prey or resources. Therefore, extending the work presented here to include some of these potential interspecific interactions would provide further insights into the role of environmental variation in species extinction risk under climate change. Finally, and perhaps most importantly, there is a need to explore the impacts of environmental noise on stage-structured populations experiencing a period of climate change. With very few exceptions, organisms life histories are stage structured. Environmental noise affects individuals by either altering their survival or changing the pattern of investment in life history (i.e. trade-off between survival to reproduce in the following year and reproduction in this year). As a result, the impacts of environmental noise on population persistence will ultimately be a function of how and where the noise affects the organisms life history and how these effects filter through the population (e.g. Greenman & Benton, 2005b; Benton, 2006). Furthermore, impacts of noise will almost certainly be lagged as a result of ‘bad years’ as a juvenile affecting adult survival and life history allocations. For example, Benton et al. (2001) showed that when the transmission of maternal environmental conditions is the cause of delayed density dependence, the population variability increases, and in a stochastic environment, this is the result of the interaction between the delayed density-dependent effects and environmental noise. Furthermore, experimental work indicates that maternal effects can cross multiple generations and vary in their impact according to density (and thus food availability) (Plaistow & Benton, 2009). Therefore, an important extension to the work presented here would be to examine the effects of noise colour and amplitude in a stage-structured population, incorporating realistic lagged effects, during a period of climate-induced range shifting.

In conclusion, we have shown that in a spatially structured population, the colour of environmental noise helps to determine the size and ultimately persistence of the occupied range during a period of climate change. Given this result and the increasing need to make conservation decisions regarding species persistence under the threats presented by multiple environmental drivers including climate change and habitat loss, future attempts to predict species responses to climate change should consider the implications of the colour of environmental stochasticity and not just mean climate projections.

Acknowledgements

KM was funded by a UKPopNet studentship. This collaboration was initiated at the workshop ‘Integrating spatially explicit statistical models and individual-based modelling using Bayesian statistics II’ organized by Dr Barbara Anderson. The authors are grateful to Mike Fowler and Lasse Ruokolainen for providing the methodology and R code to generate the coloured noise time series.

Biosketch

Karen Mustin has particular interests in conservation and spatial population ecology, and her research focuses on the interaction between anthropogenic activities and biodiversity conservation. Much of her work focuses on theoretical and empirical investigations of biogeographic range shifts under climate change.

Author contributions: K.M., C.D., T.G.B. and J.M.J.T. conceived the ideas; K.M. implemented the models and analysed the data; C.D. and J.M.J.T. provided technical advice; and K.M. led the writing, with contributions from C.D., T.G.B. and J.M.J.T.