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Confounding environmental colour and distribution shape leads to underestimation of population extinction risk.

Fowler MS, Ruokolainen L - PLoS ONE (2013)

Bottom Line: We show that changing environmental colour from white to red with 1/f models, and from white to red or blue with AR(1) models, generates coloured environmental series that are not normally distributed at finite time-scales, potentially confounding comparison with normally distributed white noise models.This work synthesises previous results and provides further insight into the processes driving extinction risk in model populations.We must let the characteristics of known natural environmental covariates (e.g., colour and distribution shape) guide us in our choice of how to best model the impact of coloured environmental variation on population dynamics.

View Article: PubMed Central - PubMed

Affiliation: Population Ecology Group, Institut Mediterrani d'Estudis Avançats (UIB-CSIC), Esporles, Illes Balears, Spain. m.s.fowler@swansea.ac.uk

ABSTRACT
The colour of environmental variability influences the size of population fluctuations when filtered through density dependent dynamics, driving extinction risk through dynamical resonance. Slow fluctuations (low frequencies) dominate in red environments, rapid fluctuations (high frequencies) in blue environments and white environments are purely random (no frequencies dominate). Two methods are commonly employed to generate the coloured spatial and/or temporal stochastic (environmental) series used in combination with population (dynamical feedback) models: autoregressive [AR(1)] and sinusoidal (1/f) models. We show that changing environmental colour from white to red with 1/f models, and from white to red or blue with AR(1) models, generates coloured environmental series that are not normally distributed at finite time-scales, potentially confounding comparison with normally distributed white noise models. Increasing variability of sample Skewness and Kurtosis and decreasing mean Kurtosis of these series alter the frequency distribution shape of the realised values of the coloured stochastic processes. These changes in distribution shape alter patterns in the probability of single and series of extreme conditions. We show that the reduced extinction risk for undercompensating (slow growing) populations in red environments previously predicted with traditional 1/f methods is an artefact of changes in the distribution shapes of the environmental series. This is demonstrated by comparison with coloured series controlled to be normally distributed using spectral mimicry. Changes in the distribution shape that arise using traditional methods lead to underestimation of extinction risk in normally distributed, red 1/f environments. AR(1) methods also underestimate extinction risks in traditionally generated red environments. This work synthesises previous results and provides further insight into the processes driving extinction risk in model populations. We must let the characteristics of known natural environmental covariates (e.g., colour and distribution shape) guide us in our choice of how to best model the impact of coloured environmental variation on population dynamics.

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Coloured stochastic time-series are not normally distributed.The proportion of coloured stochastic (environmental) time-series that fail a normality test increases as they change colour, even when the same methods produce normally distributed white noise (10,000 step series; 1000 replicates for each parameter value), under both (A) Autoregressive and (B) Spectral synthesis methods (x-axis values reversed for comparison). Inlays illustrate frequency distributions for series of εt values from sample blue (A: α≈−0.999, B: β≈1), white (α, β≈0) and red (α≈0.999, β≈−1.86) stochastic series.
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pone-0055855-g001: Coloured stochastic time-series are not normally distributed.The proportion of coloured stochastic (environmental) time-series that fail a normality test increases as they change colour, even when the same methods produce normally distributed white noise (10,000 step series; 1000 replicates for each parameter value), under both (A) Autoregressive and (B) Spectral synthesis methods (x-axis values reversed for comparison). Inlays illustrate frequency distributions for series of εt values from sample blue (A: α≈−0.999, B: β≈1), white (α, β≈0) and red (α≈0.999, β≈−1.86) stochastic series.

Mentions: Figure (1) illustrates that changing the colour of AR(1) or 1/f stochastic series also leads to a change in the distribution shape. White series are normally distributed (fail to reject the hypothesis of the Jarque-Bera test), while reddened AR(1) or 1/f series tend not to be. There is an increasing probability that blue AR(1) series fail the normality test. These results were driven by changes in the variance of the sample skewness and kurtosis statistics with colour (Fig. S1 in Supporting Information). There is also a decrease in the mean kurtosis associated with blue and reddened AR(1) and reddened 1/f models (Fig. S1). These qualitative results also held over longer (107 steps) and shorter (100 steps) series.


Confounding environmental colour and distribution shape leads to underestimation of population extinction risk.

Fowler MS, Ruokolainen L - PLoS ONE (2013)

Coloured stochastic time-series are not normally distributed.The proportion of coloured stochastic (environmental) time-series that fail a normality test increases as they change colour, even when the same methods produce normally distributed white noise (10,000 step series; 1000 replicates for each parameter value), under both (A) Autoregressive and (B) Spectral synthesis methods (x-axis values reversed for comparison). Inlays illustrate frequency distributions for series of εt values from sample blue (A: α≈−0.999, B: β≈1), white (α, β≈0) and red (α≈0.999, β≈−1.86) stochastic series.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3569452&req=5

pone-0055855-g001: Coloured stochastic time-series are not normally distributed.The proportion of coloured stochastic (environmental) time-series that fail a normality test increases as they change colour, even when the same methods produce normally distributed white noise (10,000 step series; 1000 replicates for each parameter value), under both (A) Autoregressive and (B) Spectral synthesis methods (x-axis values reversed for comparison). Inlays illustrate frequency distributions for series of εt values from sample blue (A: α≈−0.999, B: β≈1), white (α, β≈0) and red (α≈0.999, β≈−1.86) stochastic series.
Mentions: Figure (1) illustrates that changing the colour of AR(1) or 1/f stochastic series also leads to a change in the distribution shape. White series are normally distributed (fail to reject the hypothesis of the Jarque-Bera test), while reddened AR(1) or 1/f series tend not to be. There is an increasing probability that blue AR(1) series fail the normality test. These results were driven by changes in the variance of the sample skewness and kurtosis statistics with colour (Fig. S1 in Supporting Information). There is also a decrease in the mean kurtosis associated with blue and reddened AR(1) and reddened 1/f models (Fig. S1). These qualitative results also held over longer (107 steps) and shorter (100 steps) series.

Bottom Line: We show that changing environmental colour from white to red with 1/f models, and from white to red or blue with AR(1) models, generates coloured environmental series that are not normally distributed at finite time-scales, potentially confounding comparison with normally distributed white noise models.This work synthesises previous results and provides further insight into the processes driving extinction risk in model populations.We must let the characteristics of known natural environmental covariates (e.g., colour and distribution shape) guide us in our choice of how to best model the impact of coloured environmental variation on population dynamics.

View Article: PubMed Central - PubMed

Affiliation: Population Ecology Group, Institut Mediterrani d'Estudis Avançats (UIB-CSIC), Esporles, Illes Balears, Spain. m.s.fowler@swansea.ac.uk

ABSTRACT
The colour of environmental variability influences the size of population fluctuations when filtered through density dependent dynamics, driving extinction risk through dynamical resonance. Slow fluctuations (low frequencies) dominate in red environments, rapid fluctuations (high frequencies) in blue environments and white environments are purely random (no frequencies dominate). Two methods are commonly employed to generate the coloured spatial and/or temporal stochastic (environmental) series used in combination with population (dynamical feedback) models: autoregressive [AR(1)] and sinusoidal (1/f) models. We show that changing environmental colour from white to red with 1/f models, and from white to red or blue with AR(1) models, generates coloured environmental series that are not normally distributed at finite time-scales, potentially confounding comparison with normally distributed white noise models. Increasing variability of sample Skewness and Kurtosis and decreasing mean Kurtosis of these series alter the frequency distribution shape of the realised values of the coloured stochastic processes. These changes in distribution shape alter patterns in the probability of single and series of extreme conditions. We show that the reduced extinction risk for undercompensating (slow growing) populations in red environments previously predicted with traditional 1/f methods is an artefact of changes in the distribution shapes of the environmental series. This is demonstrated by comparison with coloured series controlled to be normally distributed using spectral mimicry. Changes in the distribution shape that arise using traditional methods lead to underestimation of extinction risk in normally distributed, red 1/f environments. AR(1) methods also underestimate extinction risks in traditionally generated red environments. This work synthesises previous results and provides further insight into the processes driving extinction risk in model populations. We must let the characteristics of known natural environmental covariates (e.g., colour and distribution shape) guide us in our choice of how to best model the impact of coloured environmental variation on population dynamics.

Show MeSH
Related in: MedlinePlus