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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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Population extinction risk varies with environmental colour, but changing environmental distribution shape confounds patterns.Undercompensating populations were forced by coloured environmental stochasticity modelled as either (A) 1/f, or (B) AR(1) processes. Dashed lines show the coefficient of variation (CVN) of population fluctuations based on environmental series generated with traditional methods, solid lines show results based on normally distributed series generated using spectral mimicry. Populations were iterated over 10,000 steps, forced with with σε2(T10,000) = 0.01 (blue lines), 0.1 (red lines) or 0.5 (green lines). Results show the median CVN value, based on sample environmental spectral exponents (β) binned into 25 evenly spaced groups between the limits [–2, 1], drawn from 1,000 replicates for each desired colour statistic, distributed between α = [–0.999, 0.999] and β = [–2, 1]. Population parameters: r = 1.5, b = 0.1, K = 100. Values along the x-axis have been reversed for easier comparison.
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pone-0055855-g002: Population extinction risk varies with environmental colour, but changing environmental distribution shape confounds patterns.Undercompensating populations were forced by coloured environmental stochasticity modelled as either (A) 1/f, or (B) AR(1) processes. Dashed lines show the coefficient of variation (CVN) of population fluctuations based on environmental series generated with traditional methods, solid lines show results based on normally distributed series generated using spectral mimicry. Populations were iterated over 10,000 steps, forced with with σε2(T10,000) = 0.01 (blue lines), 0.1 (red lines) or 0.5 (green lines). Results show the median CVN value, based on sample environmental spectral exponents (β) binned into 25 evenly spaced groups between the limits [–2, 1], drawn from 1,000 replicates for each desired colour statistic, distributed between α = [–0.999, 0.999] and β = [–2, 1]. Population parameters: r = 1.5, b = 0.1, K = 100. Values along the x-axis have been reversed for easier comparison.

Mentions: When the deterministic undercompensatory population model (eqn. 2) is forced by intermediate or strong environmental stochasticity [σε2(T10,000)≥0.1] generated with traditional methods, environmental reddening initially leads to an increase, followed by a decrease in the size of population fluctuations (CVN) for both 1/f and AR(1) methods (Fig. 2). This corroborates previous work using 1/f models based on slightly different assumptions [8]. It also extends the parameter space examined for AR(1) models there and elsewhere [9], [11], [12], revealing a qualitatively similar decline in population variability for very red AR(1) environments, as found with 1/f methods. Population variability tends to show an asymptotic increase with reddening for weaker environmental stochasticity [σε2(T10,000) = 0.01] in both AR(1) and 1/f models.


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

Fowler MS, Ruokolainen L - PLoS ONE (2013)

Population extinction risk varies with environmental colour, but changing environmental distribution shape confounds patterns.Undercompensating populations were forced by coloured environmental stochasticity modelled as either (A) 1/f, or (B) AR(1) processes. Dashed lines show the coefficient of variation (CVN) of population fluctuations based on environmental series generated with traditional methods, solid lines show results based on normally distributed series generated using spectral mimicry. Populations were iterated over 10,000 steps, forced with with σε2(T10,000) = 0.01 (blue lines), 0.1 (red lines) or 0.5 (green lines). Results show the median CVN value, based on sample environmental spectral exponents (β) binned into 25 evenly spaced groups between the limits [–2, 1], drawn from 1,000 replicates for each desired colour statistic, distributed between α = [–0.999, 0.999] and β = [–2, 1]. Population parameters: r = 1.5, b = 0.1, K = 100. Values along the x-axis have been reversed for easier comparison.
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3569452&req=5

pone-0055855-g002: Population extinction risk varies with environmental colour, but changing environmental distribution shape confounds patterns.Undercompensating populations were forced by coloured environmental stochasticity modelled as either (A) 1/f, or (B) AR(1) processes. Dashed lines show the coefficient of variation (CVN) of population fluctuations based on environmental series generated with traditional methods, solid lines show results based on normally distributed series generated using spectral mimicry. Populations were iterated over 10,000 steps, forced with with σε2(T10,000) = 0.01 (blue lines), 0.1 (red lines) or 0.5 (green lines). Results show the median CVN value, based on sample environmental spectral exponents (β) binned into 25 evenly spaced groups between the limits [–2, 1], drawn from 1,000 replicates for each desired colour statistic, distributed between α = [–0.999, 0.999] and β = [–2, 1]. Population parameters: r = 1.5, b = 0.1, K = 100. Values along the x-axis have been reversed for easier comparison.
Mentions: When the deterministic undercompensatory population model (eqn. 2) is forced by intermediate or strong environmental stochasticity [σε2(T10,000)≥0.1] generated with traditional methods, environmental reddening initially leads to an increase, followed by a decrease in the size of population fluctuations (CVN) for both 1/f and AR(1) methods (Fig. 2). This corroborates previous work using 1/f models based on slightly different assumptions [8]. It also extends the parameter space examined for AR(1) models there and elsewhere [9], [11], [12], revealing a qualitatively similar decline in population variability for very red AR(1) environments, as found with 1/f methods. Population variability tends to show an asymptotic increase with reddening for weaker environmental stochasticity [σε2(T10,000) = 0.01] in both AR(1) and 1/f models.

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