Measuring Ecological Stability in Systems Without Static Equilibria:�or… Chasing the Dragon, and two other methods

Adam Thomas Clark

Asst. Prof., Institute of Biology

University of Graz, Austria

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15 August 2022

adam.clark@uni-graz.at

adamclarktheecologist.com

Central Questions:

• Ecological systems are complex, and estimates of their stability are highly context dependent.
• How can we develop general methods to help us measure and forecast ecological stability with limited information?
• E.g. “How long before species X goes extinct?”; “What will happen if I reduce the size of this reserve by 50%?”
• Combining three existing tools might help…

Method 1:

Taylor’s Power Law

Taylor’s Power Law

• Sample variance and mean in many systems tend to be related by a power law:
• σ² = cμ^z
• log(σ²) = log(c) + z(log(μ))
• Found across a remarkable array of systems
• Almost any system where values are guaranteed to be positive
• Generally very strong fits
• (e.g. R² > 90%)

Taylor 1961; Fronczak & Fronczak 2010

Interpretation?

z = 2 (i.e. perfectly correlated plots)

z = 1 (i.e. perfectly uncorrelated plots)

Gradient:

Uncorrelated to correlated

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So... These dynamics are relatively strongly correlated?

Interpretation?

• Attributed to many different ecological mechanisms:
• Species aggregation in space?

Interpretation?

• Attributed to many different ecological mechanisms:
• Species aggregation in space?
• Temporal synchrony?

Interpretation?

• Attributed to many different ecological mechanisms:
• Species aggregation in space?
• Temporal synchrony?
• Fractal structure?

Interpretation?

• Attributed to many different ecological mechanisms:
• Species aggregation in space?
• Temporal synchrony?
• Fractal structure?
• Species interactions?

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Interpretation?

• Attributed to many different ecological mechanisms:
• Species aggregation in space?
• Temporal synchrony?
• Fractal structure?
• Species interactions?
• Statistical artefact?

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Chasing the Dragon…

• Lots of excitement around this pattern, but surprisingly little progress.
• People seem to be giving up…
• Potentially best to think of the Taylor Power Law as a useful statistical pattern, e.g.:
• “I know my data are normally distributed, which is helpful”
• “But, normally distributed data can arise from many different processes”

Method 2:

Coefficient of Variation (CV)

Coefficient of Variation (CV):

• Standard deviation divided by mean (σ/μ)
• Easy to measure based on empirical data
• CV gets larger as systems get more variable relative to the mean (i.e. “less stable”).

CV Across Scales:

• Leveraging statistical properties of CV (or invariability) to test how stability varies across scales.
• Links Invariability (i.e. I = 1/(CV)²) to spatial scale, based on:
• CV observed within a single patch
• Total area being considered (A)
• Mean correlation between patches (ρ)

Wang et al. 2017

CV Across Scales:

Method 3:

Stochastic (-Takens) Filtering

Partitioning (dynamical) variability:

• Three potential sources of error can influence CV:
• Observation error:

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2) System dynamics:

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3) Actual stochastic variation (“process noise”):

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Problem: How do we separate these effects?

observed values

true value

State Space Modelling:

• Based on fitting functions for two states: an “observed” variable, y, and a “latent” variable x.
• y describes the system as we observe it
• x describes the “true” state of the system

x_t+1 = f(x_t) + w

y_t+1 = x_t+1 + v

• w is process noise, v is observation error
• f(x_t) is a function that takes in the value of x at time t, and predicts its value at timestep t+1.
• Problem: outcome of this method depends on the function chosen for f(x), which we often don’t know for ecological systems.
• BUT…

Takens Theorem:

• A brilliant mathematical proof by Floris Takens:
• A general mathematical principle that allows us to predict future dynamics of a process based on its historical behavior.

Synthesis:

• The Taylor Power Law is a very general pattern that links changes in abundance to changes in variability across many real world systems.
• The Coefficient of Variation is an empirically tractable metric for quantifying variability across space, time, or species.
• Stochastic Filters can be applied to estimate CV, and separate effects of observation error, process noise, and dynamical variation.
• By joining these three method together, we can build a general purpose tool for quantifying stability in real world systems.

Kalman-Takens Filter :

• A nonparametric extension of state space modelling.
• Replace unknown function f(x) with a prediction based on Takens Theorem.

true dynamics

Hamilton et al. PLOS Com. Bio. 2017

Particle-Takens Filtering:

Data from:

Burgmer & Hillebrand Oikos 2011

Particle-Takens Filtering:

Data from:

Burgmer & Hillebrand Oikos 2011

Taylor Power Law: Intercept

Taylor Power Law: Slope

Particle-Takens Filtering:

Data from:

Burgmer & Hillebrand Oikos 2011

Next Steps:

• The Particle-Takens filter is available as an R package via GitHub (https://github.com/adamtclark/pts_r_package)
• Description will be available as a pre-print soon – check my website for updates.
• Working on several applications for these methods…

END

• Would be excited to talk more about these or related topics in ecology or dynamical systems theory.
• Contact: adam.clark@uni-graz.at
• More info: adamclarktheecologist.com
• In particular, if you know of better methods for “detrending” CV, please let me know!!
• I am especially grateful to:
• Collaborators, co-authors, and help from: S. Harpole, H. Hillebrand, L. Mühlbauer, C. Lawson, H. Ye, F. Hamilton, F. Hartig, B. Rosenbaum, A. Compagnoni, S. Munch, L. Dee, S. Miller, J. Clark, and the Harpole and Chase lab groups at iDiv.