Ecological Stability, in Theory and Practice�(… and in Nonstationary Systems)

Adam Thomas Clark

Asst. Prof., University of Graz

9 December 2021

adam.clark@uni-graz.at; adamclarktheecologist.com

Measuring Ecological Stability

Question:

• What is long-term outcomes in THIS ecological community?
• Coexistence, composition, productivity, ecosystem feedbacks, etc.

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Example:

• 24 “Old” Fields
• Abandoned from agricultural use between 1927 and 2016.
• Surveyed since 1983
• >2300 plots measuring vegetative cover and species-level plant biomass

(Isbell et al. 2019)

Broader Question:

• How do I measure and predict long-term outcomes in ecological communities?
• Empirically tractable
• Theoretically justified
• Suitable for a wide range of sites and systems
• Applicable across scales

Clark et al. AmNat 2019

Clark et al. J. Ecology 2019

Age (years)

Abundance (% cover)

r₀

Clark et al. AmNat 2019

Clark et al. Ecology Letters 2018

Problems:

• Predictions vary across:
• metrics
• models
• Scales
• There is no combination of scales and metrics that guarantees the “correct” answer!!

x = standardized abundance (N - K)

r = linearized growth rate

t = time

ζ = process noise function

disturbances ~ rNorm(0, σ)

waiting time ~ rExp(λ)

Clark et al. E. Letters 2021

Clark et al. E. Letters 2021

(Spatial and Ecological) Scaling of Stability:

• Simple function of observed values at scale b, extrapolated to scale B
• Assumes “representative” sampling

Clark et al. E. Letters 2021

Temporal Scaling:

• No change with scale for unbiased estimates
• BUT both r and σ are generally biased for “slow” sampling
• Can correct based on expected value of variance
• τ it timespan between observations, Λ is average number of disturbances over time τ

temporal scale (time between measurements)

Spatial and Ecological Scaling:

• Varies depending on covariance in abundances among species or sites
• Example above: positive site covariance, negative species covariance

ecological scale (e.g. species vs. functional groups)

spatial scale (plot size)

Caveats:

• In the real world, many different covariances are possible
• Positive between species might be most common?
• In diverse communities, r can behave strangely

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Measuring Ecological Stability

Question:

• How do I measure and predict long-term outcomes in ecological communities?

Answers:

• Choose stability metrics that can be related to scalable statistical attributes
• Sample representatively across space or species
• Apply post-hoc corrections for temporal sampling regime, or implement “fast” sampling relative to system dynamics
• Methods available on CRAN in the ecostatscale R package

Nonstationary Systems?

(Isbell et al. 2019)

Partitioning (dynamical) variability:

• Three potential sources of error can influence variability:
• 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

Partitioning (dynamical) variability:

• Variability in system dynamics can be related to stability
• e.g. species that stay rare might be more likely to go extinct
• BUT, high variability in system dynamics does not imply instability!

Blasius et al. Nature 2019

Fokker-Planck and Langevin Equations…

Arani et al. Science 2021

Takens Theorem:

• “Kalman-Takens Filter”: a nonparametric extension of state space modelling.
• “Detrends” noisy timeseries based on Takens Theorem.

Hamilton et al. PLOS Com. Bio. 2017

• Importantly! We can treat the mean trajectory as a fixed equilibrium point.
• Means we can meet the assumptions for linking variability to other theoretical metrics
• Allows us to “detrend” effects of observation error and dynamical variation.

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:
• Very briefly… identifies a general mathematical principle that allows us to predict future dynamics of a process based on its historical behavior.
• Semi-mechanistic, in that it uses lagged observations of a single variable as a proxy for the effects of unobserved variables.
• Is guaranteed to perfectly match the true underlying deterministic trajectory given an infinitely large “training” dataset.

Takens Theorem:

Takens Theorem:

Around prey_t = 5, if we see that prey_t-1 > prey_t, we know that prey_t+1 will continue to decrease. Alternatively, if prey_t-1 < prey_t, we know that prey_t+1 will continue to increase.

Particle-Takens Filtering:

Particle-Takens Filtering:

Particle-Takens Filtering:

Particle-Takens Filtering:

Particle-Takens Filtering:

• A general result from the Taylor Power Law, matching results for cross-scale comparison of CV:
• If scaling coefficient in the process noise function z > 2, then probability of extinction increases with abundance
• If z < 2, then probability of extinction decreases with abundance.
• Overall extinction risk depends on c and z from process noise function and the distribution of N abundance values.

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…