Prospects for Resilience QuantificationUsing Dynamical Systems Theory
Kate MeyerMinnesota Math Climate Seminar
March 22nd, 2016
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Outline
MotivationsEcological resilienceTranslating the qualitative to quantitative
Resilience to state variable perturbationsDefinitions from the ecology literatureMathematical tools: ε-pseudo-orbits and intensity of attraction
Resilience to parameter perturbationsDefinition from the ecology literatureMathematical tools: transient and rapid parameter changes
Future directions
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Resilience
“[T]he capacity of [a] systemto absorb change and distur-bances and still retain its basicstructure and function”
– Walker & Salt (2006)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Example
Images: http://resac.gis.umn.edu/water/regional_water_clarity/
content/project_summary.htm
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Translating to Dynamical Systems
Scheffer et al. (2001)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Translating to Dynamical Systems
Scheffer et al. (2001) Carpenter et al. (2001)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Resilience to state variable perturbations
1 Levin & Lubchenco (2008), Walker et al. (2004)2 Beisner et al. (2003a)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Resilience to state variable perturbations
3 Beisner et al. (2003b), Levin & Lubchenco (2008)4 Ives & Carpenter (2007), Pimm (1984)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
ε-pseudo-orbits
Classically defined for flows:
Fix a disturbance period, τ :
ϕ(t, x) : R× X → X ϕτ (x) : X → X
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
ε-pseudo-orbits
Classically defined for flows:
Fix a disturbance period, τ :
ϕ(t, x) : R× X → X ϕτ (x) : X → X
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Following McGehee (1988): Let X be a locally compact metricspace and φ : X → X a continuous map.
DefinitionAn ε-pseudo-orbit of length n is a sequence of points(x0, x1, . . . , xn) in the state space X satisfying d(φ(xk−1), xk) < ε,for k = 1, 2, . . . n
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Consider the multi-valued map
ϕτε : X →P(X )
x 7→ {y ∈ X : d(y , ϕτ (x)) < ε}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Let Pε(S) =∞⋃n=0
(ϕτε )n (S)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Chain intensity of attraction
DefinitionThe chain intensity of an attractor A is
µ(A) ≡ sup{ε : Pε(A) ⊂ K ⊂ D(A), where K is compact}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Chain intensity of attraction
DefinitionThe chain intensity of an attractor A is
µ(A) ≡ sup{ε : Pε(A) ⊂ K ⊂ D(A), where K is compact}
Chain intensity is amenable to numerical approximation:
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Attractor blocks
DefinitionAn attractor block for a map φ is a nonempty, compact set Bsuch that φ(B) ⊂ int(B).
TheoremFor a continuous map on a locally compact metric space,
• every attractor block contains an attractor in its interior, and
• all attractors are the maximal invariant set inside someattractor block.
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Attractor blocks
DefinitionAn attractor block for a map φ is a nonempty, compact set Bsuch that φ(B) ⊂ int(B).
TheoremFor a continuous map on a locally compact metric space,
• every attractor block contains an attractor in its interior, and
• all attractors are the maximal invariant set inside someattractor block.
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Definitionβ(B) ≡ sup{ε : ϕτε (B) ⊂ B}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Intensity of attraction
DefinitionThe intensity of A isν(A) ≡ sup{β(B) : B is an attractor block associated with A}
DefinitionThe chain intensity of an attractor A isµ(A) ≡ sup{ε : Pε(A) ⊂ K ⊂ D(A), where K is compact}
Theorem
µ(A) = ν(A)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Intensity of attraction
DefinitionThe intensity of A isν(A) ≡ sup{β(B) : B is an attractor block associated with A}
DefinitionThe chain intensity of an attractor A isµ(A) ≡ sup{ε : Pε(A) ⊂ K ⊂ D(A), where K is compact}
Theorem
µ(A) = ν(A)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Intensity of attraction
DefinitionThe intensity of A isν(A) ≡ sup{β(B) : B is an attractor block associated with A}
DefinitionThe chain intensity of an attractor A isµ(A) ≡ sup{ε : Pε(A) ⊂ K ⊂ D(A), where K is compact}
Theorem
µ(A) = ν(A)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Towards a resilience profile
Example
x = x − x3
µτ (A) = ντ (A) = supx∈(−1,0)
{x − ϕτ (x)}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Towards a resilience profile
Example
x = x − x3
µτ (A) = ντ (A) = supx∈(−1,0)
{x − ϕτ (x)}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Towards a resilience profile
Example
x = x − x3
µτ (A) = ντ (A) = supx∈(−1,0)
{x − ϕτ (x)}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Quantifying resilience to parameter changes
5 Ives & Carpenter (2007)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Example: transient change in salinity forcingof ocean circulation
http://earthobservatory.nasa.gov/Features/Paleoclimatology_
Evidence/paleoclimatology_evidence_2.php
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Cessi’s 1D model
y = p + p(t)− y[1 + µ2(1− y)2
], where p(t) =
{∆ 0 ≤ t ≤ τ0 o’wise
Cessi (1994)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Cessi’s 1D model
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Calculating critical time in terms of ∆
τcrit(∆) =
∫ τcrit
0dt =
∫ yb
ya
dy
p + ∆− y [1 + µ2(1− y)2]
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Calculating critical time in terms of ∆
τcrit(∆) =
∫ τcrit
0dt =
∫ yb
ya
dy
p + ∆− y [1 + µ2(1− y)2]
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
τcrit(∆) =
∫ τcrit
0dt =
∫ yb
ya
dy
p + ∆− y [1 + µ2(1− y)2]
Figure 3 from Cessi (1994)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Time-dependent parameter changes
Transient parameter change Rate-induced tipping
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Rate-induced tippingAn example from Ashwin et al. (2012):
dx
dt= (x + λ)2 − µ (µ > 0 fixed)
withdλ
dt= r
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
dx
dt= (x + λ)2 − µ (µ > 0 fixed)
dλ
dt= r
adapted from Figure 3 of Ashwin et al. (2012)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Asymptotically autonomous parameter shifts
An example from Ashwin et al. (2015):
dx
dt= −
((x − 0.25 + bλ)2 − 0.4 tanh(λ+ 0.3)
)(x − K/ cosh(3λ))
dλ
dt= −r(λ+ 1)(λ− 1)
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Future directions
• Develop a continuous analogue to intensity of attraction
• Generalize Cessi’s method to higher dimensions
• Generalize Cessi’s method to smooth parameter changes
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Develop a continuous analogue to intensity of attraction
Maps Flows
Space X locally compact, metric locally compact, metric
Disturbed φε(x) : X →P(X ) ϕr (t, x) : R× X →P(X )Trajectories flow on f + g , |g | < r
Region of Pε(A) Pr (A)Accessibility
Chain Intensity sup{ε : Pε(A) ⊂ K ⊂ D(A)} sup{r : Pr (A) ⊂ K ⊂ D(A)}
Attractor compact, nonempty, compact, nonempty,Block φ(B) ⊂ int(B) ϕt(B) ⊂ int(B) ∀t > 0
Intensity sup{β(B)}
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Generalize Cessi’s method to higher dimensions
Cessi’s 1D system
y = p − y[1 + µ2(1− y)2
]came from
x ′ = −(x − 1)− εx [1 + µ2(x − y)2]y ′ = ε
(p − y [1 + µ2(x − y)2]
)via fast-slow reduction.
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Generalize Cessi’s method to smooth parameter changes
e.g. p(rt) = p + ∆1+(rt)2
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
References
Ashwin et al. (2015) Parameter shifts for nonautonomous systems in lowdimension: bifurcation and rate-induced tipping. arXiv, 1506.07734v1
Ashwin et al. (2012) Tipping points in open systems: bifurcation,noise-induced and rate-dependent examples in the climate system. Phil. TransRoyal Soc., 370:1166-1184.
Beisner et al. (2003a) Variability of lakes on the landscape: roles ofphorsphorus, food webs, and dissolved organic carbon. Ecology, 84:1563-1575.
Beisner et al. (2003b) Alternative stable states in ecology. Frontiers in Ecol.Envir., 1:376-382.
Carpenter et al. (2001) From metaphor to measurement: resilience of what towhat? Ecosystems, 4:765-781.
Cessi (1994) A simple box model of stochastically forced thermohaline flow. J.Phys. Ocean. 24:1911-1920.
Ives and Carpenter (2007) Stability and diversity of ecosystems. Science,317:58-62.
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
References, cont’d
Levin and Lubchenco (2008) Resilience, robustness, and marineecosystem-based management. BioScience, 58:27-32.
McGehee (1988) Some metric properties of attractors with applications tocomputer simulations of dynamical systems. Unpuplished manuscript.
Pimm (1984) The complexity and stability of ecosystems. Nature, 307:321-326.
Scheffer et al. (2001) Catastrophic shifts in ecosystems Nature, 413:591-596.
Stommel (1961) Thermohaline convection with two stable regimes of flow.Tellus, XIII(2):224-230.
Walker et al. (2004) Resilience, adaptability and transformability insocial-ecological systems. Ecology and Society, 9:5.
Walker and Salt (2006). Resilience thinking: sustaining ecosystems and people
in a changing world. Island Press.
Motivations Resilience to state variable perturbations Resilience to parameter perturbations Future directions
Thank you.
Questions?