Prospects for Resilience Quanti cation Using … › ... › 2016-1Spring ›...

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

Resilience

“[T]he capacity of [a] systemto absorb change and distur-bances and still retain its basicstructure and function”

– Walker & Salt (2006)

Example

Images: http://resac.gis.umn.edu/water/regional_water_clarity/

content/project_summary.htm

Translating to Dynamical Systems

Scheffer et al. (2001)

Translating to Dynamical Systems

Scheffer et al. (2001) Carpenter et al. (2001)

Resilience to state variable perturbations

1 Levin & Lubchenco (2008), Walker et al. (2004)2 Beisner et al. (2003a)

Resilience to state variable perturbations

3 Beisner et al. (2003b), Levin & Lubchenco (2008)4 Ives & Carpenter (2007), Pimm (1984)

ε-pseudo-orbits

Classically defined for flows:

Fix a disturbance period, τ :

ϕ(t, x) : R× X → X ϕτ (x) : X → X

ε-pseudo-orbits

Classically defined for flows:

Fix a disturbance period, τ :

ϕ(t, x) : R× X → X ϕτ (x) : X → X

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

Consider the multi-valued map

ϕτε : X →P(X )

x 7→ {y ∈ X : d(y , ϕτ (x)) < ε}

Let Pε(S) =∞⋃n=0

(ϕτε )n (S)

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 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:

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.

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.

Definitionβ(B) ≡ sup{ε : ϕτε (B) ⊂ B}

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)

Theorem

µ(A) = ν(A)

Theorem

µ(A) = ν(A)

Towards a resilience profile

Example

x = x − x3

µτ (A) = ντ (A) = supx∈(−1,0)

{x − ϕτ (x)}

Example

x = x − x3

µτ (A) = ντ (A) = supx∈(−1,0)

{x − ϕτ (x)}

Example

x = x − x3

µτ (A) = ντ (A) = supx∈(−1,0)

{x − ϕτ (x)}

Quantifying resilience to parameter changes

5 Ives & Carpenter (2007)

Time-dependent parameter changes

Transient parameter change Rate-induced tipping

Example: transient change in salinity forcingof ocean circulation

http://earthobservatory.nasa.gov/Features/Paleoclimatology_

Evidence/paleoclimatology_evidence_2.php

Cessi’s 1D model

y = p + p(t)− y[1 + µ2(1− y)2

], where p(t) =

{∆ 0 ≤ t ≤ τ0 o’wise

Cessi (1994)

Cessi’s 1D model

Calculating critical time in terms of ∆

τcrit(∆) =

∫ τcrit

∫ yb

p + ∆− y [1 + µ2(1− y)2]

Calculating critical time in terms of ∆

τcrit(∆) =

∫ τcrit

∫ yb

p + ∆− y [1 + µ2(1− y)2]

τcrit(∆) =

∫ τcrit

∫ yb

p + ∆− y [1 + µ2(1− y)2]

Figure 3 from Cessi (1994)

Rate-induced tippingAn example from Ashwin et al. (2012):

dt= (x + λ)2 − µ (µ > 0 fixed)

withdλ

dt= (x + λ)2 − µ (µ > 0 fixed)

adapted from Figure 3 of Ashwin et al. (2012)

Asymptotically autonomous parameter shifts

An example from Ashwin et al. (2015):

dt= −

((x − 0.25 + bλ)2 − 0.4 tanh(λ+ 0.3)

)(x − K/ cosh(3λ))

dt= −r(λ+ 1)(λ− 1)

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

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)}

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.

Generalize Cessi’s method to smooth parameter changes

e.g. p(rt) = p + ∆1+(rt)2

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.

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.

Thank you.

Questions?

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