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Critical slowing down as an indicator of transitions in two-species models. Ryan Chisholm Smithsonian Tropical Research Institute Workshop on Critical Transitions in Complex Systems 21 March 2012 Imperial College London. Acknowledgements.

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critical slowing down as an indicator of transitions in two species models

Critical slowing down as an indicator of transitions in two-species models

Ryan Chisholm

Smithsonian Tropical Research Institute

Workshop on Critical Transitions in Complex Systems

21 March 2012Imperial College London

acknowledgements
Acknowledgements
  • Elise Filotas, Centre for Forest Research at the University of Quebec in Montreal
  • Simon Levin, Princeton University, Department of Ecology and Evolutionary Biology
  • Helene Muller-Landau, Smithsonian Tropical Research Institute
  • Santa Fe Institute, Complex Systems Summer School 2007: NSF Grant No. 0200500
question
Question

When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?

outline
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

outline1
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

smithsonian tropical research institute
Smithsonian Tropical Research Institute

“…dedicated to understanding biological diversity”

What determines patterns of diversity?

What factors regulate ecosystem function?

How will tropical forests respond to climate change and other anthropogenic disturbances?

smithsonian tropical research institute3
Smithsonian Tropical Research Institute
  • 1500 ha
  • 2551 mm yr-1 rainfall
  • 381 bird species
  • 102 mammal species (nearly half are bats)
  • ~100 species of amphibians and reptiles
  • 1316 plant species

Green iguana

(Iguana iguana)

Keel-billed Toucan (Ramphastossulfuratus)

Pentagoniamacrophylla

Jaguar (Pantheraonca)

Photo: Christian Ziegler

smithsonian tropical research institute4
Smithsonian Tropical Research Institute

Photo: Marcos Guerra, STRI

sciencedaily.com

Photo: Leonor Alvarez

forest resilience
Forest resilience

Staveret al. 2011 Science

outline2
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

transitions in complex systems
Transitions in complex systems

Schefferet al. 2009 Nature, Scheffer 2009 Critical Transitions in Nature and Society

Eutrophication of shallow lakes

Sahara desertification

Climate change

Shifts in public opinion

Forest-savannah transitions

critical transitions
Critical transitions

May 1977 Nature

detecting impending transitions
Detecting impending transitions

Carpenter & Brock 2006 Ecol. Lett., van Nes & Scheffer 2007 Am. Nat.,

Schefferet al. 2009 Nature

Decreasing return rate

Rising variance

Rising autocorrelation

=> All arise from critical slowing down

critical slowing down
Critical slowing down

van Nes & Scheffer 2007 Am. Nat.

Recovery rate: return rate after disturbance to the equilibrium

Critical slowing down: dominant eigenvalue tends to zero; recovery rate decreases as transition approaches

critical slowing down1
Critical slowing down

van Nes & Scheffer 2007 Am. Nat.

critical slowing down2
Critical slowing down

van Nes & Scheffer 2007 Am. Nat.

question1
Question

When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?

What is the length/duration of the warning period?

outline3
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

competition model
Competition model

Ni = abundance of species i

Ki = carrying capacity of species i

ri = intrinsic rate of increase of species i

αij= competitive impact of species j on species i

Equilibria:

Lotka 1925, 1956 Elements of Physical Biology; Chisholm & Filotas 2009 J. Theor. Biol.

competition model1
Competition model

Case 1: Interspecific competition greater than intraspecific competition

Stable

Stable

Unstable

Unstable

Chisholm & Filotas 2009 J. Theor. Biol.

question2
Question

When is critical slowing down likely to be a useful leading indicator of a critical transition in ecological models?

What is the length/duration of the warning period?

competition model2
Competition model

Ni = abundance of species i

Ki = abundance of species i

ri = intrinsic rate of increase of species i

αij= competitive impact of species j on species i

Chisholm & Filotas 2009 J. Theor. Biol.

Recovery rate:

When species 1 dominates, recovery rate begins to decline at:

competition model3
Competition model

Chisholm & Filotas 2009 J. Theor. Biol.

competition model4
Competition model

Ni = abundance of species i

Ki = abundance of species i

ri = intrinsic rate of increase of species i

αij= competitive impact of species j on species i

Chisholm & Filotas 2009 J. Theor. Biol.

Recovery rate begins to decline at:

More warning of transition if the dynamics of the rare species are slow relative to those of the dominant species

competition model5
Competition model

Case 2: Interspecific competition less than intraspecific competition

Stable

Stable

Unstable

Stable

Chisholm & Filotas 2009 J. Theor. Biol.

competition model6
Competition model

Case 2: Interspecific competition less than intraspecific competition

More warning of transition if the dynamics of the rare species are slow relative to those of the dominant species

Chisholm & Filotas 2009 J. Theor. Biol.

outline4
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

predator prey model
Predator-prey model

V = prey abundance

P = predator abundance

Rosenzweig 1971 Science

predator prey model1
Predator-prey model

h(V)

V = prey abundance

P = predator abundance

r = intrinsic rate of increase of prey

k = predation rate

J = equilibrium prey population size

A = predator-prey conversion efficiency

K = carrying capacity of prey

f(V) = effects of intra-specific competition among prey

f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0

h(V) = per-capita rate at which predators kill prey

h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0

f(V)

V

Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model2
Predator-prey model

Equilibria:

Unstable

Stable for K ≤ J

V = prey abundance

P = predator abundance

r = intrinsic rate of increase of prey

k = predation rate

J = equilibrium prey population size

A = predator-prey conversion efficiency

K = carrying capacity of prey

f(V) = effects of intra-specific competition among prey

f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0

h(V) = per-capita rate at which predators kill prey

h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0

Exists for K ≥ J

Stable for J ≤ K≤ Kcrit

Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model3
Predator-prey model

Predator isocline

V = prey abundance

P = predator abundance

r = intrinsic rate of increase of prey

k = predation rate

J = equilibrium prey population size

A = predator-prey conversion efficiency

f(V) = effects of intra-specific competition among prey

f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0

h(V) = per-capita rate at which predators kill prey

h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0

Prey isoclines

Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model4
Predator-prey model

Unstable equilibrium

V = prey abundance

P = predator abundance

r = intrinsic rate of increase of prey

k = predation rate

J = equilibrium prey population size

A = predator-prey conversion efficiency

f(V) = effects of intra-specific competition among prey

f(V) > 0; f ’(V) < 0; f(K) = 0; df/dK > 0

h(V) = per-capita rate at which predators kill prey

h(V) > 0; h’(V) > 0; h’’(V) < 0; h(0) = 0

Stable equilibrium

Rosenzweig 1971 Science,Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model5
Predator-prey model

Scheffer 1998 The Ecology of Shallow Lakes

predator prey model6
Predator-prey model

Hopf bifurcation occurs when K= Kcrit :

Critical slowing down begins when K= Kr:

predator prey model7
Predator-prey model

Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model8
Predator-prey model

Chisholm & Filotas 2009 J. Theor. Biol.

predator prey model9
Predator-prey model

Chisholm & Filotas 2009 J. Theor. Biol.

Kr and Kcrit converge as:

More warning of transition when:

  • Predator-prey conversion efficiency (A) is high
  • Predation rate (k) is high
  • Prey growth rate (r) is low
  • Prey controlled by predators rather than intrinsic density dependence
  • Increases tendency for oscillations
  • Larger K makes oscillations larger and hence rates of return slower
predator prey model10
Predator-prey model

Chisholm & Filotas 2009 J. Theor. Biol.

multi species models
Multi-species models

van Nes & Scheffer 2007 Am. Nat.

multi species models1
Multi-species models

Chisholm & Filotas 2009 J. Theor. Biol.

Expect that multi-species models will exhibit longer warning periods of transitions induced by changes in resource abundance when:

  • Dynamics of rare species are slow relative to those of the dominant species
  • Prey species are controlled by predation rather than intrinsic density dependence
outline5
Outline

Smithsonian Tropical Research Institute

Background: critical slowing down

Competition model

Predator-prey model

Grasslands model

Future work

practical utility of critical slowing down
Practical utility of critical slowing down

Chisholm & Filotas 2009 J. Theor. Biol.

“…even if an increase in variance or AR1 is detected, it provides no indication of how close to a regime shift the ecosystem is…”

Biggs et al. 2008 PNAS

western basalt plains grasslands2
Western Basalt Plains Grasslands

Williams et al. 2005 J. Ecol.; Williams et al. 2006 Ecology

grasslands invasion model
Grasslands invasion model

Agricultural fertiliser run-off

Native

grass

biomass

Sugar addition

Nutrient input rate

grasslands invasion model1
Grasslands invasion model

A = plant-available N pool

Bi = biomass of species i

ωi = N-use efficiency of species i

νi = N-use efficiency of species i

μi = N-use efficiency of species i

αij= light competition coefficients

I = abiotic N-input flux

K = soil leaching rate of plant-available N

δ = proportion of N in litterfall lost from the system

Parameterized so that species 2 (invader) has a higher uptake rate and higher turnover rate.

Chisholm & Levin in prep.; Mengeet al. 2008 PNAS

grasslands invasion model2
Grasslands invasion model

B2

Relatively safe, but higher control costs.

B1

Nutrient input

Riskier, but lower control costs.

conclusions future work
Conclusions & Future work

Critical slowing down provides an earlier indicator of transitions in two-species models where:

  • Dynamics of rare species are slow relative to those of the dominant species
  • Prey species are controlled by predation rather than intrinsic density dependence

But utility of early/late indicators depends on socio-economic considerations