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
• 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
When is critical slowing down likely to be a
useful leading indicator of a critical transition in
ecological models?
Outline
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Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
Outline
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Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
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 Institute
Panama
Smithsonian Tropical Research Institute
50 ha plot
Smithsonian Tropical Research Institute
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Green iguana
(Iguana iguana)
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
Pentagonia
macrophylla
Keel-billed Toucan
(Ramphastos
sulfuratus)
Jaguar (Panthera onca)
Photo: Christian Ziegler
Smithsonian Tropical Research Institute
Photo: Marcos Guerra, STRI
Photo: Leonor Alvarez
sciencedaily.com
Center for Tropical Forest Science
Forest resilience
Staver et al. 2011 Science
Chisholm, Condit, et al. in prep
Outline
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Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
Transitions in complex systems
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Eutrophication of shallow lakes
Sahara desertification
Climate change
Shifts in public opinion
Forest-savannah transitions
Scheffer et al. 2009 Nature, Scheffer 2009 Critical Transitions in Nature and Society
Critical transitions
May 1977 Nature
Detecting impending transitions
• Decreasing return rate
• Rising variance
• Rising autocorrelation
=> All arise from critical slowing down
Carpenter & Brock 2006 Ecol. Lett., van Nes & Scheffer 2007 Am. Nat.,
Scheffer et al. 2009 Nature
Critical slowing down
• Recovery rate: return
rate after disturbance
to the equilibrium
• Critical slowing down:
dominant eigenvalue
tends to zero; recovery
rate decreases as
transition approaches
van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
Critical slowing down
van Nes & Scheffer 2007 Am. Nat.
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?
Outline
•
•
•
•
•
•
Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
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 model
Case 1: Interspecific competition greater than intraspecific competition
Stable
Stable
Unstable
Unstable
Chisholm & Filotas 2009 J. Theor. Biol.
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 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
Recovery rate:
When species 1 dominates, recovery rate begins to decline at:
Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Chisholm & Filotas 2009 J. Theor. Biol.
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
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
Chisholm & Filotas 2009 J. Theor. Biol.
Competition model
Case 2: Interspecific competition less than intraspecific competition
Stable
Stable
Unstable
Stable
Chisholm & Filotas 2009 J. Theor. Biol.
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.
Outline
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•
•
•
•
•
Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
Predator-prey model
V = prey abundance
P = predator abundance
Rosenzweig 1971 Science
Predator-prey model
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
V = prey abundance
P = predator abundance
r = intrinsic rate of increase of prey
0
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
h(V)
f(V)
20
40
60
VV
Rosenzweig 1971 Science, Chisholm & Filotas 2009 J. Theor. Biol.
80
100
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 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 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 model
Scheffer 1998 The Ecology of Shallow Lakes
Predator-prey model
Hopf bifurcation occurs when K = Kcrit :
Critical slowing down begins when K = Kr :
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
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
Chisholm & Filotas 2009 J. Theor. Biol.
Predator-prey model
Chisholm & Filotas 2009 J. Theor. Biol.
Multi-species models
van Nes & Scheffer 2007 Am. Nat.
Multi-species models
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
Chisholm & Filotas 2009 J. Theor. Biol.
Outline
•
•
•
•
•
•
Smithsonian Tropical Research Institute
Background: critical slowing down
Competition model
Predator-prey model
Grasslands model
Future work
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 Grasslands
Western Basalt Plains Grasslands
Western Basalt Plains Grasslands
Williams et al. 2005 J. Ecol.; Williams et al. 2006 Ecology
Grasslands invasion model
Agricultural fertiliser run-off
Native
grass
biomass
Sugar addition
Nutrient input rate
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.; Menge et al. 2008 PNAS
Grasslands invasion model
B2
B1
Relatively safe, but
higher control
Nutrient input costs.
Riskier, but lower
control costs.
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