Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Environmental Limits to growth: instabilities and
dangerous rates of change
Peter Cox, Owen Kellie-Smith
School of Engineering, Computing and Mathematics
University of Exeter
Mathematics and climate change mitigation, Warwick
Mathematics Institute, 27 April 2009
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
IPCC projections dominated by emissions uncertainty
IPCC AR4 WG1 Fig 10.4
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Long Term Uncertainty is Dominated by Emissions
Cox and Stephenson, Science, 2007
Summary
Motivation
Model definition
Stability of equilibrium
Emission paths are not Optional
Example: linear damage function
Summary
Motivation
Model definition
Stability of equilibrium
Emissions Depend on Economy
http://airlineworld.wordpress.com
Example: linear damage function
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Economy Depends on Climate
http://en.wikipedia.org/wiki/Florence Owens Thompson
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Climate Depends on Emissions
http://www.awi.de/en/news/press releases/
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Fast positive feedback and slow negative feedback
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
∂ Ĉ
1 = χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
∂ Ĉ
1 = χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
IPCC AR4 WG1 Fig 10.22
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
1 ∂ Ĉ
= χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
1 ∂ Ĉ
= χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
!
!
∂ T̂
1 ∆T2∗CO2
Ĉ
=
log
− T̂ ;
τT
log 2
CPI
∂ t̂
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
1 ∂ Ĉ
= χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
!
!
∂ T̂
1 ∆T2∗CO2
Ĉ
=
log
− T̂ ;
τT
log 2
CPI
∂ t̂
∂ Ŵ
= x̂(T̂ )Ŵ ;
∂ t̂
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
1 ∂ Ĉ
= χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
!
!
∂ T̂
1 ∆T2∗CO2
Ĉ
=
log
− T̂ ;
τT
log 2
CPI
∂ t̂
∂ Ŵ
= x̂(T̂ )Ŵ ;
∂ t̂
∃T̂e : x̂(T̂e ) = µ
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Assumptions
1 ∂ Ĉ
= χe −µt̂ Ŵ −
Ĉ − CPI ;
τC
∂ t̂
!
!
∂ T̂
1 ∆T2∗CO2
Ĉ
=
log
− T̂ ;
τT
log 2
CPI
∂ t̂
∂ Ŵ
= x̂(T̂ )Ŵ ;
∂ t̂
∃T̂e : x̂(T̂e ) = µ
∂x̂
< 0.
∂ T̂
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Non-dimensional form
Ċ = W − φC
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Non-dimensional form
Ċ = W − φC
Ṫ = log (1 + C ) − T
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Non-dimensional form
Ċ = W − φC
Ṫ = log (1 + C ) − T
Ẇ = x(T )W
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Non-dimensional form
Ċ = W − φC
Ṫ = log (1 + C ) − T
Ẇ = x(T )W
∃Te : x(Te ) = 0;
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
3 Variable Model: Non-dimensional form
Ċ = W − φC
Ṫ = log (1 + C ) − T
Ẇ = x(T )W
∃Te : x(Te ) = 0;
∂x()
< 0.
∂T
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Fast positive feedback and slow negative feedback
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
3 Variable Model: Projections
Decarbonisation assumedto be 1%,τT =τC =50 years,
∆T2∗CO2 =3K, ∂∂Ŵt̂ = ξ0 1 − 0.5T̂ Ŵ
CO2 concentration
Global Warming
Global Wealth
4
600
500
400
400
300
2000
$ tr.
K
ppmv
3
500
2
1
2050
2100
Year
2150
2200
0
2000
300
200
100
2050
2100
Year
2150
2200
0
2000
2050
2100
Year
2150
2200
Background rate of economic growth ξ0 = 4% (red) and 1% (green)
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
(In)stability of equilibria
Ċ = W − φC
Ṫ = log (1 + C ) − T
Ẇ = x(T )W
Linearise, let ∆C = C − Ce etc
˙  
∆C
 ∆T  = 
∆W

−φ
1
1+Ce
0


0
1
∆C
−1
0   ∆T  + H.O.T
dx(Te )
∆W
dT We x(Te )
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Stability of Positive Equilibrium depends on pace of
Expansion
Jacobian is

−φ

1
1+Ce
0

0 1
−1 0 
dx(Te )
dT We 0
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Stability of Positive Equilibrium depends on pace of
Expansion
Jacobian is

−φ

1
1+Ce
0

0 1
−1 0 
dx(Te )
dT We 0
Characteristic polynomial is
µ (µ + φ) (µ + 1) −
dx(Te )
dT We
1 + Ce
=0
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Stability of Positive Equilibrium depends on pace of
Expansion
Jacobian is

−φ

1
1+Ce
0

0 1
−1 0 
dx(Te )
dT We 0
Characteristic polynomial is
µ (µ + φ) (µ + 1) −
dx(Te )
dT We
1 + Ce
=0
I.e.
dx(Te ) µ (µ + φ) (µ + 1) −
φ 1 − e −Te = 0
| dT {z
}
h
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
Eigenvalues of Jacobian at positive equilibrium may have
Negative or Positive Real Part
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
Eigenvalues of Jacobian at positive equilibrium may have
Negative or Positive Real Part
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Stability in terms of dimensional variables
Unstable iff
−
−1
∆T2∗CO2 dx̂(T̂e ) 1
1
+
.
1 − e −x̂ (µ) log 2/∆T2∗CO2 >
log 2
τ
τ
dT̂
T
C
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
Stability in terms of dimensional variables
Unstable iff
−
−1
∆T2∗CO2 dx̂(T̂e ) 1
1
+
.
1 − e −x̂ (µ) log 2/∆T2∗CO2 >
log 2
τ
τ
dT̂
T
C
Oscillations if
−
1
−1
∆T2∗CO2 dx̂(T̂e ) 1 − e −x̂ (µ) log 2/∆T2∗CO2 > min
log 2
4
dT̂
1 1
,
τT τC
.
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
Stability in terms of dimensional variables
Unstable iff
−
−1
∆T2∗CO2 dx̂(T̂e ) 1
1
+
.
1 − e −x̂ (µ) log 2/∆T2∗CO2 >
log 2
τ
τ
dT̂
T
C
Oscillations if
−
1
−1
∆T2∗CO2 dx̂(T̂e ) 1 − e −x̂ (µ) log 2/∆T2∗CO2 > min
log 2
4
dT̂
1 1
,
τT τC
1 1
,
τT τC
.
Soft-landing if
1
−1
∆T2∗CO2 dx̂(T̂e ) −
1 − e −x̂ (µ) log 2/∆T2∗CO2 < min
log 2
8
dT̂
.
Motivation
Example:
Model definition
∂ Ŵ
∂ t̂
Stability of equilibrium
= ξ0 1 − d T̂ Ŵ .
Example: linear damage function
Summary
Motivation
Example:
Model definition
∂ Ŵ
∂ t̂
Stability of equilibrium
= ξ0 1 − d T̂ Ŵ .
d = 0.5 (left); d = 0.1 (right)
Example: linear damage function
Summary
Motivation
Example:
Model definition
∂ Ŵ
∂ t̂
Stability of equilibrium
= ξ0 1 − d T̂ Ŵ .
Example: linear damage function
Summary
Motivation
Model definition
Stability of equilibrium
Example: linear damage function
Summary
Summary
Coupled climate-economy dynamics are very different to
uncoupled dynamics.
Under the current model assumptions, an equilibrium is
possible in climate and emissions, with economic growth equal
to the decarbonisation rate. Thus mitigation is necessary for
sustainable growth.
Decarbonisation alone does not enable a soft-landing on the
equilibrium. Adaptation and an economic slow-down is
required.
Motivation
Model definition
Thank you!
Stability of equilibrium
Example: linear damage function
Summary