Numerous scenarios of emissions growth presume that it can not grow infinitely: there are certain limits that may not or must not be exceeded. Applying the theory of pulsing logistic growth, I detected two pulses of logistic growth. This led me to the conclusion that constructing a scenario of emissions growth one should pay attention to the fact that limits to their growth do not remain constant

Gleb G. Alexandrov

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Gleb Alexandrov

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Decomposing CO2 emissions data into a pair of logistic growth pulses

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DECOMPOSING CO2 EMISSIONS DATA INTO A PAIR OF LOGISTIC GROWTH PULSES 
G.G. Alexandrov 
Faculty of Applied Mathematics Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskii 3, Moscow 
109028 Russia; g.alex2006@yahoo.com 
 
INTRODUCTION 
Carbon dioxide emissions is a heavily discussed topic 
since 1970s. It is well-recognized that their growth, 
associated with industrial development, may have 
negative consequences, and should be put under 
control [Heimann and Reichstein, 2008].  Numerous 
scenarios of emissions growth presume that it can not 
grow infinitely: there are certain limits that may not or 
must not be exceeded.  The purpose of my work is to 
analyze the limits to CO2 emissions growth during 
1850-2004 and to consider how they may change in 
the future. 
 
METHODS 
Applying the theory of pulsing logistic growth 
[Ausubel, 1996; Meyer, 1994], I assume that CO2 
emissions grow in agreement with the equation    
 
 
 
where K(t) is carrying capacity (or limit to growth) at 
the moment t, m is relative increment of y when y(t)
<<K(t).  
In order to find changing limits to growth, I find the 
general solution of the Equation 1 
 
 
 
 
approximate K(t) as follows 
 
 
 
 
 
 
and set the constant to be  
 
 
 
 
The graphics displayed at the Fig 1 corresponds to 
the case of L=155, m=0.045, y0=60, n=2, a0= -
0.000258168, a1= -0.0000440057, a2= 0.000302174, 
b1= 0.000951037, b2= 0.000375412 
 
Fig. 1  CO2 emissions and their limits to growth (carrying capacity). Units: 
MtC/y. Legend: thick dot-dashed  line  - carrying capacity, dots – CO2 emis-
sions data [Marland et al., 2008], solid line – solution of the Equation (1), 
dashed line – exponential growth (m=0.045). 
RESULTS 
Results, shown on Fig. 1, allow us to detect two pulses of logistic growth.  
The first pulse, with the carrying capacity of 2±0.8 GtC/y, culminated in 
1930s. The second pulse, with the carrying capacity of 12±2 GtC/y, recently 
entered the culmination phase. Unexpected acceleration of emissions 
growth observed in 2000-2004 [Raupach et al., 2007] may indicate to the 
start of the third pulse, which carrying capacity is even higher. 
 
CONCLUSIONS 
Obviously, constructing a scenario of emissions growth one should pay at-
tention to the fact that limits to their growth do not remain constant. They 
were and would be changing because of technological advances. The nu-
merical method, developed as a result of this study, expands opportunities 
for use of the theory of pulsing growth in constructing emissions scenarios. 
 
REFERENCES 
Ausubel, J.H. (1996), "Can Technology Spare The Earth", American Scien-
tist Magazine, 84(2):166-178. 
Heimann, M. and M. Reichstein (2008), Terrestrial ecosystem carbon dy-
namics and climate   feedbacks, Nature, 451, 289-292. 
Marland, G., T.A. Boden, and R.J. Andres (2008), Global, Regional, and Na-
tional Fossil Fuel CO2  Emissions, in Trends: A Compendium of Data on 
Global Change, Carbon Dioxide Information Analysis Center, Oak Ridge 
National Laboratory, U.S. Department of Energy, Oak Ridge, Tenn., U.S.A.  
Meyer, P. (1994), Bi-logistic growth. Technological Forecasting and Social 
Change, 47:89-102. 
Raupach, M.R., G. Marland,  P. Ciais, C. Le Quere,  J.G. Canadell,  G. 
Klepper,  C.B. Field (2007), Global and regional drivers of accelerating CO2 
emissions, PNAS, 104: 10288-10293. 
ABSTRACT 
Numerous scenarios of emissions growth presume that it can not grow infinitely: there are certain limits that may not or must not be 
exceeded.  Applying the theory of pulsing logistic growth, I detected two pulses of logistic growth. This led me to the conclusion that 
constructing a scenario of emissions growth one should pay attention to the fact that limits to their growth do not remain constant. 
 1    
𝑑𝑦
𝑑𝑡
= 𝑚 𝑦  1 −
𝑦
𝐾 𝑡 
 ;  𝑦(𝑡0) = 𝑦0  
C =
1
y0
−
1
K(t0)
 
y =
1
Ce−mt + e−mt  
memt  dt
K t 
   
K(t) =
1
a0 +  (An(t) + Bn(t))
N
n=1
 
An t = an cos  
nπ
L
  t − t0  
Bn t = bn sin  
nπ
L
 (t − t0) 
N
at
ur
e 
Pr
ec
ed
in
gs
 : 
do
i:1
0.
10
38
/n
pr
e.
20
10
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30
6.
1 
: P
os
te
d 
24
 M
ar
 2
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Can Technology Spare The Earth&quot;,

Global, Regional, and National Fossil Fuel CO2 Emissions, in Trends: A Compendium of Data on Global Change, Carbon Dioxide Information Analysis

Terrestrial ecosystem carbon dynamics and climate feedbacks,

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Decomposing CO2 emissions data into a pair of logistic growth pulses

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