Doubling time

Last updated October 27, 2024

The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate (not the absolute growth rate) is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate.

As an example, Canada's net population growth was 2.7 percent in the year 2022, dividing 72 by 2.7 gives an approximate doubling time of about 27 years. Thus if that growth rate were to remain constant, Canada's population would double from its 2023 figure of about 39 million to about 78 million by 2050. ^[2]

History

The notion of doubling time dates to interest on loans in Babylonian mathematics. Clay tablets from circa 2000 BCE include the exercise "Given an interest rate of 1/60 per month (no compounding), come the doubling time." This yields an annual interest rate of 12/60 = 20%, and hence a doubling time of 100% growth/20% growth per year = 5 years.^[3]^[4] Further, repaying double the initial amount of a loan, after a fixed time, was common commercial practice of the period: a common Assyrian loan of 1900 BCE consisted of loaning 2 minas of gold, getting back 4 in five years,^[3] and an Egyptian proverb of the time was "If wealth is placed where it bears interest, it comes back to you redoubled."^[3]^[5]

Examination

Examining the doubling time can give a more intuitive sense of the long-term impact of growth than simply viewing the percentage growth rate.

For a constant growth rate of r % within time t, the formula for the doubling time T_d is given by

T_{d}=t{\frac {\ln(2)}{\ln(1+{\frac {r}{100}})}}

A common rule-of-thumb can be derived by Taylor series expanding the denominator ln(1+x) for x=0 using $ln(1+x)\approx x$ and ignoring higher order terms.

$T_{d}=t{\frac {ln{2}}{\frac {r}{100}}}=t{\frac {100*ln{2}}{r}}$

$\approx t{\frac {69.3}{r}}\approx t{\frac {70}{r}}$

This "Rule of 70" gives accurate doubling times to within 10% for growth rates less than 25% and within 20% for rates less than 60%. Larger growth rates result in the rule underestimating the doubling time by a larger margin.

Some doubling times calculated with this formula are shown in this table.

Simple doubling time formula:

N(t)=N_{0}2^{t/T_{d}}

where

N(t) = the number of objects at time t
T_d = doubling period (time it takes for object to double in number)
N₀ = initial number of objects
t = time

Doubling times T_d given constant r% growth

r%	T_d
0.1	693.49
0.2	346.92
0.3	231.40
0.4	173.63
0.5	138.98
0.6	115.87
0.7	99.36
0.8	86.99
0.9	77.36
1.0	69.66

r%	T_d
1.1	63.64
1.2	58.11
1.3	53.66
1.4	49.86
1.5	46.56
1.6	43.67
1.7	41.12
1.8	38.85
1.9	36.83
2.0	35.00

r%	T_d
2.1	33.35
2.2	31.85
2.3	30.48
2.4	29.23
2.5	28.07
2.6	27.00
2.7	26.02
2.8	25.10
2.9	24.25
3.0	23.45

r%	T_d
3.1	22.70
3.2	22.01
3.3	21.35
3.4	20.73
3.5	20.15
3.6	19.60
3.7	19.08
3.8	18.59
3.9	18.12
4.0	17.67

r%	T_d
4.1	17.25
4.2	16.85
4.3	16.46
4.4	16.10
4.5	15.75
4.6	15.41
4.7	15.09
4.8	14.78
4.9	14.49
5.0	14.21

r%	T_d
5.5	12.95
6.0	11.90
6.5	11.01
7.0	10.24
7.5	9.58
8.0	9.01
8.5	8.50
9.0	8.04
9.5	7.64
10.0	7.27

r%	T_d
11.0	6.64
12.0	6.12
13.0	5.67
14.0	5.29
15.0	4.96
16.0	4.67
17.0	4.41
18.0	4.19
19.0	3.98
20.0	3.80

r%	T_d
21.0	3.64
22.0	3.49
23.0	3.35
24.0	3.22
25.0	3.11
26.0	3.00
27.0	2.90
28.0	2.81
29.0	2.72
30.0	2.64

r%	T_d
31.0	2.57
32.0	2.50
33.0	2.43
34.0	2.37
35.0	2.31
36.0	2.25
37.0	2.20
38.0	2.15
39.0	2.10
40.0	2.06

r%	T_d
41.0	2.02
42.0	1.98
43.0	1.94
44.0	1.90
45.0	1.87
46.0	1.83
47.0	1.80
48.0	1.77
49.0	1.74
50.0	1.71

For example, with an annual growth rate of 4.8% the doubling time is 14.78 years, and a doubling time of 10 years corresponds to a growth rate between 7% and 7.5% (actually about 7.18%).

When applied to the constant growth in consumption of a resource, the total amount consumed in one doubling period equals the total amount consumed in all previous periods. This enabled U.S. President Jimmy Carter to note in a speech in 1977 that in each of the previous two decades the world had used more oil than in all of previous history (The roughly exponential growth in world oil consumption between 1950 and 1970 had a doubling period of under a decade).

Given two measurements of a growing quantity, q₁ at time t₁ and q₂ at time t₂, and assuming a constant growth rate, the doubling time can be calculated as

T_{d}=(t_{2}-t_{1})\cdot {\frac {\ln(2)}{\ln({\frac {q_{2}}{q_{1}}})}}.

Related concepts

The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or exponential decay is the half-life.

The equivalent concept in base-e is e-folding.

Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Cell culture doubling time

Cell doubling time can be calculated in the following way using growth rate (amount of doubling in one unit of time)

Growth rate:

N(t)=N_{0}e^{rt}

or

r={\frac {\ln \left(N(t)/N_{0}\right)}{t}}

where

$N(t)$ = the number of cells at time t
$N_{0}$ = the number of cells at time 0
$r$ = growth rate
$t$ = time (usually in hours)

Doubling time:

{\text{doubling time}}={\frac {\ln(2)}{\text{growth rate}}}

The following is the known doubling time for the following cells:

Cell types	Source	Doubling time
Mesenchymal Stem Cell	Mouse	21–23 hours^[6]
Cardiac/heart stem cell	Human	29 ± 10 hours^[7]

Related Research Articles

The number $e$ is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.$

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".$

In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values. The geometric mean is defined as the $n$ th root of the product of $n$ numbers, i.e., for a collection of numbers $a 1, a 2, ..., a n$ , the geometric mean is defined as

Half-life is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential decay. For example, the medical sciences refer to the biological half-life of drugs and other chemicals in the human body. The converse of half-life is doubling time.

In mathematics, the logarithm to base $b$ is the inverse function of exponentiation with base $b$ . That means that the logarithm of a number $x$ to the base $b$ is the exponent to which $b$ must be raised to produce $x$ . For example, since $1000 = 10 3$ , the logarithm base $of 1000 is 3, or log 10 (1000) = 3 . The logarithm of x to base b is denoted as log b (x), or without parentheses, log b x . When the base is clear from the context or is irrelevant it is sometimes written log x .$

In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining the rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula. Currently, it is best seen as an empirical relationship. It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally induced processes and reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.

<span class="mw-page-title-main">Logistic function</span> S-shaped curve

A logistic function or logistic curve is a common S-shaped curve with the equation

<span class="mw-page-title-main">Exponential growth</span> Growth of quantities at rate proportional to the current amount

Exponential growth occurs when a quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

<span class="mw-page-title-main">Exponential decay</span> Decrease in value at a rate proportional to the current value

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where $N$ is the quantity and $λ$ (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant:

In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number is divided by the interest percentage per period to obtain the approximate number of periods required for doubling. Although scientific calculators and spreadsheet programs have functions to find the accurate doubling time, the rules are useful for mental calculations and when only a basic calculator is available.

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.

Specific activity is the activity per unit mass of a radionuclide and is a physical property of that radionuclide. It is usually given in units of becquerel per kilogram (Bq/kg), but another commonly used unit of specific activity is the curie per gram (Ci/g).

In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid, surface friction, radiation, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes. Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.

In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e; it is the base-e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine and theoretical physics, especially when cosmic inflation is investigated. Physicists and chemists often talk about the e-folding time scale that is determined by the proper time in which the length of a patch of space or spacetime increases by the factor e mentioned above.

The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. The right-side or future value asymptote of the function is approached much more gradually by the curve than the left-side or lower valued asymptote. This is in contrast to the simple logistic function in which both asymptotes are approached by the curve symmetrically. It is a special case of the generalised logistic function. The function was originally designed to describe human mortality, but since has been modified to be applied in biology, with regard to detailing populations.

A double exponential function is a constant raised to the power of an exponential function. The general formula is $(where a >1 and b >1), which grows much more quickly than an exponential function. For example, if a = b = 10:$

In physics and engineering, the time constant, usually denoted by the Greek letter $τ$ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. The time constant is the main characteristic unit of a first-order LTI system. It gives speed of the response.

Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.

<span class="mw-page-title-main">Pest insect population dynamics</span>

The population dynamics of pest insects is a subject of interest to farmers, agricultural economists, ecologists, and those concerned with animal welfare.

References

↑ Donella Meadows, Thinking in Systems: A Primer, Chelsea Green Publishing, 2008, page 33 (box "Hint on reinforcing feedback loops and doubling time").
↑ "The Daily — Canada's population estimates: Record-high population growth in 2022". 22 March 2023.
1 2 3 Why the “Miracle of Compound Interest” leads to Financial Crises, by Michael Hudson
↑ Have we caught your interest? by John H. Webb
↑ Miriam Lichtheim, Ancient Egyptian Literature, II:135.
↑ "Life Technologies" (PDF). Archived from the original (PDF) on 2015-04-02. Retrieved 2015-03-12.
↑ Bearzi, Claudia; et al. (August 2007). "Human cardiac stem cells". Proceedings of the National Academy of Sciences. 104 (35): 14068–14073. Bibcode:2007PNAS..10414068B. doi: 10.1073/pnas.0706760104 . PMC 1955818 . PMID 17709737.

External links

Doubling Time Calculator
http://geography.about.com/od/populationgeography/a/populationgrow.htm Archived 2006-08-28 at the Wayback Machine

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Donella Meadows, Thinking in Systems: A Primer, Chelsea Green Publishing, 2008, page 33 (box "Hint on reinforcing feedback loops and doubling time").

[2] "The Daily — Canada's population estimates: Record-high population growth in 2022". 22 March 2023.

[hudson-3] 1 2 3 Why the “Miracle of Compound Interest” leads to Financial Crises, by Michael Hudson

[4] Have we caught your interest? by John H. Webb

[5] Miriam Lichtheim, Ancient Egyptian Literature, II:135.

[6] "Life Technologies" (PDF). Archived from the original (PDF) on 2015-04-02. Retrieved 2015-03-12.

[7] Bearzi, Claudia; et al. (August 2007). "Human cardiac stem cells". Proceedings of the National Academy of Sciences. 104 (35): 14068–14073. Bibcode:2007PNAS..10414068B. doi: 10.1073/pnas.0706760104 . PMC 1955818 . PMID 17709737.

[1]

[2]

[3]

[4]

[5]

[6]

[7]