Geometry and population growth are two issue in different fields that are somewhat related. The relationship in these two topics is mainly where one relies on another to get it value; this it to mean that population growth uses geometry to measure its parameters. Geometric growth rate is one of the formulas that are used to measure the growth rate in population. In geometric growth rates, each number is multiplied by a factor, for instance, 1, 4, 16, 64, 256, 1024, 4096, 16384, …….. This geometric series has been multiplied by a factor of 4, this kind of progression or growth is sometimes called exponential growth to mean something that grows very fast (Galan, Luard and Mazzucchelli, 65).
Geometric growth can be contrasted to arithmetic growth rate, which grows in a sequence, for instance 1, 5, 9, 13, 17, 21, 25, 29, 33, 37, ….
One of the principles behind geometric growth is that the bigger a number gets, the faster it grows, this is the case with population since the larger the population becomes, more people will be available for reproduction hence the greater growth (Berlatsky, 133). For instance, a population having 500,000 people will grow five times faster than a population which has just 100, 000.
Geometric growth rate curve grows like this
In order to calculate geometric growth for population in a given time, the following formula can be used.
Assume population x depends exponentially on time t, then
x(t) = a.bt/r
where a is the initial value of the population meaning that
x.0 = a,
b is a constant growth factor and r is the time that is required for population to increase by a unit factor of b, therefore;
x(t +r) = a.bt+r/r = a.bt/r + br/r = x(t).b
From this equation, if r is greater than zero and b is greater than one, then the population will grow at exponentially, however, if r is less than zero and b is greater than one or if r is greater than one and b is between one and zero, then the population will reduce exponentially. A geometric decay curve for population would look something like this
Example
The initial population of a city is 100, 000 people, it is believed that the population doubles after every five year, find the population after 20 years and plot it in a graph.
Solution
x(t) = a.bt/r
x(20) = 100, 000 · 220/5 = 100, 000 · 24
= 100, 000 · 16 = 1, 600, 000
Therefore the population after 20 year will be 1, 600, 000 people
In a graph
Geometric (continuous growth) model
Geometric continuous time model of population growth is more realistic when it comes to determining population growth since they involve all parameters of population such as birth rates and death rate (Turchin, p96)
In order to calculate the population growth rate using the continuous growth model, the following formula can be derived. If a population has Nt individuals where t is time in years, the number of children being born in a year is a fraction represented by the symbol β and the number of people that die in a year is a fraction that is represented by the symbol γ.
Solving this equation further would give the following final equation
These geometric progression equations have been used for a long time to calculate population growth rates of countries or of the world at large, for instance, the following figure show a curve of world growth rate that was calculated from the available data and backward projections of population.
Retrieved from: http://www.learner.org/courses/envsci/unit/text.php?unit=5&secNum=2.
Geometric progression has been use in population growth and without it, calculating projections on growth rates would be very difficult, the reason why it has been used in calculating population growth is because of the fact that population grows at an exponential rate with a certain factor from the initial population. Several formulas can be derived for calculating population growth rate depending on whether they are discrete or continuous models, however, continuous models are the most appropriate since they are more realistic.
Works cited
Berlatsky, Noah. Population Growth. Detroit: Greenhaven Press, 2009. Print.
Galan, Marcius, Luard, Honey and Mazzucchelli, Kiki. Marcius Galan: Geometric Progression. London: White Cube, 2013. Print.
Turchin, Peter. Complex Population Dynamics: A Theoretical/empirical Synthesis. Princeton: Princeton University Press, 2013. Internet resource.
