In this video I go over further into differential equations, and this time look at the proportionality constant used in population growth. The proportionality constant represents the relative growth rate and states that, for the example of population, it is constant. This basically means that the rate at which a population grows is relatively the same as another population, but scaled up due to size. For example, two populations can be growing at the same relative rate, i.e. 2% per year, while the larger one in fact grows at an absolute rate faster than the smaller population because 2% of a larger population is also larger. This is a very interesting video on the proportionality constant and will serve as a basis for future example videos on population growth so make sure to watch it!
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Population Growth
What is the significance of the proportionality constant k?
In the context of population growth, we can write:
The quantity:
is the growth rate divided by the population size and it is relative growth rate.
According to the governing differential equation above instead of saying "the growth rate is proportional to population size" we could say "the relative growth rate is constant."
Then, since the solution to this differential equation is an exponential function (as explained in my earlier video), the population with a constant relative growth rate must grow exponentially.
Notice that the relative growth rate k appears as the coefficient of t in the exponential function y0ekt.
For instance, if:
and t is measured in years, then the relative growth rate is k = 0.02 (units of 1/year) and the population grows at a relative rate of 2% per year.
If the population at time 0 is P0, then the expression for the population is:
