I recently took the AP Calculus BC exam and I learned a lot of new concepts. While studying for the exam, I enjoyed the topics and wanted to learn in-depth about them. There were major concepts like The Mean Value Theorem, Fundamental theorems of calculus, Reimann sums for approximation, Logistic growth models, and Taylor series. After researching about uses and applications of Logistic Growth Models I realised its importance and vast uses in economics and biology. Today, I am going to share a bit of what I have learned about Logistic Growth models.
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Logistic growth models are models and functions used to predict the population of an organism through any stage of growth. In theory, an organism can take over the planet by reproducing infinitely. Since that has not happened yet, we realise something is limiting the growth. As the population increases, there are more organisms to reproduce, and therefore the rate of growth increases. This is trivial, but one would need to account for the limiting or carrying capacity.
Let P(t) be a function of time that represents the population. At t = 0, P(t) must be greater than 0 or there is no population to reproduce.
The limiting population can be determined by many factors such as famines, natural disasters, limited resources, etc. As the population gets closer and closer to the limiting population the rate of growth decreases. This is an example of what the graph of the population would look like.
Although the actual population would fluctuate up and down from the carrying capacity,
for a model it is pretty accurate.
Why are logistical growth models important?
A typical application of the logistic equation is a common model of population growth originally due to Pierre-François Verhulst in 1838. The maximum value of any continuous function can be modelled through logistic growth graphs. It can be used to represent the growth of companies where the limiting value depends upon the market share and the total market for a product. The most relevant example of its use today: Predicting, approximating, and analysing the maximum population and growth of the spread of COVID-19 disease caused by the SARS CoV-2 coronavirus. Many mathematicians and doctors have used logistic growth models and exponential models for analysing the growth of COVID-19, although the differential equations are much more complex and the limiting value is dependent upon numerous factors.
Mathematics of the model.
Differential Calculus is used heavily to understand these models.
The growth of the population
The increase in population is represented as dP/dt.
dP/dt is proportional to the current population, as well as how close it is to the limiting population.
The differential equation can be represented as :
dP/dt = KP(L-P) or KP(1-P/L)
[K is the proportionality constant which is different for both representations].
As t approaches infinity, P(t) approaches L.
This can be proved by using the concept of absolute maximum of a function.
As the population is continuous,
The extrema value theorem guarantees a maximum value on a continuous interval.
Absolute maximum will be when the rate of growth becomes 0 (Population stops growing).
dP/dt = 0, at P = 0 or (L-P) = 0.
as P is not 0, the absolute maximum occurs at P(t) = L.
As P approaches L, the growth rate approaches 0.
Thus as time passes on, the population gets closer to the Limiting capacity but never reaches it .
Thus we define the carrying limit as:
L = Lim P(t) as t -> infinity
This can also be represented as a horizontal asymptote of P(t) at P = L.
Rate of change in the growth of P.
The rate at which the population increases is represented by the first derivative of P(t).
P'(t) = KP(L-P).
P'(t) = KPL - KP^2.
But the rate at which the growth of population changes is represented by the second derivative of P(t). It is essentially the increasing or decreasing rate at which Population increases.
P''(t) = [ KL - 2KP ](dP/dt) = [ KL - 2KP ] * [KP(L-P)].
P(t) is increasing at the maximum rate when P'(t) is maximum and P''(t) = 0.
P''(t) = [ KL - 2KP ] * [KP(L-P)] = 0,
at [KP(L-P)] = 0 or [ KL - 2KP ] = 0.
As [KP(L-P)] = P'(t) which we need to be the maximum,
P(t) is increasing at the maximum rate when [ KL - 2KP ] = 0.
or P(t) = L/2.
This can visibly be noticed from the graph of P(t) which increases at the maximum rate when P=L/2. (At P = L/2, the concavity of P(t) changes from concave up to concave down).