Exponential Models (Explaining Calculus #11)

We’ve used calculus for a couple different applications at this point. I’d like to tack on another application to our list, perhaps the most important one – at least from the perspective of a society very reliant on technology, engineering, and science. The application I have in mind goes under the very broad title of mathematical modelling.

Mathematical Modelling

Mathematical modelling is an extremely broad term – so broad that it is a bit difficult to define. I would argue that large sections of physics, chemistry, engineering, and statistics (including polling in the social sciences) fall in one sense or another under this category. If I had to give a definition, I would say that mathematical modelling is the practice of framing patterns observed in the external world in mathematical language. So, mathematical modelling would involve first gathering some sort of data from the real world and subsequently attempting to explain that data by a single theory that utilizes some mathematical formulas.

Large portions of modern science were formulated this way. For example, because of the great ancient philosopher Aristotle, for more than a thousand years it was believed that heavy objects fall faster than lighter objects. This was believed not on the basis of gathering data, but on intuition. Perhaps you might think something like “well, since it is more difficult to prevent a heavy object from falling, it would make sense that a heavy object ought to fall faster”. And there is certainly intuitive plausibility here, but it doesn’t count as mathematical modelling for two main reasons. For one, there is not an equation given telling us how much quicker heavy objects fall. Secondly, the theory was not formulated by data.

This isn’t necessarily a bad thing for Aristotle’s theory – Einstein did not form his theory of relativity based on observational data either, but on more philosophical/physical intuitions. So Einstein was not engaged in mathematical modelling, yet his theory ended up being perfectly aligned with data once data was collected. But Aristotle’s theory turned out to conflict with data – notably Galileo did famous experiments that proved this wrong. Galileo’s work is much closer to mathematical modelling, because his theory arose directly from his data.

But even better would be a formulation of the data-based theory of Galileo in formulas. This is more or less what Isaac Newton did when he formulated his theory of gravity. Newton used the language of calculus to describe exactly what it means that objects of different weights fall at the same speed. Using the mathematical model of Newton, you can make predictions about a large range of moving objects – from footballs to planets.

Notice what has been done here. We start with Galileo dropping objects that weight different amounts and noticing they hit the ground at the same time. Obviously Galileo didn’t drop every possible object (who has the time for that), but he didn’t need to. From the experiments he did do, he formulated a theory – a model – that predicts that all objects always fall at the same speed towards the ground when dropped on Earth. Newton took that model, made it more precise mathematically, and expanded its scope to correctly predict much more data than Galileo’s theory did. This is one example of the way of thinking embodied by mathematical models.

In a later post, I plan to show how Newton’s model works. But before we go into that model, I want to talk about what I would consider to be a simpler model – one we can formulate together from the ground up.

Example: Growing Population

The example we will look at is the growth in population of living beings. The same basic model will work for any creature – whether bacteria, rabbits, or humans. I opt here to use human population growth. To begin, we’ll look at a graph of the estimated growth of the population I found.

The growth in human population over time (from Wikipedia Commons)

This is a rather odd graph. It is basically completely flat for a long time, then it just explodes. How do we understand a graph like this? Well, it all begins with a simple principle.

The more humans there are, the more babies they will have.

This seems rather obvious as a general rule. Since individual women are the ones that give birth to new humans, it would naturally follow that the more women there are, the more babies there can be in the future. And yet it is all we need to get a mathematical model started. How? Well, this is where mathematical modelling comes in.

Making the Model

How do we turn out observation into a model? Well, since we want a mathematical equation, let’s start by defining a mathematical symbol we can use to count the population of all humans. Let’s say

P(t) = Number of people alive t year after the first humans existed.

We are building a mathematical model based on the observation “the more humans there are, the more babies they will have“. In order to make things easier for us, we are going to pretend that only one generation of people is alive at a time. This just makes our counting job easier. Later on we will talk about how a mathematical modeler would go about making their model better able to account for multiple generations at a time. For now, we will leave

Ok. Now, let’s sit down and think about to take our observation that “the more humans there are, the more babies they will have” and make it into a formula. To see how this works, let’s imagine that there are currently 1000 humans alive on earth. That would be about 500 women – let’s just say exactly 500 women. Now, we reach a decision point in our model – how many children are these women having? We don’t really have to ask about each of the 500 women – an average will be good enough. Let’s imagine that, on average, each woman has 3 children in their life. How many people will be in the next generation? Well, that would be the number of women times the average number of children per woman – which leads us to 500 * 3 = 1500 people in the next generation (750 women). If the average number of children per woman stays at 3, then the next generation will have 750*3 = 2250 people. If we keep up this process, we will get bigger and bigger population sizes.

Now, how might be try to put this into a formula? This is where calculus becomes so important. Because what we are trying to figure out is how the total population changes over time – and change over time is exactly what calculus is meant to help us with! Notice now what we’ve done in the previous paragraph. There were two factors that told us how to predict the size of the next generation of humanity:

  1. The number of people alive currently.
  2. The average number of babies each woman has.

In the spirit of making this into an equation, we should express both of these in a mathematical way. We’ve already set up the function P(t) as the number of people alive at a specific time, so (1) is given to us by P(t). For the second one, we can use the letter B (short for ‘babies’) to stand in for the average number of babies each woman has. Now, when we calculated the size of each generation earlier, we multiplied the number of women by the average number of babies per woman. If our current generation is P(t) people, that would be about \dfrac{1}{2} P(t) women, and so the number of people in the next generation would be about

\dfrac{1}{2} P(t) * B = \bigg( \dfrac{B}{2} \bigg) P(t).

This leads us to an interesting formula. What if we want to know how quickly the human population is growing? Well, let’s do this. If we want to know how much bigger (or smaller) the new generation is compared to the old one, we just subtract these values from each other. In the example earlier, the first and second generation differed by 1500 – 1000 = 500 people, and the second and third differed by 2250 – 1500 = 750 people. If we use our big-picture formulas, along with the derivative P^\prime(t) to stand in for the change in population (since derivatives literally are changes over time), then we wind up with a formula “Change in Population = New Generation Size – Old Generation Size”, which if we put it in the language of calculus, leads us to

P^\prime(t) = \bigg( \dfrac{B}{2} P(t) \bigg) - P(t) = \bigg( \dfrac{B}{2} - 1 \bigg) P(t).

So, if we know B, then we can use this new formula to approximate how the population of humans will change over time.

Solving the Key Equation

We now have a key equation we want to solve. We’ve found an equation for the change over time that the population exhibits:

P^\prime(t) = \bigg( \dfrac{B}{2} - 1 \bigg) P(t).

We now want to solve this equation. This equation is basically just “Derivative = Constant * Population”, so it will be a bit easier to just write a single letter C instead of the more cumbersome \dfrac{B}{2} - 1. So, we will think about the equation

P^\prime(t) = C * P(t).

Now, this is an equation in the world of calculus. It would then be natural, as a mathematical modeler, to ask if we know about any functions P(t) that solve this equation. We actually don’t quite have the tools yet to solve this equation without already knowing the answer (we need something called antiderivatives for that), but in the examples of derivatives we’ve already talked about in previous posts, we already have an example of a function that works – the exponential functions. From the ‘calculating derivatives’ posts, I showed that if f(x) = b^x for some constant b, then f^\prime(x) = \log{b} * b^x = \log{b} * f(x). So, if we choose the value of b so that C = \log{b}, then f^\prime(x) = C * f(x). If we solve for b, then we find that b = e^C, and therefore we conclude that

f(x) = b^x = (e^C)^x = e^{Cx}

is a solution to the population growth equation P^\prime(t) = C * P(t). Now, does this actually work? Well, if you graph functions like b^x for values of b larger than 1, then the shape you get is a lot like the graph I started the post with – initially very flat, but with a sudden explosion later on. So we seem to have done a pretty good job.

Application to Population Growth

What does this mean for predicting future population growth? Well, we’ve already established that if you know the average number of babies each woman has, we can use that to form an equation predicting how quickly the population of the world will grow. So, the main thing we have to keep track of is this average.

But we can go even deeper. Recall the original formula we came up with – if B is the average number of children per woman and P(t) is the population, then we found the equation

P^\prime(t) = \bigg( \dfrac{B}{2} - 1 \bigg) P(t).

Perhaps you are wondering whether the global population is increasing or decreasing. We can use the value of B to predict this. If the value of \dfrac{B}{2} - 1 is positive, then the right-hand side of the above formula is a positive number, and this means the population is increasing. If on the other hand \dfrac{B}{2} - 1 is negative, then the right-hand side is negative, so the population will be decreasing. It is fairly easy to figure out that growing populations happen when B > 2 and shrinking populations happen when B < 2. This mathematical model has given us a quite interesting result then – we can determine whether the population is growing or shrinking without actually counting the population. That is a pretty neat consequence that we likely didn’t know or wouldn’t have thought of before we made our model.

We could draw tons of other conclusions from this model. I won’t do so here. My point is just to show how calculus played a role in developing this theory of population growth.

How to Improve Your Model

There are, of course, lots of ways we could improve the model. We could account for the fact that some people die before having children, we can account for the fact that multiple generations are alive at the same time, we can account for medical advancements that make it easier for some people to have children, we can account for cultural factors that change people’s desires about having children either up or down over time, we can factor in random events like plagues that cause huge decreases in population over short times. If we factor in those new observations into our mathematical model, our predictions will get better and better.

Lots of people have done this. There are plenty of models out there that take many of these factors into account. But, in the end, all of them are still basically exponential – they look very much like b^x. I’ll give one example of how this works. Say I want to add to my observation that only people in a certain age range can have babies. Then the equation you end up with is something like

“Babies this year = Babies from 20 year-olds + Babies from 21 year-olds + … + Babies from 45 year-olds.”

What ends up happening is you get something like the famous Fibonacci sequence. This sequence begins with F_1 = 1, F_2 = 1, and in order to get the next item in the list you add together the previous two. The first several numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. In my series on the Fibonacci numbers, I showed that these number are very, very close to the exponential function \bigg( \dfrac{1 + \sqrt{5}}{2} \bigg)^x. Check out that series if you want to see how you could figure out something like this.

Modifications like these do have significant impacts on the model, but they don’t prevent it from being like b^x – more often they cause changes to the value of b, which still does radically alter how quickly the population grows. Despite this change, these modifications don’t tend to change the shape of the graph overall – it will still start very flat and then explode. The alterations tend to cause slight delays in the explosion or slightly slow down the explosion in the long-term.

There is a major exception that is worth addressing. One is that the observation we made earlier about whether B is greater than or less than 2 isn’t actually quite right. It isn’t quite right because some children die before they reach adulthood, and so never even have the option to have children. In reality, if you look at academic sociological polling (like Pew Research), you’ll see the number 2.1 in place of two here. What you’ll also realize is that different cultures tend to have different values of B – countries with lots of Muslims, for example, tend to have very high values of B compared with more secularized countries. These cultural differences actually tend to, over time, skew the overall, global value of B towards the higher side, since the cultures with high values of B grow faster than cultures with low values of B. These kinds of cultural considerations can actually change the value of B from one generation to the next, and so this has a significant impact on the graph. On the other hand, if enough cultures wind up with a value of B smaller than 2.1, then the graph will stop exploding altogether and start leveling off, as if there were an ‘imaginary ceiling’ the graph isn’t allowed to cross.

So, if you want to make predictions about population growth, probably the most important factor you need to account for is the different numbers of children being born in different geographical areas and cultures.


I’ve been a bit vague at points in this discussion. This is partly because I do not specialize in mathematical modeling – I would want to leave really specific details to professionals. My goal is not to develop a sophisticated mathematical model – my point is to show how calculus can be useful in formulating and studying these mathematical models and the kinds of ways you might think about trying to build your own model. As we go on in the calculus series, it is important to keep in mind the importance of the concepts in calculus in putting into mathematical language concepts about growth and change that come up so often in trying to make predictions about the world around us.

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