# Population growth

Are we doomed? Maths says no!

# Debrief

Last post I published a guide on quotient structures in mathematics, this time I will write about ordinary differential equations and specifically use simple models to describe population growth.

The question that we will solve this time is of major importance society wise and has caused major controversy in politics and media, the exact question is:

How can we model population growth?

Turns out maths can be used to solve this question, however we have to put some things in line first.

# Non-Trivial facts about population

**Let's talk about some cool non-intuitive facts about population data:**

**The entire world population could fit inside Texas**: While it’s technically true that the world’s population could physically fit within the land area of Texas, this doesn’t account for the resources and infrastructure needed to support such a large number of people.**Religion is the main cause of rapid population growth**: Although certain religious institutions may oppose birth control, the actual behaviour of followers often differs. For example, countries with large Catholic populations like Italy and Spain have some of the lowest fertility rates in the world1.**Slower population growth harms the economy**: While some argue that a declining population can lead to economic challenges, others point out that sustainable development and technological advancements can offset these issues.**The world population growth rate is under 1%, so it’s barely growing**: Even a seemingly small growth rate can result in significant increases in population over time. For instance, a 1.2% growth rate adds roughly 80 million people to the global population each year.**Overpopulation will inevitably lead to resource scarcity**: This is a common fear, but human innovation and technological progress have historically found ways to increase resource efficiency and production.**Population growth will stop soon anyway**: While growth rates are slowing in many parts of the world due to better access to education, healthcare, and family planning, global population is still expected to increase significantly before stabilizing

Here is a picture about human population worldwide:

and here is a human population growth graph of some of the biggest countries population wise.

As we can see, the tendency is not really to go on an exponential rise, but a somewhat stabilisation in the future (so space for everyone !). We want to model this data with maths, for this we use a model called **Logistic Growth Law**.

# Ordinary Differential Equations (ODE)

## Disclaimer

If you have not learned about mathematical derivatives, then give it a check here.

ODE are very interesting and extremely difficult to solve analytically. We use them to model pretty much everything in nature, after all nature follows differential equations, but what are they?

An **Ordinary differential equation (ODE)** is a mathematical equation that involves the derivatives of an unknown function y with respect to a single independent variable x.

Formally, it can be expressed as: F(x, y, y′, y′′, … , y^(n))=0 where:

- x is the independent variable
- y is the dependent variable
- y’, y’’, … , y^(n) are the first, second, and higher-order derivatives of y with respect to x

For example, we might have the differential equation given by: y’’ + 2y - 1=0 by itself this equation does not tell us too much about it, however we if we try to graph the vector field associated with it then it is given by:

To solve the equation mentioned above, you should drop a particle in one of the arrows and just imagine it flowing overtime in the direction of the arrows. (What happens in the origin?)

there are some fascinating models when it comes to ODE that we might explore in next articles 👀 but for now let's just stick with the Logistic Growth Law.

# Logistic Growth Law

The Logistic Growth Law is given by any function which respects the following ODE

And this equation turns out to model reasonably well the way population grows, in particular if you put P(t) as number people, B(t) as the birth rate and the D(t) as the death rate then it is not completely unreasonable to define B(t) = P(t) and D(t) = P²(t) as with more people comes a higher birth and death rate, the balance between birth and death rate are given by B(t)-D(t) and so we might say that the rate of change of the population in given by:

this ODE puts us in a reasonable foot to model population growth, its solution is given by

where L is the carrying capacity, the supremum of the values of the function; k is the logistic growth rate, the steepness of the curve; and x_0 is the x value of the function’s midpoint. A solution graph is given here:

If you want to play with the parameters, go here to my desmos animation.

# Conclusion

In this article, we’ve explored the complex issue of population growth using mathematical models. By applying the Logistic Growth Law, an ordinary differential equation, we’ve demonstrated that while population growth is a significant concern, it doesn’t necessarily doom us to resource scarcity or societal collapse.

The logistic model suggests that population growth tends to stabilize over time, reaching a carrying capacity determined by environmental factors. This implies that human ingenuity and technological advancements can help us adapt to a growing population while ensuring sustainable resource management.

While the model provides a valuable framework for understanding population dynamics, it’s essential to acknowledge its limitations. Factors such as climate change, geopolitical tensions, and disease outbreaks can significantly impact population growth and resource availability.

In conclusion, while the future of population growth presents challenges, mathematical modelling offers a valuable tool for understanding and addressing these issues. By combining scientific knowledge, technological innovation, and responsible governance, we can work towards a sustainable and equitable future for all.