Circle of Life Brady and Butler

The Circle of Life: The Mathematics of

Predator-Prey Dynamics
Rebecca M Brady1,2, John S Butler1,2, *
1 School of Mathematical Sciences, Technological University Dublin, City Campus, Dublin
2 ESHI (Environmental Sustainability and Health Institute), Technological University Dublin,
City Campus, Dublin
* corresponding author
John.s.butler@tudublin.ie

Word count
Abstract: 104 words
Main document: 2415 words

Number of Figures: 3

Number of References: 6

Abstract

Some animals hunt other animals in order to feed themselves; these are called predators.
Animals who are hunted and eaten are known as prey. What do you think would happen if a
predator was introduced to an ecosystem where the prey previously lived without fear of
being hunted? Would the new predator eat all the prey animals until they go extinct?
Actually, the relationship between predator and prey is far more interesting than this. In this
article, we show what this relationship looks like and explain how scientists can make
predictions about future population levels all using simple mathematics like addition,
subtraction and multiplication.

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Introduction
Why do we study animal populations?

Scientists need to collect information so they can understand how to protect the
environment and the animals who live in it. To do this they might want to understand how
animal populations change over time (are there more births or deaths happening?). They
might want to know how this species of animal is interacting with other animals and their
environment. They sometimes use mathematics to test theories they have about the
animals or even try to predict the future! This is called mathematical modeling. Investigating
and modeling the relationship between predator and prey helps scientists understand the
natural rise and fall of a population and can let them know when the animal could be at risk
of extinction. To make a successful model Scientists need to collect data, so they can see
what is happening now in the environment. They can then use mathematics to model and
predict what might happen. In this paper we will show how some simple mathematics like
addition, subtraction and multiplication can be used to model the predator-prey
relationship seen in the wild.

Data and Models

But first, we need data!

Good models start with good data. Data is any information that can be recorded, such as
facts or numbers. In the case of a predator and prey relationship, data could be a record of
how many animals are in each population at any given time. Interestingly one of the first
observations of the rise and fall of predator-prey populations was by a company that was
hunting both the predator and prey for their fur in the 19th and 20th century. The Hudson
Bay Company made yearly records of the amount of Snow Lynx and Snowshoe Hare pelts
they collected shown in Figure 1. When the Snowshoe Hare population was large the
Hudson Bay Company would collect more hare pelts and when the population was small,
they would collect fewer hare pelts. The same is true for the Snow Lynx population. This
means the data they collected can tell us about the population of each of the animals.
Figure 1 shows data for the number of hare pelts in green over time in the top row and the
lynx pelts over time in red on the bottom row. (Can you see the rise and fall of the

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populations in the graphs?) The data showed that some years there were more lynxes
(predators) but less hares (prey) like 1927 while other years, like 1932 there were more
hares but less lynxes.

Figure 1. Hudson Bay Company data from 1895 to 1935: The top panel shows the change in hare (green) population in tens
of thousands over the time. The bottom panel shows the lynx (red) population in tens of thousands.

What does the Hudson Bay Data tell us?

The rise and fall in the hare and lynx population over time suggests that there is a
relationship between the two animals, which makes sense as we know that lynxes eat hares.
Can you see in Figure 1 that the population of the lynxes and the hares fall at around the
same time and rise at the same time? Why do you think this happens?
When there are more lynxes, they eat more hares which decreases the hare population. But
when the hare population is low it is hard for the lynxes to find the hares which means less
food for the lynxes and results in a decrease in the lynx population. When, the lynx
population decreases, the hare population will increase again, and the up-down cycle will
continue. If the predator and prey populations are balanced, they will go up and down over
time. Up and down like waves, the question a mathematician asks is, “can I explain this

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using addition, subtraction and multiplication (not even division) and can I predict the
future?”.

Explaining the Relationship with Mathematical Models

Mathematicians often use equations to describe what people see in the world. They use
data collected from the real world to inform their equations. In this paper we will show how
differential equations can be used to model the populations of lynxes (predators) and hares
(preys) using the data from the Hudson Bay Company above.

The predator prey relationship was first modelled in the 1920s by Alfred J Lotka and Vito
Volterra. They both noticed the relationship between the rise and fall in populations and
they wanted to see if they could use mathematics to explain the general predator prey
relationship. To do this, Lotka and Volterra created a mathematical model using differential
equations which we will use to model the hare and lynx populations.

Now, when you first come across these differential equations, they can look very
complicated but all they are is a way for mathematicians to describe how, what and why
populations change. A famous mathematician called Leonhard Euler (1707-1783) showed
that differential equations could just be written as plusses, minuses with a bit of
multiplication and that is what we will show in the next sections.

Modelling the Snowshoe Hare Population

Before a mathematician develops an equation, they think about the world. So, let’s think
about what is happening to the hare population over time. If there were no lynxes, then the
hare population in the future, HF, is just equal to the current hare population, HC, plus births
minus deaths, which we call the growth rate rgrowth and multiply it by the current population.
This equation is shown in Figure 2A.
Let’s use the Hudson Bay data to put numbers into the equation. We can read our initial
values for the populations directly. When the data starts in 1895, the hare population is 85,
so we will let HC = 85 hares. We can estimate a value for a growth rate of 0.9 in in a year this
gives the future hare population HF in 1896 as,

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HF= HC + (rgrowth× HC),
HF= 85 + (0.9×85) =161.5.

If the population keeps growing like this the entire world will be covered in hares, which is
called exponential growth. This happened when rabbits were introduced to Australia in
1859 and their population grew exponentially as they had no natural predators causing
havoc on the ecosystem.

Figure 2 A. The mathematics used to predict future hare population if there were no lynxes which results in exponential
growth. B. The equations to predict the lynx population if there are no hares, resulting in an exponential decline. C. The
equations for the hare and lynx populations when both animals are interacting with each other. This results in fluctuating
population levels for both hares and lynxes.

Modelling the Snow Lynx Population

Let’s next consider what is happening to the lynx population over time. If there are no
hares, then the lynx population will go down as without hares to eat, the lynxes would have
no food. To model this, we use subtraction as is shown in Figure 2B. Can you see why this is
different from the hare equation? The future number of lynxes is equal to the current
number of lynxes minus the death rate, rdeath, times the current number of lynxes.

The lynx population in 1895 is LC=51 and we set the death rate to 0.25 then the equation
gives,

LF= LC - (rdeath×LC),

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LF =51 - (0.25×51) =38.25.

This is known as an exponential decline, shown in the top left of Figure 2, and if it keeps
happening after a while there will be no lynxes.

Modelling the Hare and Lynx Interaction

Let’s now consider how the hare and lynx populations interact with each other and affect
the equations as shown in Figure 2C. Lynxes eat the hares and so decrease the hare
population; therefore, we can use subtraction to model this. We will now create a term for
the eat rate, reaten., which is multiplied by the lynx and hare populations in order to model
how many hares are hunted and eaten by lynxes. By the same logic, the hares are a food
source for the lynxes, so we use addition in the lynx equation. The current hare population
is included by multiplying them by the current lynx population times an eat rate, rfood.

Again, we can put numbers into the equations using the Hudson Bay data. The values of
HC=85 hares, rgrowth=0.9, LC=51 lynxes and the rDeath=0.25 remain the same. We get an eat
rate of reaten=0.024 and a food rate of rfood =0.005 by fitting the data.

Putting these numbers into the equation for predicting the future number of hares gives,

HF= HC+ (rgrowth× HC) - (reaten × LC× HC),

HF= 85+ (0.9×85) – (0.024×85×51) =59.24,

And the equation for predicting the future number of lynxes now gives,

LF= LC - (rDeath×LC ) + (rfood ×LC × HC ),

LF= 51 - (0.025×51) + (0.005×51×85) =59.925.

That means when we consider the interaction of the two animals, we calculate a decrease in
the hare population and an increase in the lynx population in in 1896 just as shown in Figure
3 by the green and red arrows.

Using these equations for the lynx and hare populations we can model how they change
over time by taking steps in time. The equations are set out above, so now we can use the
Hudson Bay company data to estimate the rgrowth , rdeath, reaten, and rfood, using some computer

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code by de Silva, Champion, Quade, Loiseau, Kutz, & Brunton (2020). This process alone
could fill a whole paper! But for now, let’s see what our model looks like when we solve and
graph it.

Visualizing the Predator Prey-Relationship

Using the Lotka-Volterra equations from above and some computer code (available in
python Brady and Butler, 2020), we simulate both the hare and lynx populations and make
our own data showing how the predator prey relationship evolves over time in Figure 3
(left). The plots for the hare population (green) and lynx population (red) show the rise and
fall that we saw earlier in the real data collected by the Hudson Bay company.
.

Figure 3 The modelled hare and lynx population using the Lotka-Volterra model and the Hudson-Bay Company data from
1895 to 1935. The top left panel shows the modelled hare (green) population the bottom panel shows the modelled lynx
population. The panel on the right plots how the lynx population changes for different hare populations hare population.
The x is the mean lynx and hare population which the two populations orbit. The arrows and numbers indicate the
calculations in the text from 1895 to 1896.

Another way we can investigate how two species are linked is to plot how the hare
population and lynx population change in relation to each other as shown in Figure 3 on the
right. If you know there are 50 hares in a population this graph tells us there will be either

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24 or 60 lynxes alive at the same point in time. The reason there are two answers is because
when the lynx population is at 24 it is about to grow as there is loads of food, or when the
lynx population is high at 60, it is about to go down as there is not enough food. The x in the
middle is the average (equilibrium point) hare and average lynx population. It looks a bit like
the orbit of the earth around the sun. The populations go around and round which is the up
and down over time. In a balanced ecosystem, like the one modeled here, the orbits stay
stable but if it starts spiraling in or out could be an early warning sign.

Discussion

Models are not exact

When mathematicians try to describe something complicated, they simplify things, and it is
the same for these equations. The simplifications we made mean that the predictions and
simulations do not perfectly follow the original data. Here is some information we left out of
our model:
1. There is more than one predator of the Snowshoe Hare, what about Snow Foxes and
large birds?
2. Snow Lynxes hunt more than just Snowshoe Hares, they can also eat other animals
to survive, like fish and squirrels.
3. In our model, the Snowshoe Hares do not run out of food, which is definitely not
true in winter.
4. What about human fur hunters, who hunt both lynxes and hares?

To make the equations work for all these other situations you can include extra equations
and more pluses and minuses. If you have all the data, you could perfectly model the future.
Even with these simplifications the mathematics still does a good job modelling the hare
and lynx populations.

How else can this model be used?

The mathematics in the predator-prey model can be used to model lots of other populations
and make predictions. You could change the words Snow Lynx to Shark and Snowshoe Hare

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to fish and the mathematics will still work with the right data (Volterra 1926). You could
even use the same equations and change the word lynx to Zombie and hare to human. This
has already been used in a computer game!
The mathematics can also be used to predict the impact of re-introducing a previously
extinct animal back into an area like in 1995 when wolves were re-introduced into
Yellowstone park with surprising results watch the video (Sustainable Human 2014).
Inspired by these amazing observations, there has been a game developed that uses the
same mathematics described in this paper. Go to https://ecobuildergame.org to play.
The predator prey relationship can be expanded further outside of animal populations and
can be used to model how companies interact, how chemical reactions occur and how
viruses spread. You can read more about this in another Frontiers for Young Minds paper by
Brooks, Kanjanasaratool, Kureh, & Mason (2021) which talks about using similar
mathematics to model and understand the spread of COVID 19

Summary
To develop models of the real world, mathematicians need to start with good data. This
means it is vital for scientists, conservationists - and even fur hunters! - to collect
information from the environment around them. Using data, we can spot patterns in the
relationship and then use mathematics to recreate these patterns and create new data to
represent and predict the future of these relationships. Hopefully from this article you can
see how all it takes is the simple mathematics of adding, subtracting and multiplication - and
a bit of clever thinking - to model and predict the populations of a predator, its prey and
much more.

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References

Brooks, H. Z., Kanjanasaratool, U., Kureh, Y. H. & Mason. P. Disease Detectives: Using
Mathematics to Forecast the Spread of Infectious Diseases Frontiers for Young Minds (2021)

Butler, J. S., & Brady, R. M. (2020, January 1). Predator prey code for Young-Minds. GitHub.
https://github.com/john-s-butler-dit/Predator-Prey-for-Young-Minds

Lotka, A. J. Analytical Note on Certain Rhythmic Relations in Organic Systems. Proc. Natl.
Acad. Sci. 6, 410–415 (1920).

de Silva, B. M., Champion, K., Quade, M., Loiseau, J. C., Kutz, J. N., & Brunton, S. L. (2020).
PySINDy: A Python package for the sparse identification of nonlinear dynamics from data.
arXiv preprint arXiv:2004.08424.

Sustainable Human. (2014, February 13). How Wolves Change Rivers. YouTube.
https://www.youtube.com/watch?v=ysa5OBhXz-Q&feature=emb_logo

Volterra, V. Fluctuations in the Abundance of a Species considered Mathematically. Nature

118, 558–560 (1926).

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Glossary

Differential Equations: Differential equations are mathematical equations that describe

how populations change over time. They can describe many other processes, such as how a
helicopter flies, how planets orbit a star or how blood flows in our veins.

Estimate: A value that is close enough to the exact answer, usually created based on some
knowledge you have of the system or by preforming a calculation.

Exponentially: Exponentially is a mathematical way of describing the steady and rapid

growth or decay of something.

Orbit: An orbit is the path a planet takes when it circles a sun. The same can be said for
balanced predator-prey populations as their path circles the average, never growing too
large or too small.

Equilibrium Point: This is the balance point in the two populations. When a system is in
equilibrium, we say is it stable or unchanging.

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Author Biographies

John S. Butler, PhD

I am a Mathematics Lecturer in the Technological University Dublin, Ireland. I use my dual
backgrounds in mathematics and neuroscience to design experiments and analysis methods
to understand the world using maths. I have researched how the brain uses sensory
information for walking, driving and flying using robots to fly people through virtual reality.

Rebecca Brady, BSc

I am a PhD student of Mathematical Sciences also based in Technological University Dublin,
Ireland. My research focuses on creating mathematical models to describe how we use
sensory information taken in from the world around us. I’m particularly interested in how
this process differs in people who have problems with their movement, like those with
Parkinson’s disease. Outside my research, I like to spend my time reading, creating art and
spending quality time with my family, friends and my dog, Pippa.

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