A summary of Dr Helen Christodoulidi’s “Lotka-Volterra Systems: Dynamics and Applications”
18/11/2025
Imagine you’ve just arrived to work only to find out your boss has appointed you with the surprise task of predicting the future population sizes of animals in your local ecosystem. You might ask yourself, “I’m only a barista. What in the world is going on?”, but after your boss says that your job is on the line otherwise, you may then ask yourself, “How do I start to predict such a thing”. The answer to that would be to conveniently continue reading this blog on explaining the ‘Lotka-Volterra Model’!
So, what is this so-called ‘model’, and why is this one so useful? Dr Helen Christodoulidi’s seminar entitled “Lotka-Volterra Systems: Dynamics and Applications” explains what a Lotka-Volterra Model is and how it can be used to mathematically predict the population sizes of multiple species, for example: foxes and rabbits.
A real-world problem portrayed using mathematics is called a ‘model’. The Lotka-Volterra Model is a ‘Predator-Prey Model’, meaning it predicts the population sizes of a so-called ‘predator’ and its ‘prey’. You can think of the rabbit-fox example: the more rabbits there are, the more abundant the food supply for the foxes there will be, so the foxes can support more offspring, in turn increasing the fox’s population. However, as this happens eventually the foxes will be eating too many rabbits, decreasing the rabbit population, meaning the future foxes will have less abundant food around, and in turn decrease the fox population.
So, you can imagine that the population size of foxes will decrease just after the population size of rabbits decreases, which will in turn soon increase the population of rabbits, allowing the foxes to be more abundant and letting the cycle continue again. This cycle of population increase and decrease can be modelled (described) by the Lotka-Volterra Model, showing how the population increases and decreases at regular intervals. The time it takes for this cycle to start and end is called the period of the cycle (period of time). At this period, the cycle will repeat itself. Dr Christodoulidi highlights two graphs showing possible variations of this cycle, highlighting the trends that have just been mentioned:
[Figures taken from given seminar slides]
Lotka-Volterra Models don’t only model the populations of animals, however. In fact, using the modelling skills we have just used, Dr Christodoulidi shows that with a tweak to the initial conditions of the model we can apply the Lotka-Volterra Model to many different fields, anywhere from Ecology to Biology, and even to Finance! Din (2013) states “The Lotka-Volterra models have many applications in applied sciences.” [1], highlighting the practical uses across disciplines. For example, we can use the model to describe an epidemic’s reduced effect over time, as more people catch the infection and become immune.
Dr Christodoulidi highlights her work on ‘4D Lotka-Volterra Systems’ where instead of modelling 2 species (a predator and prey), you can model 4 species (The ‘4’ in ‘4D’), all with varying levels of predator and prey. A graph can be made of these, where each arrow away from a species shows what the species is a predator to, and each arrow towards a species shows what that species is prey to.
This graph taken from the seminar shows one interpretation of a 4-species predator-prey model. You can see that some species may be predators to all others, such as x4, while some may be prey to all others, such as x1. Others may have more complicated interactions with other species, such as x2 and x3.
Shown in the seminar is another graph showing the trends when 4 species like this interact over time in a predictable way. Here we can see that the two species at the top will continue having a steady population, however two of the species will die off. This isn’t conducive to real-life however, as the real world is far more complicated and chaotic than this.
‘Chaos’ in mathematics is actually a surprisingly intricate topic. So, what is chaos? Why is it useful here? Chaos can be seen as the behaviour of something being unpredictable (like the butterfly effect), even if the behaviour is deterministic (able to be calculated exactly) [2]. The previous systems have all been non-chaotic, and predictable. Dr Christodoulidi shows that in a chaotic 4D Lotka-Volterra Model you can actually get all four species to co-exist stably together. The chaotic nature of this highlights the immense complexity of the real world’s high number of species coexisting.
So hopefully after reading all of that your theoretical boss is off your theoretical back and you can breathe a theoretical sigh of relief - knowing now the implications that the Lotka-Volterra Model has on a range of fields affecting the wider world!
References:
[1] Din, Q. Dynamics of a discrete Lotka-Volterra model. Adv Differ Equ 2013, 95 (2013). https://doi.org/10.1186/1687-1847-2013-95
[2] https://plato.stanford.edu/entries/chaos/
90561
Just realised the pictures didn’t copy-paste into the web page, my bad. First graphs should be two of the sine graphs of population size, second figure should be the 4D node graph, and the final figure should be the 4-species population graph, where 2 of the species die out.
I think you have explained this really well, you have explained terms precisely and the paragraphs flow. It also has parts with a subtle undertone of casualness which you’d hope to find in these blog-style reports
26990
You have an original, gripping intro and the casual tone makes it very easy to carry on reading!
You’ve explained how the system works well, especially for a lay audience, using relevant examples from the seminar.
Just double check your second reference as it is just a link :).
60795
This reads very well and holds my interest throughout. It may be beneficial to look at trying to fit in another reference somewhere for variety.