LotkaVolterra Systems: Dynamics and Applications Dr Helen Christodoulidi 19 November 2025
Dr Helen Christodoulidi delivered a fascinating seminar exploring the LotkaVolterra systems, a cornerstone of mathematical modelling used to describe predatorprey dynamics and their wider applications. The session combined theoretical foundations with practical examples, offering both depth and accessibility.
The LotkaVolterra Model can be thought of as a mathematical a way to model cycles of growth and decline between two interacting groups. Let’s think through an example of rabbits (prey) and foxes (predators). If there are lots of rabbits, foxes thrive because they have plenty to eat. But if foxes eat too many rabbits, the rabbit population drops, and then the foxes’ struggle. As rabbits recover, the cycle starts again. The model uses two equations to describe how the numbers of predators and prey change over time. It’s not just about animals the same idea can be applied to biology, ecology, or even finance, wherever two groups depend on each other in a pushandpull way.
The HartmanGrobman Theorem says that if you have a really complicated system, you don’t always need to study the whole thing. Instead, if you look at the system right around a balance point (where things are steady), the complicated system behaves almost the same as a simpler, straightline system. A very simple analogy would be to imagine a bumpy, twisting road. If you zoom in on just a small stretch near a flat section, it looks and acts like a straight road. This makes it much easier to understand and predict what happens in that small area. Thus, nonlinear systems (complicated ones with curves and twists) are hard to study directly. This theorem says: that ‘Two autonomous systems of differential equations are said to be topologically equivalent in a neighbourhood of the origin or to have the same qualitive structure near the origin’[1]. In other words, zoom in close enough, and the messy system behaves like a neat, linear one. That makes it much easier to predict what will happen, because linear systems are simpler to analyse. Dr Christodoulidi showed this with diagrams, where the behaviour of the system could be
One compelling application discussed was during the COVID19 pandemic, where the model was adapted to represent susceptible versus infected populations. Dr Christodoulidi demonstrated how this framework helped predict the decline of infection rates over time, supported by visual data from the UK government [2].
After showing how the LotkaVolterra model could be applied to the COVID pandemic, Dr Helen Christodoulidi moved on to explore how the system behaves in different dimensions.
Two Dimensions: This is the classic predatorprey case (like rabbits and foxes). The equations can be solved, and the outcome shows that both species can survive in a repeating cycle, sometimes one grows while the other shrinks, but neither disappears completely.
Four Dimensions: Things get more complex.
In chaotic but bounded cases, the populations fluctuate unpredictably but remain within limits, meaning coexistence is still possible.
In organised but unbounded cases, the system spirals out of control. Here, not all species survive, and extinction of one or more groups becomes likely.
Dr Christodoulidi explained these scenarios clearly, showing how moving from simple twospecies interactions to more complex systems can reveal very different outcomes, from stable coexistence to potential collapse.
Dr Christodoulidi’s seminar was both insightful and engaging, blending rigorous mathematics with practical relevance. Her ability to connect abstract theory to realworld scenarios made the session particularly valuable, leaving the audience with a deeper appreciation of how dynamical systems can illuminate complex phenomena across disciplines.
Reference:
[1] L. Perko Differential Equations and Dynamical Systems, Springer (2000)
[2] Gov.uk, “COVID-19 | UKHSA data dashboard,” Data.gov.uk, 2024. https://ukhsa-dashboard.data.gov.uk/respiratory-viruses/covid-19
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In the third paragraph you repeat yourself a bit when talking about looking at smaller sections - it might be worth removing or rewording some of the sentances here.
The second reference doesn’t appear to be particularly relevant to the blog? Data from it is shown in the lecture but not here, it may be worth adding a diagram comparing the predicted cases against the actual ones.
Overall, the whole blog flows well and feels cohesive with no obvious gramatical mistakes. The complex ideas presented in the seminar are delivered here in a very approachable manner.