Blog report on the Seminar ‘Lotka-Volterra Systems: Dynamics and Applications’ by Dr Helen Christodoulidi
Mathematical modelling has always been a great tool to utilize when it comes to describing large, complex systems. One particular model, known as the Lotka-Volterra system, was discussed in detail during the seminar ‘Lotka-Volterra Systems: Dynamics and Applications’ delivered on the 19th of November 2025 by Dr Helen Christodoulidi. Within this seminar, Dr Christodoulidi discussed the mathematics behind the system and where it came from, before going on to describe how the system can be applied to real-life examples.
The system was introduced independently by two scientists: Alfred J. Lotka (1880-1949) and Vito Volterra (1860-1940). Lotka’s work investigated coupled differential equations relating to chemical and ecological dynamics [1]. His principle was as follows: the competition among organisms for available energy is a natural selection. Volterra on the other hand was focussed more on the application of mathematics to mathematical ecology. During his research on the population dynamics of fisheries, he ended up publishing an article containing the same equations that Lotka had previously published in his paper ‘Elements of Physical Biology’ (1925). After becoming aware of the similarities, the two publishers acknowledged the others’ research, while continuing to work independently. Volterra stated “Working independently, the one from the other, we have found some common results, and this confirms the exactitude and the interest in the position of the problem. I agree with (Lotka) in his conclusions that these studies and these methods of research deserve to receive greated attention from scholars, and should give rise to important applications.” [1]. Since then, as Volterra predicted, the model has been found to be applicable to many important areas of research, and is still relevant to this day.
So what can this system be used for? It can be applied to systems in ecology, epidemics and even finance. Its most common use is for simulating predator-prey population models, describing the evolution dynamics of both species. Before we go deeper into the applications of the model, let us lay out the foundations required to form such a model in the first place. We can begin by assigning variables - the prey to the x-variable and the predator to the y-variable. (eg. rabbits and foxes). We can then go about describing the system’s dynamics using a set of differential equations, where dx/dt is the rate of population growth of the prey species, and dy/dt is the rate of population growth of the predator species. For a predator-prey dynamic, we get the following pair of first-order, non-linear differential equations:
(Eqn. 1.1)
(Eqn. 1.2)
Where a, �, ?, d are constants that allow for flexibility to fit real data. In this model, the population size of the predator negatively affects the prey population, while the prey population positively influences the predator population [2]. In other words, as the predator population grows, the prey population will decrease as they are hunted for food. As more prey are hunted for food, their numbers decrease and thus there will be a shorter supply of food for the predator population, so their numbers will also begin to decrease. As the predator population decreases, less prey are being hunted, allowing for their numbers to rise again. As the number of prey rises, the predators will have access to more food, allowing their numbers to grow once again, and the cycle repeats. Over time, the population of both species will continue to fluctuate but the cycles always balance out in the end.
We can go about solving these equations using the Hartman-Grobman Theorem, which states that near a hyperbolic equilibrium point x, the nonlinear system ? = f(x) has the same qualitative structure as the linear system ? = Ax where A = Df(x). In other words, near a hyperbolic equilibrium point (a point at which a model is “well-behaved” and easy to predict), a complex, nonlinear system like the Lotka-Volterra model behaves qualitatively the same as its simpler, linearized approximation. Solving Equations 1.1 and 1.2 using this method results in the following system of equations:
(Eqn. 2.1)
(Eqn. 2.2)
Which, when solved, gives the solutions x = 0 or x = ?/d, and y = 0 or y = a/�.
The points (0,0) and (?/d, a/�) can be plotted on a set of axes. By using eigenvalues, which tell us how the population will respond to small disturbances near a balance point, the behaviour of the system can be sketched out as a phase space, resulting in a plot that looks something like Fig.1, which utilises the previous example of rabbits and foxes. The plot shows us how the population of a species can fluctuate depending on the number of the other population, visualising the relationship between the two.
Fig.1: A phase space plot of the population of foxes against rabbits using Matlab [3]
This solution can at least be applied to the case where n=2, n being the number of variables in a system. Dr Christodoulidi’s recent research has been investigating whether the Lotka-Volterra model can be applied to higher dimensions. For more than two variables the complexity of the system increases, and things are no longer as simple as a single solution. In 4-dimensional systems, with 4 different species populations, the relationship dynamics between these species can differ and produce varying results each time. The most common outcome resulted in instances where entire populations would go extinct. Though for some values, there were also instances where all four populations could co-exist together. Dr Christodoulidi concluded that in all examples for higher dimensional systems, the system has more chaos, meaning results become more unpredictable. In other words, as n increases, the chaos of a system also increases. For the case of n=4, the system can exhibit both organised and chaotic behaviour. 4D organised cases are unbounded, and at least two of the four species will go extinct. On the other hand, 4D chaotic cases are bounded, and all four species can co-exist at the same time and display complexity. Comparing this to the original case of n=2, there is no chaos in the system, meaning it is completely solvable, and it is entirely possible that both species survive.
One of the main downsides of the model is its simplicity. Though it works well enough for large enough datasets, the model contains the minimum possible number of variables and assumptions, meaning that it is somewhat unrealistic when it comes to being applied to real-world scenarios. It is purely a theoretical model, utilising real-world concepts and simplifying them to allow for the construction of a generalised mathematical model. Despite this, the model still remains relevant as it gives us an overall idea of the dynamics of a population and, as previously mentioned, tends to be fairly accurate for large enough sets of data.
[1] Allesina, S. (n.d.). 2 Generalized Lotka-Volterra | A Tour of the Generalized Lotka-Volterra Model. [online] stefanoallesina.github.io. Available at: https://stefanoallesina.github.io/Sao_Paulo_School/intro.html.
[2] Shim, H. and Fishwick, P.A. (2008). Lotka-Volterra Model - an overview | ScienceDirect Topics. [online] www.sciencedirect.com. Available at: https://www.sciencedirect.com/topics/earth-and-planetary-sciences/lotka-volterra-model.
[3] (Fig.1) scipy-cookbook.readthedocs.io. (n.d.). Matplotlib: lotka volterra tutorial SciPy Cookbook documentation. [online] Available at: https://scipy-cookbook.readthedocs.io/items/LoktaVolterraTutorial.html.
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The presentation is well put together but to improve it you could fix some of the grammatical mistakes and it has slightly informal phrasing. (2/3)
You have understood the context of the talk well but to improve you could expand on the Lotka-Volterra system and begin to talk about the model. (2/3)
You have stated about the talk about the models but not as to why the actual models matter for society, and you also could have made a connection with the Hartman-Grobman better instead of it being mentioned generically. (1/3)
You have stated an external quote from an external source but to improve you could perhaps maybe critically assess the disscusion instead of restating it. (3/3)
You have used language that is both able to be understood by both scientific and non-scientific audiences, however some terms still maybe misunderstood by non-scientific audiences such as phase portrait. (2/3)
Overall a good start to a blog but could do with a few adjustments (10/15)
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I agree with the previous comment that the presentation is clear and concise, it flows well and is evident that you fully understand the topic. However, respectfully, this is not in lay terms - the average person will not know what a “hyperbolic equilibrium point” is or what “linearised approximation” means - just as 2 examples. It feels like most of the mathematical terms are quite specific and undefined. Other than that, it looks good to me.
90561
Very informative however I’m not sure how well it would read for a lay person due the the complicated mathematics involved. A statement such as “first-order, non-linear differential equations” may seem simple to a maths or physics student, however this terminology would alienate anyone outside of the mathematics sphere and would likely lose a lay person’s comprehension; this too goes for the mathematical equations involved. Ignoring this though, it is very well structured and in-depth, with great resources such as graphs and external references backing this up.
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Overall Blog Presentation: The title, name of speaker, and date are all included. However, some of the sentences drag on and could do with being split up to make reading easier. 2/3
Accurate Reporting: You know the subject and have accurately reported it. 3/3
Accurate Contexualisation: You’ve explained why the research is important. 3/3
External Source: References have been made and quoted.
Writing Style and Technical Level: You have a clear understanding of what is going on, but the technical level of this is almost too high for the lay man to actually understand it. If you are mentioning complex concepts, it is best to add at least a basic explanation for the lay man to follow along. 1/3
Overall: 12/15. I like it, but it’s just a little too technical to be a report for the lay man. Really good work, you just need to make it a bit more accessible and break up the lengthy sentences.