The Lynx and the Hare

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While my training is in pure mathematics, I also have an appreciation of the way applied mathematicians make use of it. Maths can be deployed to understand, explain and predict the behaviour of real-world situations, and it’s fundamental to the study of practically all other areas of science.

Given a situation in the real world, you might measure and make observations about what happens – counting the number of animals in a population at given time intervals, measuring the temperature of a chemical reaction, or collecting data about any variable which changes with time or given different initial conditions. But while measurements are useful to tell you about what’s happening now, they don’t tell you much about what’s going to happen in the future.

Mathematical modelling allows you to construct a model – a virtual version of the system, which you can use to predict how it’s going to evolve. You could, for instance, study your observations and notice a correlation between two things – when one of them increases, so does the other. If you’re studying the population of an animal (p), you might notice that at times when the amount of food available (f) is high, the population of the animal also tends to be higher.

Looking at the data you’ve collected would allow you to determine the coefficient, c, which describes the correlation – if the quantity of food goes up, how much does the population go up by? This coefficient would be part of your model, and allow you to predict what the effect of a given amount of food being added would be, on the size of the population.

You can use a correlation in a model even if the two things turn out not to be directly related. For example, it’s been observed that cows which are given individual names tend to produce more milk – but that doesn’t mean it’s the name that’s causing Daisy’s milk production to increase. It’s the presence of a third factor: a farmer that cares enough about their cows to name them, and who also looks after them well. (If you’re a fan of spurious correlations, with or without a connecting reason, there are some who make a hobby of collecting them). But you can still use an indirect relationship to make predictions – if you know the two things will increase in tandem, regardless of why, the model is still useful.

Of course, the model given above is very simple. There will almost certainly be other factors affecting the size of a population, and there might be a limit to how many animals can live in a given area – you can’t keep increasing the amount of food indefinitely and expect the population to always keep up, so your model might only work for certain ranges of values.

Despite all this, population dynamics remains one of the easiest types of system to model – the variables involved tend to be fairly simple, and other minor confounding factors often don’t have a large impact on the final values. Once nice set of equations developed for modelling the interactions between two species – the predator-prey equations, also called the Lotka-Volterra equations, give a lovely approximation to the dynamics of two species in competition.

Generally, for modelling an animal population, you need to take into account a number of factors, including the rate of reproduction – how quickly that particular animal creates new animals; and the rate of death, which might depend on the average lifespan, but also other hazards which might cause them to die – in particular, the number of predators in the area, and how many of their prey each predator needs to sustain themselves.

The Lotka-Volterra equations describe the relationship between two populations as follows:

In these equations, x is the number of prey animals in the population, and y is the number of predators. Δ here is a Greek letter which mathematicians use to represent change – so Δx is the change in x. What these equations are saying is that the rate at which the prey population changes will depend on the size of the prey population, but also on the size of the predator population; and vice versa.

The values of a, b, c and d in these equations will depend on the particular animals involved – their rates of reproduction, how frequently they prey on each other, how much food the predators need to live – observing them and recording some values will allow you to establish what these values might be for a given pair of species, and then you can use this model to make predictions of what will happen in the future.

Looking at the equation for Δx, you might notice the term which depends on the number of predators (y) is negative – meaning the more predators you have, the more this will reduce the prey population; in the Δy equation, the term dependent on the number of prey (x) is positive, because the more prey there are around, the more food for predators so the population will go up more quickly.

Since these two values both depend on each other, you can end up with some interesting results – if there are more prey around relative to the number of predators, that will increase the number of predators, which will in turn reduce the number of prey, which will have a knock-on effect on the number of predators, and so on. The graphs produced by these models often involve two similar-looking oscillating curves, one of which follows the other with a delay (as effects take time to manifest in the population numbers).

If the populations are nicely balanced when you start your model, this system can be relatively stable. (There’s obviously one other stable system you can have here, which is when x and y are zero, but that’s boring). For stability, you ideally want

as your starting states. Any changes to the system – maybe a flood which wipes out a chunk of the predator population, or an introduction of more animals from another area – will destabilise the equilibrium, and the equations will let you model what might happen.

One of my favourite things in maths is when something you’ve predicted comes true, and there’s a wonderful example of the patterns predicted by this model being exactly replicated in a real situation. Around the early 1900s, the Hudson Bay Company traded in fur pelts – from lynxes and hares trapped locally. What they might not have realised at the time was that in doing so, they were collecting data about the local hare and lynx populations – which were predator and prey to each other.

In the early 20th century, biologist Charles Gordon Hewitt analysed the data and discovered that the pattern of increase and decrease in the two populations matched the Lotka-Volterra model pretty well. The two populations were in equilibrium, with their sizes varying from year to year in a way that could be quite nicely predicted, and shows the characteristic delayed peaks of predator and prey.

This model is relatively simple, but mathematical modelling in general is a constant balancing act – if you can model the system with a simple set of equations, and not lose too much of the detail, that’s great. If your model is too simple, you’ll be able to compute your predictions quickly, but they’re less likely to be accurate, as you won’t have taken into account all the possible factors that could affect what’s going on. There’s a famous saying in statistical modelling: “All models are wrong, but some are useful”. The compromise you’re aiming for is a model which is the least possible amount of wrong, while still being reasonably useful.

Most real-world systems are more complicated than this simple population example, and couldn’t be captured with such a minimal model. Meteorologists spend their time predicting the weather, using many overlapping and detailed models – but a true model to accurately predict the entire weather system would need billions of variables and would probably take so long to compute, it’d be quicker just to wait and look out of the window.

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is a mathematician based in Manchester, who gives talks and workshops on different areas of maths. She finished her PhD in 2011, and since then has talked about maths in schools, at science festivals, on BBC radio, at music festivals, as part of theatre shows and on the internet. Katie writes blog posts and editorials for The Aperiodical, a semi-regular maths news site.

1 comment

  1. Katie Steckles wrote (18. June 2019):
    > […] The Lotka-Volterra equations describe the relationship between two populations as follows:

    > Δx = a x – b x y
    > Δy = c x y – d y

    > In these equations, x is the number of prey animals in the population, and y is the number of predators. […]

    (See however the use of parameter labels “α, β, γ and δ” in this statement of the Lotka-Volterra equations.)

    > […] For stability, you ideally want

    > y = a / b,     x = c / d

    > as your starting states. […]

    Inserting “ y = a / b ” results in “Δx = 0” for the first LV equation above.

    However, “ x = c / d ” gives “Δy = y (c^2 / d – d)” for the second LV equation …

    So, perhaps, for stability, you ideally want instead:

    y = a / b,     x = d / c .

    p.s. — SciLogs comments LaTeX test:

    “\( y = \frac{a}{b} \)” is rendered as “\( y = \frac{a}{b} \)”.


    Joachim Schulz schrieb (26. Juni 2018):
    > Der Duden definiert System in den Naturwissenschaften als “Gesamtheit von Objekten, die sich in einem ganzheitlichen Zusammenhang befinden und durch die Wechselbeziehungen untereinander gegenüber ihrer Umgebung abzugrenzen sind”. Diese Definition ist richtig, […]

    … und sie entspricht insbesondere z.B. der Definition eines “Inertialsystems” (nach W. Rindler):

    Ein Inertialsystem ist einfach eine bestimmte unendliche Menge von Punktteilchen, die räumlich gegenüber einander ruhen.

    sowie der (entsprechend verallgemeinerten) Definition eines “Bezugssystems”:

    Ein bestimmtes Bezugssystem eines bestimmten Beteiligten ist eine Menge von Beteiligten (einschl. des erstgenannten), die alle durchwegs gegenseitige geometrische und kinematische Beziehungen untereinander ermitteln können, und von denen sich insbesondere keine zwei jemals trafen.

    Mit beiden ist auch eine (implizite) Forderung nach “(iterativer) größtmöglicher Ausdehnung” verbunden, nämlich:

    – dass für alle Ereignisse, für die sich ein teilnehmendes Punktteilchen denken oder sogar auffinden lässt, dass bzgl. allen schon gegebenen bzw. in Betracht gezogenen anderen Mitgliedern des betreffenden Inertialsystems durchwegs ruhte, dann auch dieses zu diesem Inertialsystem gehört; bzw.

    – dass für alle (Mengen von) Ereignisse(n), an denen (mindestens) ein Beteiligter sich denken oder sogar auffinden lässt, dem gegenüber sich bzgl. allen schon gegebenen bzw. in Betracht gezogenen anderen Mitgliedern des betreffenden Bezugssystems durchwegs gegenseitige geometrische und kinematische Beziehungen ermitteln lassen, dann jeweils genau einer davon ebenfalls zu diesem Bezugssystem gehört.

    > […] Tatsächlich ist mit dem Raketensystem aber nicht das mechanische System Rakete gemeint.

    Ganz recht; und auch ganz im Sinne der genannten Definitionen und damit verbundenen Forderung.

    > Inertialsysteme sind globale Koordinatensysteme

    Nein, denn die o.g. Definition eines Inertialsystems ist offensichtlich ohne Bezugnahme auf irgendwelche Koordinaten formuliert. W. Rindler weist ausdrücklich darauf hin:

    Streng genommen sollten wir zwischen einem Inertialsystem und einem inertialen Koordinatensystem unterscheiden.

    Ein “inertiales Koordinatensystem” ist demnach ein bestimmtes Inertialsystem entsprechend der o.g. Definition zusammen mit (irgend-)einer (eindeutigen, umkehrbaren) Zuordnung von Koordinaten (d.h. Tupeln reeller Zahlen) zu den Mitglieder dieses Inertialsystems (und hinsichtlich einer “t”-Koordinate, jeweils zu den Anzeigen dieser Mitglieder).

    > Das Raketensystem ist das Ruhesystem der Rakete.

    Diese Wortwahl kann zu Missverständnissen führen, falls die Vorstellung einer Rakete damit verbunden ist, dass die Rakete durchwegs beschleunigte (also nicht “an sich ruhte”; und folglich auch nicht gegenüber irgendwelchen anderen Beteiligten ruht, sondern “bestenfalls” gegenüber bestimmten anderen Beteiligten starr war und blieb; vgl. [[Hyperbolic motion]] usw.).
    Es lässt sich stattdessen jedenfalls von einem Bezugssystem der Rakete sprechen.

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