The Logistic Map: a Simple Model with Rich Dynamics

Suppose the population of the world $latex {P(t)}&fg=000000$ is described by the equation $latex \displaystyle \frac{\mathrm{d}P} {\mathrm{d}t} = a P \,. &fg=000000$ Then $latex {P(t)}&fg=000000$ grows exponentially: $latex {P(t) = P_0 \exp(at)}&fg=000000$. This was the nightmare prediction of Thomas Robert Malthus. Taking a value $latex {a=0.02\ \mathrm{yr}^{-1}}&fg=000000$ for the growth rate, we get a doubling … Continue reading The Logistic Map: a Simple Model with Rich Dynamics →

Following the Money around the Eurozone

Take a fistful of euro coins and examine the obverse sides; you may be surprised at the wide variety of designs. The eurozone is a monetary union of 19 member states of the European Union that have adopted the euro as their primary currency. In addition to these countries, Andorra, Monaco, San Marino and Vatican … Continue reading Following the Money around the Eurozone →

Phantom traffic-jams are all too real

Driving along the motorway on a busy day, you see brake-lights ahead and slow down until the flow grinds to a halt. The traffic stutters forward for five minutes or so until, mysteriously, the way ahead is clear again. But, before long, you arrive at the back of another stagnant queue. Hold-ups like this, with … Continue reading Phantom traffic-jams are all too real →

Using Maths to Reduce Aircraft Noise

If you have ever tried to sleep under a flight-path near an airport, you will know how serious the problem of aircraft noise can be. Aircraft noise is amongst the loudest sounds produced by human activities. The noise is over a broad range of frequencies, extending well beyond the range of hearing. The problem of … Continue reading Using Maths to Reduce Aircraft Noise →

A Zero-Order Front

Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions. Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed. Life-cycle … Continue reading A Zero-Order Front →

Sigmoid Functions: Gudermannian and Gompertz Curves

The Gudermannian is named after Christoph Gudermann (1798--1852). The Gompertz function is named after Benjamin Gompertz (1779--1865). These are two amongst several sigmoid functions. Sigmoid functions find applications in many areas, including population dynamics, artificial neural networks, cartography, control systems and probability theory. We will look at several examples in this class of functions. Sigmoid … Continue reading Sigmoid Functions: Gudermannian and Gompertz Curves →

El Niño likely this Winter

This week’s That’s Maths column in The Irish Times (TM056 or search for “thatsmaths” at irishtimes.com) is about El Niño and the ENSO phenomenon. In 1997-98, abnormally high ocean temperatures off South America caused a collapse of the anchovy fisheries. Anchovies are a vital link in the food-chain and shortages can bring great hardship. Weather … Continue reading El Niño likely this Winter →

Sunflowers and Fibonacci: Models of Efficiency

The article in this week’s That’s Maths column in The Irish Times ( TM046 ) is about the maths behind the efficient packing of sunflowers and many other plants Strolling along Baggot Street in Dublin recently, I noticed a plaque at the entrance to the Ibec head office. It showed a circular pattern of dots, … Continue reading Sunflowers and Fibonacci: Models of Efficiency →

Clothoids Drive Us Round the Bend

The article in this week’s That’s Maths column in the Irish Times ( TM043 ) is about the mathematical curves called clothoids, used in the design of motorways. * * * Next time you travel on a motorway, take heed of the graceful curves and elegant dips and crests of the road. Every twist and … Continue reading Clothoids Drive Us Round the Bend →

Robots & Biology

The article in this week’s That’s Maths column in the Irish Times ( TM037 ) is about connections between robotics and biological systems via mechanics. The application of mathematics in biology is a flourishing research field. Most living organisms are far too complex to be modelled in their entirety, but great progress is under way … Continue reading Robots & Biology →

A Simple Growth Function

Three Styles of Growth Early models of population growth represented the number of people as an exponential function of time, $latex \displaystyle N(t) = N_0 \exp(t/\tau) &fg=000000$ where $latex {\tau}&fg=000000$ is the e-folding time. For every period of length $latex {\tau}&fg=000000$, the population increases by a factor $latex {e\approx 2.7}&fg=000000$. Exponential growth was assumed by … Continue reading A Simple Growth Function →

Spots and Stripes

How do leopards get their spots? Mathematics gives us a better answer than the one offered by Rudyard Kipling in Just So Stories. This is the topic of That's Maths this week ( TM019 ). Turing's Morphogenesis paper The information to form a fully-grown animal is encoded in its DNA, so there is a lot … Continue reading Spots and Stripes →