Getting Bigger, with an 'S'

"All that we can do, is to keep steadily in mind that each organic being is striving to increase at a geometrical ratio."

Charles Darwin.

Osprey chicks at approx 28 days from hatch
[(c) BGGW Project 2014]

What do hurricanes, the Greek alphabet, share movements on the Stock Market, the war in Afghanistan and old movies have to do with growing osprey chicks? Okay – I know that most of the people who read this blog are smart, so you're probably way ahead of me already.

This is another one of those “things”, right?

Yes, it is.

The thing in question is a graph, having a very specific shape – that of a letter “S”. The name “sigmoid curve” derives from the character Σ which is sigma in Greek. Of course it doesn't look anything like the shape of the graph but we will dismiss that minor detail with an airy wave and press on regardless...

An osprey egg is approximately the same size as a standard domestic hen's egg. Osprey chicks are born small and have to grow fast. Very fast. But their RATE of growth is not a constant. They start to develop slowly but, as time passes, the speed of growth accelerates until it hits a maximum at about 20 days – at which point, something interesting happens.

Female ospreys (and most other female birds of prey) are larger than adult males. But this size difference – technically knows as sexual dimorphism – is only implemented during the latter stages of the growth spurt during pre-fledge development. A version of the graph (thanks to University of Montana Geosciences) shows this clearly. [1]

General graph of weight / days from hatching, (as modified by E. Evans, MWT)

The sigmoid curve seems to be something that is very fundamental in natural processes. We can model it mathematically using a function called the logistic equation which has the typical form

Since it was first discovered by Belgian mathematician Pierre Verhulst in 1838 as a possible descriptor for population growth, the logistic equation and its derivatives have been spotted all over the place. Hurricanes and other weather systems develop according to this pattern. The capacity of troops to sustain combat, the share price of a newly-floated company on the Stock Exchange, the relationship between foxes and rabbits as predator and prey... all can be found to fit the same behaviour.

In medicine, cancerous tumours show the sigmoid curve while growing – and this discovery means that mass screening and early diagnosis are critical in the successful treatment of such conditions. It has even been found that some non-biological chemical reactions work this way too, leading to methods of salvaging old film stock that has been deteriorating over time. 

But back to the ospreys...

Osprey chicks at 40 hours (left) and 4 days from hatch (right).
[(c) Montgomeryshore Wildlife Trust]

The fact that growth rates conform to this version of the sigmoid curve means that we can't tell (by size alone) if a chick is male or female, until they are older than about 30 days from hatching.  After that, it should become clearer.

Notes:

[1] The graph shown was drawn from data taken in North America, where the resident ospreys are a different sub-species. This version was modified by Emyr Evans (MWT) to more properly reflect the observed rates in European birds.

The plot itself is compressed along the horizontal axis for clarity. The actual curve would, of course, be much shallower than this.