How close is a system to collapse?
Occasionally this subject is fun for a moment.

I do not remember a specific cartoon, but it used to be a cliché to show a couple animals behaving somewhat like humans punching each other.  The beast taking the damage would respond to the first buffet aggressively but on the second his response time was longer.  This continued until he was wobbling but otherwise motionless.  Then the final punch would come.

This violent but clever bit of drama has been formalized into a way to quantify how close a stressed system is to collapse.  Suppose you are counting rabbits in a field over a long period of time.  The number of foxes (for illustration only – they say foxes almost never catch rabbits eagerly though they try) is slowly increasing.  Eventually if nothing else happens the rabbits will be gone.  The test is to follow small perturbations in the rabbit population.  The numbers rise and fall.  So long as recovery times are relatively short and do not change the rabbits as a population (but of course not many unhappy individuals) are deemed to be in good shape.  But when the recovery times start progressively lengthening, some sort of intervention is indicated if the rabbits are to be kept.

It makes good sense and is intuitively clear if you saw enough cartoons as a child.

But suppose you are called in to evaluate a species for risk of collapse and nobody has been keeping records.  You don’t know what the last recovery time was or whether it was the same as prior recovery times although you may have a clue from observations of the species elsewhere.  It would be nice if you could work out what was going on from observations that could be made quickly.

Well a paper has now described (Lei Dai et al Slower Recovery in Space before Collapse of Connected Populations NATURE vol. 496 no. 7445 April 18, 2013 page 496 and reviewed by Stephen Carpenter as “Spatial Signatures of Resilience on page 308 in the same issue) how to proceed.  To change the metaphor, let us consider shellfish (Is there a pun in there?  Something about the genera of shellfish?) on the flat floor of the sea.  They move about slowly but by and large are equally distributed over the area of interest.  Look around and find an area that is not good for the shellfish, a deep hole maybe.  Since shellfish fall into the hole but never manage to get out there will be a halo around the hole with reduced numbers.  This halo drops off with distance from the hole in a fashion that can be quantified mathematically. 

And this is the good part.  If the halo is thin, then the population is capable of rapid recovery from a loss.  If the halo is thick then the population is stressed.

I think that’s wonderful.  It’s abstract reasoning at its best.  And it will probably be very useful.  My congratulations to Lei Dai.  If I were still a child I would immediately run out and annoy my playmates by giving them just enough clues so they couldn’t quite figure out how to do it.   

That was the fun part.  Can we take that insight and apply it to human demographics?  Alas I don’t think the spatial signature would be easy to find.  I was once having a bit to eat of an evening with friends at a table on the sidewalk in front of a little restaurant.  Somebody mentioned that we were in a very dangerous city.  I looked around and said I didn’t feel any threat at all.  My friend agreed but went on that within a few blocks there were sidewalks where the death rate by violence was higher than that for soldiers in a war zone.  The difference was good policing.  So demographic changes occur on a very tight scale.

Yet people, particularly in Florida, will travel long distances.  I have known friends to drive hundreds of miles to parties I used to have.  William Bartram in his book Bartram’s Travels in the eighteenth century describes encountering a group of people in Florida walking, all dressed up, the men toting guns and carrying food in big cauldrons slung from poles on their shoulders.  The people said that they were from Jacksonville and they were going to a party in Ocala.  That’s ninety miles as the crow flies.  It would take them days.  Jet travel carries people farther and faster. 

So I went back to look for the time signature of stress.  Let’s see.  Populations experience a fall in birth rate which stabilizes but never recovers.  Then they experience a rise in age at first marriage for women which never reverses, neither stops nor slows.  It accelerates. 

Oh dear.  This is not good.

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I wrote Lei Dai to tell him about this article and recieved a prompt and cheerful acknowledgment.