Patterns and proofs

Patterns and proofs

Spotting the collapse of a species before it happens

‘Are we all doomed?’ Maurice McDonald/PA Archive/Press Association Images

In the summer of 1992, the north Atlantic cod population collapsed. For five hundred years, the offshore fishery had been a central part of life on the Canadian coast. But overfishing had led to a sudden decline in stocks, with numbers falling to less than 1% of their earlier levels.

Is it possible to anticipate such a collapse? Recent work by physicists at MIT suggests it might be. In April, they published a paper in Nature showing that patterns in the density of a yeast population can reveal whether it is at risk of collapsing.

The paper helps address a problem that has puzzled researchers since the 1970s. Using ideas from the newly created field of “catastrophe theory”, mathematicians wanted to understand how biological systems undergo abrupt, dramatic changes. They began by trying to describe through equations how animal populations vary over time in response to external factors like environmental stress.

When stress levels were low, the population ended up in a stable state. In other words, if a disturbance reduced animal numbers slightly, the population would grow back until it reached its previous size again.

Likewise, if a few more animals were added, they would compete over resources until their numbers dropped back down to the stable state. For a specific level of stress, the behaviour looks like this:

The orange dot shows a population that has been reduced from the stable population size – represented by the red line – and the green dot shows a population that has been “nudged” in the other direction. The arrows show that even if we nudge the population size up or down, it will eventually settle back to its original position on the red line. The same pattern crops up for other low levels of stress:

But what happens if the size of the nudge increases, or the amount of external stress grows? Even when stress levels were fairly low, the mathematicians discovered that the population was unable to recover if numbers were reduced too far. They also noticed that if the stress level grew too large, the population would no longer be able to sustain a stable state. The system’s behaviour in such situations followed a “fold” shape:

The solid line again represents the stable state, and the dashed line the size from which the population cannot rebound. The point of the fold shows the stress level that will lead to a population collapse.

Interestingly, catastrophe theory suggests that it should be possible to anticipate this point. When the stress level is just to the left of the critical point, the hypothetical population takes longer to recover from nudges than it would if stress levels were lower. This process is known as “critical slowing down”.

Of course, this is only a prediction from a mathematical model. But last year, the same researchers at MIT tested the theory on a real yeast population. Subjecting the yeast to increasing levels of stress – in the form of reduced food – they found that the organisms behaved just as catastrophe theory suggested they would, eventually falling victim to the “fold catastrophe” shown in the final graph above.

The researchers also spotted the critical slowing down before a population collapse: as the population was put under more and more stress it took longer and longer to recover from small disturbances. Unfortunately, it sometimes took a long time for the yeast to recover. Clearly this is a big drawback for a potential catastrophe-alarm: by the time the critical point is identified, the population may have already collapsed.

So the team decided to look at a different aspect of how yeast populations recover from disturbances. Rather than studying recovery time, they focused on “recovery length”.

From forests to marine reserves, populations living near inhospitable areas tend to be affected by the poor quality of their neighbouring habitat. Further away from an undesirable location, populations suffer less and after a certain distance – the recovery length – organisms return to the familiar stable state.

By linking several sets of yeast cultures, including one “bad patch” that had very little food, the MIT team explored how recovery length varied as the population was put under an increasing amount of stress. They found that the larger the distance from an undesirable location needed to return the population to normal, the nearer the population was to collapse.

Things can be carefully controlled in a laboratory, but in real life ecosystems can suffer from external shocks. It is therefore unlikely that differences in recovery length could identify precisely when a population in the real world will collapse.

Changes in recovery time or recovery length may also have other causes, unrelated to population stress. So rather than being used to generate predictions, the method should instead be seen as a warning signal that something is awry.

As well as putting together more complex experiments, the researchers now plan to look at real life ecosystems such as fisheries and forests. Although we might never know when a population will tumble off a cliff, recovery length could prove a useful way of spotting whether it is near the edge.

The Conversation is a non-profit + your donation is tax deductible. Help knowledge-based, ethical journalism today.