Planet Earth appears to be on course for the start of a sixth mass extinction of life by about 2100 because of the amount of carbon being pumped into the atmosphere, according to a mathematical study of the five previous events in the last 540 million years.

Professor Daniel Rothman, co-director of the Massachusetts Institute of Technology’s Lorenz Centre, theorised that disturbances in the natural cycle of carbon through the atmosphere, oceans, plant and animal life played a role in mass die-offs of animals and plants.

So he studied 31 times when there had been such changes and found four out of the five previous mass extinctions took place when the disruption crossed a “threshold of catastrophic change”.

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The worst mass extinction of all – the so-called Great Dying some 248 million years ago when 96 per cent of species died out – breached one of these thresholds by the greatest margin.

Based on his analysis of these mass extinctions, Professor Rothman developed a mathematical formula to help predict how much extra carbon could be added to the oceans – which absorb vast amounts from the atmosphere – before triggering a sixth one.

The answer was alarming.

For the figure of 310 gigatons is just 10 gigatons above the figure expected to be emitted by 2100 under the best-case scenario forecast by the Intergovernmental Panel on Climate Change. The worst-case scenario would result in more than 500 gigatons.

Some scientists argue that the sixth mass extinction has already effectively begun. While the total number of species that have disappeared from the planet comes nowhere near the most apocalyptic events of the past, the rate of species loss is comparable.

Professor Rothman stressed that mass extinctions did not necessarily involve dramatic changes to the carbon cycle – as shown by the absence of this during the Late Devonian extinction more than 360 million years ago.

Writing in the journal Science Advances, he noted that events such as volcanic eruptions, climate change and other environmental factors could also play a role.

But he said changes to the carbon cycle – such as the burning of vast amounts of carbon in the form of oil, coal and gas laid down over millions of years – should also be considered.

“The history of the Earth system is a story of change. Some changes are gradual and benign, but others, especially those associated with catastrophic mass extinction, are relatively abrupt and destructive,” Professor Rothman wrote.

“What sets one group apart from the other? Here, I hypothesize that perturbations of Earth’s carbon cycle lead to mass extinction if they exceed either a critical rate at long time scales or a critical size at short time scales.

“By analyzing 31 carbon isotopic events during the past 542 million years, I identify the critical rate with a limit imposed by mass conservation.

“The modern critical size for the marine carbon cycle is roughly similar to the mass of carbon that human activities will likely have added to the oceans by the year 2100.”

Shape Created with Sketch. Animals in decline Show all 8 left Created with Sketch. right Created with Sketch. Shape Created with Sketch. Animals in decline 1/8 Harbour seal (Phoca vitulina) Where: Orkney Islands. What: Between 2001-2006, numbers in Orkney declined by 40 per cent. Why: epidemics of the phocine distemper virus are thought to have caused major declines, but the killing of seals in the Moray Firth to protect salmon farms may have an impact. Alamy 2/8 African lion (Panthera leo) Where: Ghana. What: In Ghana’s Mole National Park, lion numbers have declined by more than 90 per cent in 40 years. Why: local conflicts are thought to have contributed to the slaughter of lions and are a worrying example of the status of the animal in Western and Central Africa. 3/8 Leatherback turtle (Dermochelys coriacea) Where: Indonesia, Malaysia, Mexico, Costa Rica. What: Numbers are down in both the Atlantic and Pacific. It declined by 95 per cent between 1989-2002 in Costa Rica. Why: mainly due to them being caught as bycatch, but they’ve also been affected by local developments. Alamy 4/8 Wandering albatross (Diomedea exulans) Where: South Atlantic. What: A rapid decline. One population, from Bird Island, South Georgia, declined by 50 per cent between 1972-2010, according to the British Antarctic Survey. Why: being caught in various commercial longline fisheries. Alamy 5/8 Saiga Antelope (Saiga tatarica) Where: Kazakhstan, Mongolia, Russia, Turkmenistan, Uzbekistan. What: fall in populations has been dramatic. In the early 1990s numbers were over a million, but are now estimated to be around 50,000. Why: the break up of the former USSR led to uncontrolled hunting. Increased rural poverty means the species is hunted for its meat 6/8 Swordfish (Xiphias gladius) Where: found worldwide in tropical, subtropical and temperate seas. Why: at risk from overfishing and as a target in recreational fishing. A significant number of swordfish are also caught by illegal driftnet fisheries in the Mediterranean 7/8 Argali Sheep (Ovis mammon) Where: Central and Southern Asian mountains,usually at 3,000-5,000 metres altitude. Why: domesticated herds of sheep competing for grazing grounds. Over-hunting and poaching. 8/8 Humphead Wrasse (Cheilinus undulatus) Where: the Indo-Pacific, from the Red Sea to South Africa and to the Tuamoto Islands (Polynesia), north to the Ryukyu Islands (south-west Japan), and south to New Caledonia. Why: Illegal, Unregulated and Unreported (IUU) fishing and trading of the species 1/8 Harbour seal (Phoca vitulina) Where: Orkney Islands. What: Between 2001-2006, numbers in Orkney declined by 40 per cent. Why: epidemics of the phocine distemper virus are thought to have caused major declines, but the killing of seals in the Moray Firth to protect salmon farms may have an impact. Alamy 2/8 African lion (Panthera leo) Where: Ghana. What: In Ghana’s Mole National Park, lion numbers have declined by more than 90 per cent in 40 years. Why: local conflicts are thought to have contributed to the slaughter of lions and are a worrying example of the status of the animal in Western and Central Africa. 3/8 Leatherback turtle (Dermochelys coriacea) Where: Indonesia, Malaysia, Mexico, Costa Rica. What: Numbers are down in both the Atlantic and Pacific. It declined by 95 per cent between 1989-2002 in Costa Rica. Why: mainly due to them being caught as bycatch, but they’ve also been affected by local developments. Alamy 4/8 Wandering albatross (Diomedea exulans) Where: South Atlantic. What: A rapid decline. One population, from Bird Island, South Georgia, declined by 50 per cent between 1972-2010, according to the British Antarctic Survey. Why: being caught in various commercial longline fisheries. Alamy 5/8 Saiga Antelope (Saiga tatarica) Where: Kazakhstan, Mongolia, Russia, Turkmenistan, Uzbekistan. What: fall in populations has been dramatic. In the early 1990s numbers were over a million, but are now estimated to be around 50,000. Why: the break up of the former USSR led to uncontrolled hunting. Increased rural poverty means the species is hunted for its meat 6/8 Swordfish (Xiphias gladius) Where: found worldwide in tropical, subtropical and temperate seas. Why: at risk from overfishing and as a target in recreational fishing. A significant number of swordfish are also caught by illegal driftnet fisheries in the Mediterranean 7/8 Argali Sheep (Ovis mammon) Where: Central and Southern Asian mountains,usually at 3,000-5,000 metres altitude. Why: domesticated herds of sheep competing for grazing grounds. Over-hunting and poaching. 8/8 Humphead Wrasse (Cheilinus undulatus) Where: the Indo-Pacific, from the Red Sea to South Africa and to the Tuamoto Islands (Polynesia), north to the Ryukyu Islands (south-west Japan), and south to New Caledonia. Why: Illegal, Unregulated and Unreported (IUU) fishing and trading of the species

The idea that mass extinctions are caused by major environmental changes was suggested about 200 years ago by the famed French naturalist Georges Cuvier.

If these changes are too rapid for the evolution of species, they may die out either as a result of being out-competed by those that can adapt or because they are unable to cope in the new environment.

Already some species of tree are in trouble because the temperature is rising so quickly that they cannot gradually “migrate” – by seeding saplings – uphill to cooler climates.

But scientists also recently described how the Atlantic killifish had evolved extremely quickly to be able to survive toxic pollution off the US east coast that would normally have killed them. They noted that “unfortunately” most species “we care about” were unlikely to be able to do the same.

Professor Rothman said that during his research into previous mass extinctions “it became evident that there was a characteristic rate of change that the system basically didn’t like to go past”.

He said it would probably take thousands of years for any ecological disaster to play out, but that 2100 could be the tipping point at which the world entered “unknown territory”.

“This is not saying that disaster occurs the next day,” Professor Rothman said.

“It’s saying that, if left unchecked, the carbon cycle would move into a realm which would be no longer stable, and would behave in a way that would be difficult to predict.

“In the geologic past, this type of behaviour is associated with mass extinction.”