Four monitor screens reveal the interior of the compound, a lattice of enclosures with high wire fencing folded inwards at the top and electrified. These units are surrounded by a windowless perimeter, maximising security and keeping access to a minimum. The volunteer in front of the screens flicks between them, dragging a mouse to move the cameras around the enclosures, which are scrubland habitats in miniature. She picks out their occupants, stocky felines whose purposeful demeanour is offset by the jaunty tufts they wear on their ears. These are Iberian lynxes, the most endangered cats in the world.

It’s feeding time. A partridge enters an enclosure and commences a tour of inspection; a lynx hurtles at it, legs stretching out almost flat, leaps and brings it down. These scenes are kept off the El Acebuche lynx centre’s live web stream, though they are critically important to the captive breeding project that is under way here in southern Spain. This lunchtime is a test for one cub, who has never seen a partridge before. She seizes and dispatches the bird in a flash: a promising tick for her survival skill-set.

Partridges are a once-a-fortnight treat at El Acebuche. Iberian lynxes, which are about half the size of Eurasian lynxes and comparable to American bobcats, subsist almost entirely on rabbits and cannot survive without them. Rabbits have been hard hit in the Iberian region by viral diseases, including myxomatosis, deliberately introduced into Europe in 1952. Iberian lynxes were themselves persecuted by Provincial Boards for the Extermination of Vermin in the 1950s. Extermination gave way to protection, but poaching continued, and the toll on the roads increased as traffic grew and lynxes were forced to roam wider in search of rabbits. By the mid-2000s there were fewer than 150 adult Iberian lynxes alive. The International Union for the Conservation of Nature (IUCN), which keeps the ledger of threatened species known as the Red List, classes the Iberian lynx as "critically endangered".

A 2010 census found that the numbers in the wild had increased to 250, and there are about 70 more in the captive breeding programme’s five centres. Some 30 are in El Acebuche: this one compound thus contains more or less a tenth of all the Iberian lynxes in existence, protecting them from pathogens and humans. The monitoring continues round the clock. Once an hour the observer on duty notes what each lynx is doing and records the details on a central screen, to build up individual behaviour profiles. These offer indications as to how the animals might fare if released to live free. One of the danger signs is being friendly to their keepers.

For Iberian lynxes, living free does not really mean living in the wild. They have to fend for themselves in terrain on which humans have one claim after another. Although the Doñana area that includes El Acebuche is designated a national park, it is not the kind of epic wilderness that the term suggests in the United States. It’s a reserve, hemmed and dissected, where some lines have been drawn in the sand. Although Iberian lynxes are exceptional in their scarcity, their conditions of existence are entirely typical on a planet that is being ever more densely colonised by its human population. The greater our dominance becomes, the more species will become as scarce as the Iberian lynx is today. Very few of them will benefit from the science, the technology, the volunteers, the political backing and the charisma that are combining to pull the lynx back from the brink.

Already a quarter of the mammal species surveyed by the IUCN are classed as threatened. So are 41% of the amphibians, 13% of the birds, a third of coral species and a third of conifers. And that may not be the half of it.

Five times in the history of life on Earth, mass extinctions have eliminated at least three-quarters of the species that were present before each episode began. The likely exterminators were volcanoes, noxious gases, climatic upheavals and the asteroid that did for the dinosaurs. Now a single species threatens to wipe out most of the others that surround it. We are faced with the realisation, as the ecologist Robert May puts it, that we "can now do things which are on the scale of being hit by an asteroid".

The lynx became top predator in Doñana after the last wolf was shot in 1951. That is how it goes with predators and large animals. The bigger they are, the sooner they tend to vanish. Among mammals, the risk of extinction rises sharply for species that weigh more than three kilograms—about as much as a small pet cat. Big creatures need more food and more space to find it in than small ones; they are slower to reproduce, and are apt to get on the wrong side of humans. "The species that tend to go extinct first tend to be the big-bodied things, and the tasty things," says Rob Ewers of Imperial College London. He is talking about the Amazon forests, but it’s a general truth.

Big animals, particularly those at the top of food chains, "are really fundamentally important to holding ecosystems together," says Jim Estes, a biologist based at the University of California, Santa Cruz. When they go, ecosystems unravel and reorganise, removing more species in the process. "Apex consumers" can take whole habitats with them. Wolves may protect forests by preying on the deer that browse saplings. If the wolves are wiped out, the deer multiply at the expense of the trees, preventing the forest from renewing itself: the end-point, as on the once-forested Scottish island of Rùm, is a treeless landscape. Globally, the result is the "downgrading of Planet Earth", as Estes put it in an article for the journal Science in 2011.

The exits began long before roads or rifles were devised. Nearly three-quarters of North American and a third of Eurasian megafauna disappeared between 50,000 and 10,000 years ago. Woolly mammoths, woolly rhinoceroses, giant sloths and sabre-toothed cats were among the species that vanished from the face of the Earth. While climate change was one part of the story, human expansion was another. The selective disappearance of large animals marks this period out from other extinction episodes, and was the start of what Estes and his fellow authors suggested "is arguably humankind’s most pervasive influence on the natural world". For Estes, it was the beginning of the sixth mass extinction.

If that was the opening phase, the second distinctive spike was the wave of extinctions in historic times that took place on islands colonised by humans and the animals that came with them on their voyages. Species after species ended up as dead as the dodo, which succumbed a few decades after people began to settle the Indian Ocean island of Mauritius in the 17th century.

Today Mauritius is the humanised world in a nutshell. Smaller than many English counties, it has been cleared for cane fields, strung with towns, dotted with resorts, factories and call centres. A mountainous outcrop endures as a national park, and the fragility of island nature is given poignant expression on the Ile aux Aigrettes, an offshore microdot the size of a large town park from which the rats and non-native plants have been cleared. Its 26 hectares are now covered in a low net of native vegetation that offers shelter to threatened native creatures. A pink pigeon huddles beneath a bush: it’s one of fewer than a hundred on the Ile, and of fewer than 500 in the world. A little bird called a Mauritius olive white-eye darts among the leaves: there are a couple of dozen here, and a couple of hundred in all.

The pink pigeon demonstrates even more dramatically than the Iberian lynx how emergency intervention can throw a species a lifeline—in 1990, there were just ten left in the wild. Turnarounds like these affirm the value of conservation efforts. But they may also induce complacency about the broader sweep of extinction. As a child I was distressed by the thought of species disappearing for ever, thanks largely to a small album entitled "Wildlife in Danger", filled with cards given away in cartons of Brooke Bond tea. Written by the ornithologist Peter Scott, it was published in 1963 and reissued ten years later. None of the 50 animals featured on the cards is yet officially classed as extinct—though one of them, the North American ivory-billed woodpecker, very probably is—and three or four are now out of danger. In the case of the Javan rhino, it is almost as though extinction has been put on pause. The album gives the population figure as just 40. Today’s estimates are pretty much the same—and the album has been superseded by a website showing 35 of them in individual video clips. The last of the species have become something like reality-TV stars. And my generation, the first to grow up with a background sense of ecological crisis, has reached middle age without having to read many obituaries of species.

That isn’t just because special efforts have been made for charismatic creatures, or because naturalists prefer to regard a species as missing until they are quite sure that it must be dead. About 800 extinctions have been recorded since 1500, a low figure even allowing for the likelihood that there are several unknown species for each one that has been given a Latin name, and most of them were on islands. Now that the island phase has largely run its course, the big question is what will happen on the land masses, where species are more vulnerable than in the oceans.

Estimates of future extinction levels are usually based on the relationship between the area of a habitat and the number of species in it. A rough rule of thumb is that if the area shrinks by 90%, 50% of its species will be lost. They may not go straight away, though. The difference between the number of species remaining and the number predicted is regarded as an "extinction debt" that will be paid in the long run. Rob Ewers took part in a study that found 80% of local extinctions in Amazon forests were still to come.

The accuracy of such predictions was challenged in 2011 by Fangliang He and Stephen Hubbell, who argued in a theoretical paper for the journal Nature that they always over-estimate extinction rates. Nevertheless, the researchers agreed that "the sixth mass extinction might already be upon us or imminent." And reports from remaining fragments of Brazil’s Atlantic-coast forests show that the situation on the ground may be much worse than it would appear from the graphs. "These habitat patches are essentially sitting ducks," says Carlos Peres of the University of East Anglia.

He and his colleagues surveyed 200 forest fragments across an area the size of Britain, wrecking four 4x4 vehicles in the process. They found what they called a "staggering" rate of extinctions among medium-sized and large mammals. Four-fifths of the populations had gone, although species-area calculations predicted that up to four-fifths would still be there. Fragmentation had left patches exposed to hunters and fire; their effects interacted "in a very perverse way", as Peres puts it, with those of area loss and isolation. Similar perverse synergies can be expected elsewhere. "I think that the processes we describe are actually quite ubiquitous," he says.

Peres regards the death of the last member of a species as "relatively trivial". What matters is the decline in population that leads to it. "People only care about those very terminal patients once the very last of a species dies out," he observes. "They hardly ever care about the long march towards global-scale extinction."

That view of nature, as an assembly of examples, does have a powerful hold on the imagination. We seem prone to a kind of Noah delusion: as long as we save a pair of each kind, we have fulfilled our responsibilities. But although a species that is down to its last few members is not extinct, it is not fully alive either, because it is no longer part of society. It is no longer contributing, competing, or helping to shape a larger living system. "Nature is not like a museum collection of the world’s species," says Georgina Mace of University College London. "It’s not just a matter of naming and keeping every one of those things. We should care about keeping the parts of the system. Can they still interact with each other? Can they still migrate, disperse, adapt, evolve?"

Mass extinction is thus more than the loss of kinds. It is the loss of abundance, of range, of populations. It is local extinction, the attrition of diversity as ranges shrink to enclaves, as well as global extinction. And, as the ecologist Daniel Janzen recognised in the 1970s, it is "what escapes the eye…a much more insidious kind of extinction: the extinction of ecological interactions."

Extinctions of this kind are surely happening all around us. Will they develop into a sixth mass extinction on the scale of the "Big Five", in which three-quarters of living kinds vanish? Extinctions among mammals, birds and amphibians are already running at higher rates than those which led to mass extinctions in the past, according to a study led by Anthony Barnosky, of the University of California, Berkeley. His team calculated that if all currently threatened species were to disappear within a century—the likelihood of which "would be quite high if we continue doing business as usual," Barnosky says—and extinctions carried on at the same rate, they could reach "Big Five" levels around 300 years from now.

In a later study, Barnosky and his colleagues pointed out that humans have already transformed over 40% of the Earth’s land ecosystems, and are on course to hit the 50% mark around 2025. They warn that this could trigger a radical change of state in the remaining lands. Human activity may be pushing the planet towards a tipping point that could "transform Earth rapidly and irreversibly into a state unknown in human experience."

And then there is climate change, which may take the planet past a whole series of tipping points. By 2080 its victims could include 40% of the world’s lizard populations and 20% of lizard species, according to a study which modelled the way the reptiles’ need to shelter from rising temperatures would interfere with their ability to forage for food. A survey of over 300 studies, covering 300 different kinds of organism, found that the average estimated risk of extinction by 2100 was over 10%. Climate change looks capable of causing mass extinctions on its own—and it won’t be acting on its own, but in concert with all the other forces humankind is exerting on the rest of the living world. Habitat destruction will block off climate escape routes. Lizards making for cooler regions will hit concrete highways; butterflies will flutter into agro-prairies devoid of flowers.

Late in the afternoon, back in Spain, I set out on a broad sand road along the edge of a Mediterranean pinewood, thrumming with herb scents, occupied by a female lynx and her two cubs. It is almost dark and I am almost back to where I started when I hear a single harsh-throated call, unfamiliar but unmistakably that of a carnivore. A little farther on, a low bolt hurls itself across the path; I glimpse a full-stretched limb like those I saw on the screens earlier in the day as the captive lynxes seized the partridges.

Beyond it, two figures are approaching; women out for an evening stroll, gently extending the life of the nearby town into the lynx’s territory. This, perhaps, is the best that can be hoped for: a local equilibrium achieved by moderating human claims enough to maintain a little space for natural variety on the fringes. All it needs is science, spadework, legislation, money and public consent.

It also requires that management, political support and popular concern are sustained indefinitely. Past mass extinctions, with the probable exception of the dinosaur-destroying one, unfolded over geological timescales. In historical times several hundred years of extinction passed before the emergence of widespread concern for the fate of other species. We can be quite sure that human pressures on other species will continue to grow, but we cannot assume that people will keep wanting to offset those pressures.

After each of the five mass extinctions of the past, life on Earth attained a new stable state. Although scientists have no way to model what kind of stable state would follow a sixth extinction, the vision that emerges from what we can see around us is that of a radically simplified biosphere, centred on our own species. "One thing that we’ll see is an increasing homogenisation," suggests Eline Lorenzen of the University of California, Berkeley, whose research into the prehistoric fate of large woolly beasts sent her prospecting for bones in Siberia. Cities will share the same limited stock of species, and outside them "it will basically be farmland".

It looks much the same to Georgina Mace, who foresees "a homogeneous world with the same relatively small number of species everywhere in rather large numbers. Wherever you went you’d see the same rats, the same blackbirds." It would be like today’s urban fringes, on continental scales. The age of dinosaurs was replaced by the age of mammals, and now we seem to be setting up an age of suburbs.

Humans will undoubtedly incur costs by degrading the living world, but there is little reason to suppose that the extinction of other species might threaten the position of our own. Whether the world turned to suburbs or wastelands, a sixth mass extinction would probably leave humankind dominating the biosphere more than ever. Whether our descendants would enjoy their hollow dominance is another matter.