With his leopard-like spots, Navarro - a male lynx - calls out during mating season as he walks towards a camera trap.

Just short of 100cm (39 inches) in length and 45cm in height, the Iberian lynx is a rare sight. But there are now more than 2,000 in the wild across Spain and Portugal, so you're much more likely to see them than you were 20 years ago.

"The Iberian lynx was very, very close to extinction," says Rodrigo Serra, who runs the reproduction programme across Spain and Portugal.

At the lowest point there were fewer than 100 lynxes left in two populations that didn't interact, and only 25 of them were females of reproductive age.

"The only feline species that was threatened at this level was the sabre tooth tiger thousands of years ago."

The decline of the lynx population was partly down to more and more land being used for agriculture, a rise in fatalities on the roads, and a struggle for food.

Wild rabbits are essential prey for the lynx and two pandemics led to a 95% fall in their number.

By 2005, Portugal had no lynxes left, but it was also the year that Spain saw the first litter born in captivity.

It took another three years before Portugal decided on a national conservation action plan to save the species. A National Breeding Centre for Iberian lynxes was built in Silves in the Algarve.

Here they are monitored 24 hours a day. The aim is twofold – to prepare them for life in the wild and to pair them for reproduction.

Serra speaks in a whisper, because even from a distance of 200m you can cause stress to the animals in the 16 pens where most of the animals are kept.

Sometimes, though, stress is exactly what the lynxes need.

"When we notice a litter is becoming a bit more confident, we go in and chase them and make noise so they are scared again and climb the fences," says Serra. "We're training them not to get close to people in the wild."

That's partly for their own protection, but also so they stay away from people and their animals. "A lynx should be a lynx, not be treated like a house cat."

So the lynxes never associate food with people, they are fed through a tunnel system at the centre.

Then, when the time comes, they are released into the wild.

Genetics determines where they end up, to diminish the risks of inbreeding or disease. Even if a lynx was born in Portugal it might be taken to Spain.

Pedro Sarmento is responsible for reintroducing the lynx in Portugal and has studied the Iberian lynx for 30 years.

"As a biologist there are two things that strike me when I'm handling a lynx. It's an animal with a fairly small head for its body and extraordinarily wide paws. That gives them an impulse and ability to jump which are rare."

The breeding programme and the return of the lynx have been hailed as great successes, but as their numbers climb there may be problems too.

As lynxes are often released on private land in Portugal, the organisers of the reproduction programme have to reach an agreement with the owners first.

Where the animals go after that is up to them, and although there have been some attacks on chicken coops, Sarmento says there have not been many.

"This can lead to uneasiness within locals. We've been strengthening the coops so lynxes can't access them, and in some cases we keep monitoring the lynxes and scare them off if needed."

He recounts the story of Lítio, one of the first lynxes released in Portugal.

For six months Lítio stayed in the same area but then the team lost track of him.

He eventually made his way to Doñana, a national park in southern Spain where he had come from originally.

As Lítio was sick, he was treated and then returned to the reproduction team in the Algarve.

Within days of his release from the centre he began heading back to Doñana, swimming across the Guadiana river to reach Spain.

For a time he disappeared, but eventually he was brought back to the Algarve.

When he was released for a third time, Lítio did not venture back to Spain but instead he walked 3km (two miles), found a female and never moved again.

"He is the oldest lynx we have here, and he's fathered plenty of cubs ever since," says Sarmento.

Three decades after Spain decided to save the lynx, the species is no longer endangered, and Sarmento hopes it'll reach a favourable conservation status by 2035.

For that to happen, the numbers need to reach 5,000-6,000 in the wild.

"I saw the species disappearing. It's surreal that we're in a place where we can see lynxes in nature or through camera trapping almost daily," says Sarmento.

The reproduction team are not being complacent and there are risks involved in their work. Last year 80% of lynx deaths took place on the roads.

For now, though, they feel confident the Iberian lynx has been saved.