"They were there on game days, within 50 metres of hundreds of people – and no one knew," says the centre's deputy director, Dr Rodney van der Ree. "They're able to live in very close proximity to us without getting spotted."

The European or red fox – Vulpes vulpes – is a handsome animal. It has a pointed muzzle, auburn coat and bushy tail. In Quentin Blake's famous illustrations for Roald Dahl's Fantastic Mr Fox, the titular mammal wears a waistcoat.

But they are one of the most invasive species we've got. Foxes are considered a threat to 76 kinds of native birds, mammals, reptiles and amphibians, including the critically endangered orange-bellied parrot, spotted quail-thrush and western swamp tortoise.

"The fox and the cat, between them, have been responsible for the decline and extinction of many species of native mammals," van der Ree says.

But we are still fascinated by them; fox mad, even. Our championship-winning netball team is the Melbourne Vixens (last month, for the 2014 flag, they made mincemeat of Queensland's Firebirds).

Search the internet for foxes in Melbourne and you'll see the city declared "the fox capital of the western world" on the authority of the RSPCA, no less. Contacted by Fairfax Media this week, the RSPCA could not confirm it had ever made such a claim.

In fact, we can't be sure how many foxes there are in the state. The last estimate came from CSIRO research in the early 1990s.

John Matthews, from the Department of Environment and Primary Industry, says there's no reason to think numbers have changed significantly. But there are more than you think: in country Victoria, foxes number between 1 and 4 per square kilometre. But in the city, where the living is easy, there are four times as many.

Around the wharves and wastelands of Port Melbourne, the vulpine population is at its peak: as many as 20 foxes prowl every square kilometre.

Most sightings happen in autumn, when naive pups leave the den in search of food and territory. On the citizen science website FoxScan, where people can list glimpses of foxes, the number of reports has been steady.

"They're quite timid and shy, and they do whatever they can to avoid any interaction with humans," says Matthews, who manages the state's control programs for feral foxes, pigs, goats and rabbits from Casterton, in the Western District.

Nevertheless, sometimes we glimpse their secret lives at night: stalking along the train tracks in Elsternwick, slinking across a road in Box Hill, or padding over the flatland by a bridge in Essendon.

Sometimes, we even spy them in the day. Recently, two foxes were photographed on a rooftop in Mount Waverley after lunchtime. More often, however, we see them as roadkill in the morning: in the absence of urban predators, cars are their biggest threat.

Greater Melbourne may or may not be the fox capital of the western world, but it is the first place they called home in Australia.

The new colony's gentlemen brought them out for sport. In 1860, Edward Wilson, the owner of The Argus newspaper, lauded the benefits of fox hunting for youth. It "tends to prevent them from sinking into mere dawdlers in an opera box or loungers in a cafe", he wrote.

A recent historical study by zoologist Ian Abbott contends that it took several attempts over decades before a fox population took hold.

He believes the successful culprits were the wealthy Chirnside brothers – Thomas and Andrew – who were among the largest landholders in the colony. They had already established a herd of deer, and by 1870 began importing foxes for hunting expeditions on their Werribee estate, where, soon after, they began constructing the mansion that still stands today.

Within two decades, foxes were declared a pest in Victoria, and within 40 years they pawed all the way into Western Australia. They'd long since conquered NSW and Queensland. Now, the red fox ranges over three-quarters of the continent.

Surprisingly, however, there were no recorded sightings in the centre of Melbourne until the middle of the 20th century. In 1948, a fox den was spotted at the cemetery in Parkville. Now, they're thought to occupy the entire metropolitan area.

"There's ideal habitat in the city," says Matthews. "There's ample food – lots of bins – and there's really no predation."

They're deterred by dogs, but if there are none around, they'll hide under houses or in tight gaps between buildings, in rock heaps, culverts or up cypress trees. They dig dens in parks or coastal scrub.

Foxes want warmth, security and good shelter, and they don't want to give it up. They mark their territory with urine and scat.

"It's a unique smell. But don't ask me to describe it!" says Matthews. "Farmers know it. They'll get out to the gate post in the morning and say: 'Hmm, a fox was here last night'."

Despite the stench, the animals pose little threat to humans – other than as a carrier of worms or, in the case of an outbreak, rabies. They are, however, a mortal danger to suburban chickens.

Many a suburban homesteader has woken to a grisly crime scene, after the one-and-only night they forgot to shut the henhouse gate. Often, the dead chooks remain uneaten. "Foxes are thrill-killers," Matthews explains. "It's instinctive. With the clucking and feathers going everywhere, they can get into a frenzy."

In the countryside, foxes exact a toll on farmers' flocks. But their impact on native species is far greater than their cost to agriculture.

In the city, there are now very few threatened species – largely because foxes and cats have already wiped them out. But on the city fringe, foxes are still eating endangered species, including southern brown bandicoots, eastern barred bandicoots, growling grass frogs and swamp skinks.

For the authorities in cities and towns, it's very hard to control their numbers – some municipalities try trapping, or netting and fumigating dens. Citizens can help reduce fox numbers by making sure pet food isn't left outside, clearing fallen fruit from trees and using a secure compost bin.

Elsewhere around the state, the Department of Environment and Primary Industries and Parks Victoria conduct an extensive baiting program, using the poison known as 1080. And in 2011, the state government introduced a $10 bounty for fox scalps. Since then, 285,000 have been accepted from hunters.

Neither method has escaped criticism. Many landholders are wary of 1080. For its part, the RSPCA states that 1080 "is not a humane poison". It argues we must conduct more research into alternatives, so the technique can be phased out.

The bounty, meanwhile, has been roundly criticised by experts, who argue it's a waste of money because foxes breed too quickly. Control programs need to wipe out two-thirds of a fox population over a large area to have a lasting effect.

In any case, if foxes were once the universal bad guy, now their role is much more ambiguous, especially in the city.

"Even in modified ecological systems, top-order predators are still important to keep prey populations in check," says van der Ree.

In urban areas, native predators are missing: the quolls have gone, there are no goannas and few powerful owls. That leaves the fox to help limit the populations of feral cats, rats and mice, and even possums.

"If you get rid of all the foxes then feral cat numbers can increase. You need to control foxes, cats and rabbits together. It's got to be done strategically, over a large scale, otherwise they'll just reinvade the area," he said.

It's a controversial, but increasingly prevalent question: do we need foxes in our cities?

"No one would argue that foxes are not a damaging species, but sometimes they may help us as well," says Dr Euan Ritchie, an ecologist from Deakin University, whose speciality is the role of predators in ecosystems.

We need to rethink the way we deal with pest species more generally, he says. "A lot of the things we do now are expensive and they're not working. Are there ways to better coexist?"

Part of the answer, he says, is encouraging predator species, but doing so in conjunction with guardian animals on farms. Alpacas or Maremma sheepdogs are increasingly used to protect livestock from dingoes.

Ritchie advocates the re-introduction of Tasmanian Devils to Victoria, beginning with Wilsons Promontory. They'd help threatened species and improve biodiversity, he says, by controlling the numbers of foxes, cats, swamp wallabies and wombats. "The way people often describe it is: My enemy's enemy is my friend."

It's an argument made famous in 1949 by the pioneering American ecologist Aldo Leopold in his classic book, A Sand County Almanac. He wrote that he'd seen "state after state extirpate its wolves", and subsequently, observed their landscapes overrun and defoliated and eroded by deer.

"I now suspect that just as a deer herd lives in mortal fear of its wolves, so does a mountain live in mortal fear of its deer," he wrote.

Ritchie says the scientific community is only just catching on. There's now a growing push for "rewilding": restoring habitats by introducing or reintroducing key species.

He is co-ordinating the Australian Mammal Society's annual conference, at the Melbourne Zoo from July 7-10. One whole day will be devoted to the ecological roles of predators.

"In the last 200 years, Australia has arguably the worst record in the world for mammal conservation. And globally we're in the midst of a biodiversity extinction crisis comparable to some of the biggest mass extinction events we've had in geological history.

"If we can find any way to start reversing that, then we should prioritise it. We really need to be more bold, because if we keep going down the same path, many more species will disappear."

The Royal Botanic Gardens in Cranbourne has a feral-proof fence: it is nine kilometres in circumference and 1.8 metres tall, with a floppy top to prevent foxes climbing over.

At the base, wire has been laid below ground level and set back from the fence line, to stop foxes digging under.

"Once a week, we drive right around the perimeter to make sure there are no new holes," explains Ricardo Simao, the gardens' land and infrastructure manager.

"The danger is that, sure, we might be keeping the foxes out, but we're also stopping all other animals from moving through."

So the fence has another trick: allowing native animals to pass. Simao's team have crafted gates for bandicoots, wombats and tortoises. Now they're working on one for swamp wallabies, whose numbers have risen fast in the absence of predation.

The bandicoots' gates consist of PVC piping with flaps at either end: foxes can't fit and rabbits aren't curious enough to brave the flaps, Simao explains.

Southern brown bandicoots were abundant in the south-east when Melbourne was colonised. "Everyone used to see them – the animal wasn't particularly shy," he says. But gradually, the animals lost their habitat to urbanisation and their heads to foxes.

In the decade since the fence was built, however, the bandicoots' numbers have increased and become "quite healthy", Simao says. "But the next phase is to make sure it's not an isolated population."

Michael Green is a Melbourne journalist