Every year, the swollen Okavango River floods the Kalahari Desert and the area bristles with fearsome big game. Time for that Potteresque cloak of invisibility or perhaps a secret whistle

Anyone who has been on safari must be familiar with the sensation of invisibility.

It accompanies the moment when the likelihood of being trampled, mauled or simply eaten seems imminent – for example, when a lion or two stride purposefully up to you, seated and helpless in an open Land Cruiser with the engine switched off. "I am invisible," you repeat inwardly while clenching your suddenly unreliable bowels. Amazingly the predators really seem unable to see you and amble past within touching distance, uninterested in easy meat. It is tempting to conclude that safari tourists are protected by a Potteresque invisibility cloak. That would be unwise.

John Fox, a chemical engineer from Newcastle, was on day four of his first safari when he found out the hard way. "The elephant came into view from behind a tree. They are cunning masters of disguise," he laughs. John and his wife, Louise, have returned to Eagle Island Camp in the Okavango Delta – a year on from a very unpleasant encounter. "We saw him and we started walking back towards the room as you're supposed to do. Somebody shouted 'you're too close'. The elephant heard it, turned, saw us – I guess got scared – put his trunk up, ears back, trumpeted and charged."

The elephants had been grazing inside the camp on their favourite fruit-bearing jackalberry trees; they (in common with most of the other fauna) don't recognise the camp boundaries. The rogue male was wandering between tents when he was startled by the Geordie couple. John was lucky to survive the charge, but suffered a painful fracture of his tibial plateau (knee) as he tried to make his escape; he has had eight months of reconstructive surgery and rehab since. Despite that trauma the couple are back, which seems a bit counterintuitive.

"The first three and a half days were absolutely fantastic," says Louise. "Probably the best holiday we'd ever had. That amount of wildlife, the birds, the scenery, we'd enjoyed so much – we felt we had to come back."

The Okavango is an unlikely Eden. Essentially it is a 15,000sq km flat pan of sand in the Kalahari Desert. Every year it is transformed into a lush swamp by the Okavango River which empties into it from the watersheds of Angola, creating an inland delta. Where there's water, there's life. It comes in a bewildering range of shapes, sizes and colours – from the granite grey mass of elephants feeding in the shallows to the tiny and ubiquitous painted reed frogs that cling, as their name suggests, to papyrus reeds.

Game viewing is done from power boats or dug-out canoes called mokoros. The chance of sighting big cats from the boats is remote but this is a birder's paradise. Matt, at the controls of our boat, is the kind of guide who can spot a chinspot batis from 100 yards by the merest hint of its call; a descending "teuu-teuu-teuu" to the tune of three blind mice, since you ask. Smartly uniformed fish eagles occupy sentry posts in the apex of trees; pied kingfishers hover above the water flashing monochrome arcs into the rippling currents when they strike. The sunset, always a spectacle in Africa, is twice as nice on the Okavango's mirrored surfaces. Twilight descends and Matt glides us into a noisy heron nursery – countless herons, as well as nesting egrets and reed cormorants flap and flail in the swaying reeds as our boat skims past.

The safari experience at Khwai River Lodge, my next stop, is on terra firma. Manager Jeff Gush has a luxurious blond mane that evokes something Tarzanish; it seems conceivable he might wrestle crocodiles for fun. Next to the bar and restaurant terrace he points out a dark patch on the lawn. "That was where a pack of wild dogs killed an impala two weeks ago while guests were having breakfast." I picture guests passing the muesli while the unfortunate animal was ripped to bits a few yards away.

My guide here has decided to call himself "Leopard" because "it is more suitable to my work". His real name, Kashanga, seems adequate to me and has the benefit of not causing a false dawn of excitement (Leopard!) every time someone tries to catch his attention. He fails to find a leopard but ... he knows where the lions are.

Khwai River Lodge is in a concession rather than within a national park and, as such, the game drives are regulated with a lighter touch. Guides are allowed to drive off-track if they spot something interesting. In this case, it is a large bush that shelters a substantial pride of lions.

We drive up to within a few feet. The bloody ribcage of a large animal is inside the bush – the remains of a giraffe that the pride killed a few days ago. There are two males, a bunch of females and about eight cubs. They all look very smug and their immediate plans don't seem to extend beyond a good yawn and stretch, and then possibly another stretch and yawn. The cubs are playing like kittens, creeping up on each other and pouncing. They try out this rough and tumble on their alpha male dad, who sports a mane even more magnificent than Tarzan's. The king lion could swat them aside with a brush of his huge paw, but he is very indulgent with his progeny allowing them to muss up his hair. If a lion could smile paternally this one is doing it.

The lodge itself, another in the Orient Express trio of camps I am visiting, is surreally comfortable. My tent is pitched on a wooden platform and is sympathetically designed not to look out of place on the banks of the Khwai. It is palatial and has hot and cold running water, electricity, air conditioning, two wash basins, a sit-down toilet, stand-up shower, a hair dryer and a phone. There is a manifest eagerness to round off the edges of adventure with creature comforts.

After dinner, sitting at the fire pit between the internet lounge and the floodlit swimming pool, I ask Jeff if the modern safari is perhaps too slick. Surprisingly, he agrees. "I think secretly, in the back of their minds, a lot of people who come on safari wish it could be like it was in the 1930s, 40s and 50s. You can still do that in a few places – small owner-run operations were the norm a few years ago but they've now become archaic." Jeff, a South African who's spent the past 20 years in the bush in Botswana, becomes wistful: "You'd have to accept there's no Wi-Fi and no electricity – just a fire, great food and the wilderness."

There is a shuffling noise just behind us in the reeds that cuts through the background metallic tinkling music of thousands of painted reed frogs. Jeff gestures to me to be quiet and reaches for his torch, which he rakes across the darkness. The beams catch three hippos, out of the water, grazing right next to our terrace – they look like primeval swamp creatures sent to remind us of our vanities.

From the light aircraft on the short flight from Khwai to Savute Elephant Camp you can see the havoc wrought by elephants from 1,000ft above. Acres of woodland lie flattened in what resembles hurricane damage.

The Savute Camp is ringed by an electric fence to keep the voracious pachyderms out. They had a taste for the local camelthorn trees that grow between the tents. Initially the camp management tried to protect the vegetation by wrapping chicken wire around the tree trunks but the elephants would not be denied and had to be excluded. The electric fence is porous to most other animals that wander through at night and they leave tantalising footprints in the sand.

The camp, sited on the Savute Channel, is experiencing a miracle of nature this year. A notice tells guests about the "mystery" of the channel which has the ability to turn its flow on and off – dry since 1982 it says, adding rather plaintively that "one can only imagine a river here with crocodiles and hippos". Imagine no more. The sands have shifted, or rather the tectonic plates beneath the Kalahari have (according to one theory) and the channel is flowing again for the first time in 29 years. The result is an instantly revivified ecosystem, pulsating with life. The change is so recent that only three pioneering hippos are known to have recolonised the river and word has yet to spread to the crocodile community.

Savute is within the Chobe National Park, and under its rules my guide, Rodgers, must not leave the designated tracks during game drives. It seems unlikely that, under such constraints, we will have great sightings. No matter, Chobe is extraordinarily beautiful and offers a huge range of habitats: riverine, desert, savannah, mopane woodlands, hills and floodplains and, this year, something seems to have electrified the wildlife. First Rodgers produces the best elephant sighting of my life – I count 33 in the extended herd – frolicking at the edge of the Chobe River. This is followed by a pride of lions. They seem out of reach, hundreds of yards away on the edge of the Savute marsh. But as soon as they spot the safari vehicles they come running to entertain us – lying down right next to the vehicles striking a variety of photogenic poses for the best part of an hour.

Later we come upon a leopard surveying the grasslands from a termite mound. Leopards are solitary and notoriously shy animals, but this one is as raucously confident as Def Leppard. Instead of slinking off into cover it makes a beeline for us, rocking with a swagger between the Land Cruisers, and close enough to be stroked. It sets off pandemonium among its fans.

I ask Rodgers how he does it. He doesn't miss a beat. "I have a whistle. It's a secret whistle that you can't hear but the animals can. I blow, they come." And Rodgers bursts into a great big life-affirming cackle.

Compact Facts

How to get there

Sankha Guha travelled with Cox & Kings (020-7873 5000; coxandkings .co.uk). The nine-day/six-night Great Botswana Safari costs from £4,575 per person including return BA flights from London, transfers, game drives and all-inclusive accommodation at Orient Express lodges.