In January, Mr and Mrs Darwall were given the green light to take the case to the Supreme Court.
Hedge fund manager Mr Darwall said ahead of the Supreme Court hearing: “No-one is suggesting restricting public access to walk or ride in the national park.
“But the extent of recreational activity is an important issue if we want to preserve the open moor for the long term.”
He said campfires on dry ground were dangerous and campers often left litter and human excrement.
Campaigners The Stars are for Everyone, who were part of a demonstration at Hound Tor on Sunday, said whatever the outcome, the Darwalls’ legal action had “succeeded in unleashing something powerful and ancient, a commitment to our commons, our love for the land, our right to belong”.
The hearing is due to last one day and a judgment is expected at a later date.
