Then they carry on walking again, quite often turning around to parallel walk the way they came.
This gives the males a chance to size each other up and most often the smaller male will concede without fighting.
"The less dominant stag will just run away, because he thinks he's going to get beaten," says Mr Morris.
"They are only going to fight if they think they have a good chance of winning."
Battle of strength
Even when they do fight, deaths are rare.
"They are fighting to breed which is the ultimate prize and obviously being killed isn't the aim," explains Mr Morris.
"The antlers are designed that they can fight and push against and interlock and it's a battle of strength."
The Rum Red Deer project assesses the lives of the wild red deer
Titus and Percy had previously faced each another in a significant battle that left both stags unharmed, a fight that was filmed by the BBC crew. The rivalry then turned fatal.
Mr Morris thinks that Titus only died because of the location of his last fight.
"I think the reason that this stag died is that he's unlucky. If they hadn't have been on the wet rock and he hadn't have slipped it probably wouldn't have ended the way it did."
The battle between Percy and Titus was the first mortal combat Mr Morris has seen in his years spent observing red deer on Rum.
"It's very rare to actually see it. It's not an annual event in the study area," he says.
Rum is owned and administered by Scottish Natural Heritage, and scientific studies of the ecology of the behaviour of the deer on the island are undertaken by the Rum Red Deer Project, a collaboration between the University of Cambridge and the University of Edinburgh, UK.
While red deer research has been conducted on the island since 1953, since 1972 a specific study of deer has been conducted in the Kilmory area in the north of Rum, known as the North Block.
Here approximately 12 main stags service 200 hinds.
Red deer stags on Rum usually live to an age of around 12 years, while hinds can live to around 14 or 15 years.
Percy, at around 14 years old, is one of the oldest stags on the island.
A more detailed analysis of the encounter and the implications of the deer rut will be broadcast on
on Friday 9 October at 2100BST on BBC Two.
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