The Irish Deer Society has called for the complete closure of the Phoenix Park in Dublin. It would be a precaution against the possible spread of foot-and-mouth disease and for the protection of the deer herd there.
Mr Liam McGarry, chairman of the Leinster branch of the society, said it had complained to the Department of Agriculture about allowing traffic through the park.
"As we see it, the Phoenix Park is a huge farm and it has 400 to 500 fallow deer in it and it should be treated like any other farm," he said.
It was "illogical" to allow vehicles to pass through an area where one of the at-risk species which could contract and spread the disease was located.
"What logic is there to a situation where you have cars from all over the country, and I even presume from the North of Ireland, travelling through the park where they could carry the disease to the animals?" he said.
"Apart from the obvious danger the deer could have as a role in spreading the disease, this park herd is very important from a scientific point of view," he said.
"The deer would have to be destroyed if there was an outbreak of the disease anywhere near the park."
Duchas, which has responsibility for the park, said it had allowed cars through on veterinary advice from the Department of Agriculture.
Meanwhile, the Mountaineering Council of Ireland, which represents hill-walkers, climbers and ramblers, has condemned the activities of some members who, it said, were behaving irresponsibly by ignoring instructions to stay out of the countryside.
The council's chairman, Mr Peter O'Neill, confirmed that some hill-walkers had flouted the safety warnings on Saturday by participating in outdoor activities. He said, however: "The message is now getting through", and only a handful of individuals could be found yesterday on the Wicklow hills.
Up to a dozen volunteers from the council manned entry points to parks and walkways in north Wicklow over the weekend to try to prevent any walking activity.
"The reports we are getting is that lots of day-trippers, including foreign visitors, are coming up, rather than hill-walkers," Mr O'Neill said.
Most people were happy to turn back when the situation was explained to them, he said.