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Even with a modern bow it’s still really difficult to sneak close enough to a deer to reliably make a kill shot. You’re not going to sneak close enough to poke it with a spear and with game that size, throwing rocks is not really an option either because that wont kill it. Something like axis deer is quick enough to even dodge a modern arrow.
The reality is that the animal will notice you and it will out-sprint you as well but it wont outrun a human on a long distance. When the animal is exhausted and no more able to run, then you can then stick your spear in it.
Which is why bow hunters typically scout ahead to determine where deer frequent, then hide and use calls and scents to get the deer to come to them.