Scenario A: A chain split occurs due to a bug or contentious rule change (i.e. the miner incentives aren't changed in any way).
Both coins initially start out with the same difficulty (and won't change for another 2016 blocks, in the worst case), but their market price will vary.
A rational economic miner will only mine the chain with the highest reward, which is the chain with the higher market price. Users of the lower value chain can offer high fees to compensate, but this is costly.
Scenario B: A hard fork chain split occurs where one chain has decreased difficulty and/or higher block rewards (not touching the 21 million limit, but pushing the release schedule forward).
Unless the new chain has a really low market value, miners are highly incentivized to mine it. It will be hard for the old chain to compete.
The question for both scenarios: How will the scenario play out? Can both chains coexist or will the less popular chain die from a negative reinforcement loop (lower value -> less blocks mined -> even lower value) and be forced to do a PoW hard fork?
And if both chains can't coexist, does that mean that any chain that tries to out-compete bitcoin for the same PoW is a threat to the network?