I was studying the existence of two species in a ecosystem. I was thinking if there could be two consecutive stable equilibrium points. I don't have a valid proof in this regard.
But if geometrically we think, suppose a species is currently in a stable equilibrium. A small perturbation is applied and the species moves away from its previous steady point and reaches to a new steady point.
If the previous one was stable, then the species would tend to reach that steady point and the new one so became unstable.
If I assume that the perturbation was quite large to push the species far away from its first steady point, then the species would likely to go to the next steady point. Can this new steady point be a stable one again? If so, then possibly the species will not extinct in long run.
Moreover, if two consecutive steady states are stable, can one of them be stable and the other one be asymptotically stable?
Correct me please if I am wrong.