To answer the question what an equilibrium point biologically means, I found the following knowledge on internet and I made a gist out of it.
When we say $N^*$ is an equilibrium point, we understand that there is no population growth in the population density. Or in other words, the population has attained its saturation level. If the initial population is much closer to the equilibrium point and it moves towards the equilibrium point as time increases, we say that the equilibrium is stable. If it is asymptotically stable, then all the initial populations will tend to move towards the equilibrium point as time increases. But if the population density is at unstable equilibrium, then eventually the population will move away from the equilibrium point.
Can I say that abovementioned lines biologically imply an equilibrium point? As far my concern, there is no biology involved in it.
Edit:
As Paichu mentioned in his answer, I am adding some more context to my question.
When we analysis a model, e.g. Malthus model or Logistic growth model or Allee effect, we find the equilibrium points first and then we find out whether the equilibrium point is ASE or an unstable equilibrium point. So what does an ASE point biologically mean? Or what does an unstable equilibrium point mean biologically?
If you're looking for a realistic biological interpretation of equilibrium, you need to give a specific scenario. In population dynamics in general, equilibria are just points where the rate of changes of all variables are zeros, meaning for instance, the number of birth and death are the same at that point. But say in a human body, equilibrium can refer to the homeostasis of the system (everything is changing, but the aggregated change does not affect the core property of the system).
Once you have a good interpretation of the equilibrium, then you can start discussing the implications of stable and unstable equilibria. For example, why is an equilibrium unstable from a biological perspective? To answer this type of question, you would need to look at the parameters involved in the condition for the existence and stability of the equilibrium.