What is the need for finding the equilibrium points in the mathematical models?

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What is the need for finding the equilibrium points in the mathematical models? Why we check the stability of those equilibrium points?What benefit we will get in real life after doing such calculations?

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The concept of equilibrium of a dynamical system is associated with the attractors of the system, which characterise the long-term behaviour

Any dynamical system may have no, one or several equilibrium points, each of which may either be stable or unstable. A plain mechanical example is a simple pendulum. If the pendulum is at the downward position, the system will remain there forever, and thus, the downward position is an equilibrium point. Obviously, this equilibrium is stable (if friction is considered). Likewise, it is easy to see that the upward position is an unstable equilibrium.

Understanding these equilibria tells you important system behaviours that characterise the model. Understanding these, leads one to understand the model fully.