Equilibria
An equilibrium is a state that does not change with time under the model dynamics.
One-dimensional systems
For
\[\frac{dx}{dt}=f(x),\]an equilibrium \(x^*\) satisfies
\[f(x^*)=0.\]At this point the instantaneous rate of change is zero.
Systems of equations
For
\[\frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x}),\]an equilibrium \(\mathbf{x}^*\) satisfies
\[\mathbf{F}(\mathbf{x}^*)=\mathbf{0}.\]Biological meaning
An equilibrium may represent extinction, coexistence, a disease-free state, an endemic state, or another constant biological condition.
Key idea. An equilibrium identifies a possible steady state of the model, but whether nearby trajectories approach or move away from it is a separate question of stability.