How is a steady state defined in a mathematical model?

In mathematical modeling, a steady state refers to a situation in which the variables of the system do not change over time. This means that their values remain constant, leading to a stable equilibrium within the model. In various fields such as physics, economics, and biology, the steady state is often used to describe situations where all the influencing factors are balanced, leading to consistent outputs or behaviors in the system.

For example, consider a population model where the birth rate equals the death rate, leading to a constant population size; this scenario exemplifies a steady state. In contrast, conditions that involve fluctuations, maximum changes, or random variations do not fit the definition of a steady state, as they indicate ongoing dynamics rather than a stable condition. Thus, defining a steady state as a situation where values remain unchanged over time accurately captures the essence of this concept in mathematical models.