dN1dt

dN1dt is the time derivative of N1, representing the rate of change of the quantity N1 with respect to time. In many scientific fields, N1 denotes the number of individuals or items in a particular class, stage, or compartment of a dynamical system. The expression dN1dt thus measures how that quantity increases or decreases over time.

In practice, dN1dt is obtained from a differential equation that describes inflows and outflows for the N1 category. A general form is dN1dt = inputs − outputs, where inputs may include births or transitions into class 1, and outputs may include deaths or transitions out of class 1. A simple two-stage population example is dN1dt = b1 N1 − μ1 N1 − σ12 N1 + σ21 N2, where b1 is the birth rate into stage 1, μ1 is mortality in stage 1, σ12 is progression from stage 1 to stage 2, and σ21 represents any return flow from stage 2 to stage 1. In purely forward-transition models, the equation reduces to dN1dt = births − losses due to death or maturation.

dN1dt is central to ecology, epidemiology, and systems biology, where N1 may represent age classes, life stages, or compartments. Solutions to the system of equations {dNi/dt} describe how the distribution across classes evolves over time. The term carries units of individuals (or items) per unit time. Notation can vary, with N1′(t) or dN1/dt commonly used interchangeably.