stepgrowth

Stepgrowth is a term used in several disciplines to describe growth that progresses in discrete increments rather than continuously. In a stepgrowth model, the state variable y evolves through a sequence of steps, with abrupt changes Δy occurring at specific times or in response to thresholds. The approach contrasts with continuous growth models where the rate of change is defined for every moment in time, such as exponential or logistic growth. Stepgrowth can capture processes that advance only when certain conditions are met or at scheduled updates.

Mathematically, a simple stepgrowth representation can be expressed as y(t) = y0 + Σ_{k=1}^{N(t)} s_k, where N(t) is the number of steps that have occurred by time t and s_k is the size of the k-th step. If steps occur at fixed intervals Δt, then N(t) ≈ floor(t/Δt). More complex variants allow step sizes s_k to depend on the current state or time since the last step, or to be triggered by reaching a threshold in an auxiliary variable.

Applications include ecological models of population bursts triggered by resource thresholds, economic models of investment or adoption that occur at discrete moments, and computational or digital processes that update in discrete blocks. In chemistry and materials science, the term step-growth appears in separate contexts, such as step-growth polymerization, which is a different framework from discrete-time stepgrowth dynamics.

Limitations include potential sensitivity to the choice of step timing and difficulty in parameter estimation from intermittently observed data. See also discrete-time models, threshold models, piecewise-constant growth.