Discrete exponential growth model

Mathematical statement

Fix a time step and a nonnegative constant r. The discrete exponential model is x_(n+1)=r*x_n. From initial x_0, repeated substitution or induction gives x_n=x_0*r^n. This is an algebraic result within the specified model, not proof that observed data fits it.

Representative formula (exact TeX)

x(n+1)=M(r,x(n))

Assumptions

• Time steps are fixed, n starts at 0, and r remains constant throughout the modeled period.
• Exercises use exact rational initial values and nonnegative rational factors, not a continuous-time model or empirical biological population forecast.

Symbol scope

• x — modeled quantity
• n — time-step index
• r — fixed per-step factor

Intuition, definition, and misconceptions

A factor r>1 scales each step by more than 1, r=1 leaves the factor neutral, and 0≤r<1 scales down. A zero initial value remains zero.

The recurrence and closed form can be compared step by step and proved equivalent by natural-number induction.

Real biological systems can involve limited resources, randomness, time variation and measurement error. A fixed-r model alone establishes no empirical fit.

Do not infer an eternally fixed factor from a few growing periods or identify a discrete multiplier directly with a continuous growth rate.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Sequence — Discrete states are ordered by natural-number time.
• Natural-number powers and principal square roots — The closed form uses natural powers of r.
• Arithmetic operations — Each recurrence step multiplies by the same factor.

Verification boundary

This slice checks only the exact fixed-step, fixed-nonnegative-factor model; it is not an empirical prediction.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Develop logistic models, time-varying factors, calibration and residuals.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.