Discrete logistic increment model

Mathematical statement

This discrete logistic increment model fixes K>0 and 0≤r≤1 and updates x_(n+1)=x_n+r*x_n*(1−x_n/K). If 0≤x_n≤K, the next state remains in [0,K]; 0 and K are fixed points. This is the specified discrete model, not an analytic solution of the continuous logistic equation.

Representative formula (exact TeX)

x(n+1)=G(x(n))

Assumptions

• Time step, K and r remain fixed; the initial state lies in [0,K] and every computation uses exact rationals.
• Exercises expand only short finite orbits and infer no stability, convergence or real-population fit from them.

Symbol scope

• x — model state
• K — fixed capacity
• r — per-step increment parameter

Intuition, definition, and misconceptions

Near x=0 the capacity factor is near 1; at x=K the increment is zero.

The bounds 0≤r≤1 and 0≤x≤K make one update preserve the interval. This does not automatically cover other parameter domains or definitions of a discrete logistic model.

Real populations face randomness, time-varying capacity, migration and measurement effects. A fixed model’s finite orbit is not empirical validation.

Do not conflate different logistic-map forms or treat a short orbit’s trend as long-term stability or a measured forecast.

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

• Discrete exponential growth model — The logistic increment adds a capacity-suppression factor to fixed-factor growth.
• Sequence — Discrete time indexes the model states.
• Arithmetic operations — The update uses exact addition, subtraction, multiplication and division.

Verification boundary

This slice fixes the increment model and parameter domain; it asserts neither stability, convergence nor biological fit.

not_formalized

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

Next obligation

Compare the standard logistic map, Euler discretization, equilibrium stability and data calibration.

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.