Question 2389: One Exact Capacity-Limited Discrete Increment

Original question and givens

Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).

• G001 — Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-discrete-logistic-growth

Step-by-step derivation

0≤r=3/4≤1; K=8>0; 0≤x=13/5≤K

T(x)=x+r·x·(1−x/K)

x/K at step1 := div(13/5, 8)

x/K at step1 = 13/40

1−x/K at step1 := sub(1, 13/40)

1−x/K at step1 = 27/40

x(1−x/K) at step1 := mul(13/5, 27/40)

x(1−x/K) at step1 = 351/200

increment at step1 := mul(3/4, 351/200)

increment at step1 = 1053/800

next state at step1 := sum(13/5, 1053/800)

next state at step1 = 3133/800

T(x)−x=r·x·(1−x/K)≥0; K−T(x)=(K−x)(1−r·x/K)≥0

0≤3133/800≤8

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute next state at step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute increment at step1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute x(1−x/K) at step1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute 1−x/K at step1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute x/K at step1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(13/5, 3/4, 8, 3133/800)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.