Question 2354: A Short Exact Orbit of the Specified Logistic Increment Model

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. initial=3/25, rate=3/4, capacity=8/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.

• 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. initial=3/25, rate=3/4, capacity=8/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.

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=3/25≤K

T(x)=x+r·x·(1−x/K)

initial=3/25; steps=1

retain the initial state before updating

x/K at step1 := div(3/25, 8)

x/K at step1 = 3/200

1−x/K at step1 := sub(1, 3/200)

1−x/K at step1 = 197/200

x(1−x/K) at step1 := mul(3/25, 197/200)

x(1−x/K) at step1 = 591/5000

increment at step1 := mul(3/4, 591/5000)

increment at step1 = 1773/20000

next state at step1 := sum(3/25, 1773/20000)

next state at step1 = 4173/20000

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

0≤4173/20000≤8

orbit=[3/25, 4173/20000]

entries=2=steps+1

Reverse checks

• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute next state at step1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute increment at step1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute x(1−x/K) at step1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute 1−x/K at step1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute x/K at step1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(3/25, 3/4, 8, 1, [3/25, 4173/20000])

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.