Question 2361: 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=10/25, rate=0/1, capacity=5/1, updates=2. 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=10/25, rate=0/1, capacity=5/1, updates=2. 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=0≤1; K=5>0; 0≤x=2/5≤K

T(x)=x+r·x·(1−x/K)

initial=2/5; steps=2

retain the initial state before updating

x/K at step1 := div(2/5, 5)

x/K at step1 = 2/25

1−x/K at step1 := sub(1, 2/25)

1−x/K at step1 = 23/25

x(1−x/K) at step1 := mul(2/5, 23/25)

x(1−x/K) at step1 = 46/125

increment at step1 := mul(0, 46/125)

increment at step1 = 0

next state at step1 := sum(2/5, 0)

next state at step1 = 2/5

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

0≤2/5≤5

x/K at step2 := div(2/5, 5)

x/K at step2 = 2/25

1−x/K at step2 := sub(1, 2/25)

1−x/K at step2 = 23/25

x(1−x/K) at step2 := mul(2/5, 23/25)

x(1−x/K) at step2 = 46/125

increment at step2 := mul(0, 46/125)

increment at step2 = 0

next state at step2 := sum(2/5, 0)

next state at step2 = 2/5

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

0≤2/5≤5

orbit=[2/5, 2/5, 2/5]

entries=3=steps+1

Reverse checks

• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute next state at step2 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute increment at step2 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute x(1−x/K) at step2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute 1−x/K at step2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute x/K at step2 with exact rational arithmetic, not rounded decimals.
• 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

(2/5, 0, 5, 2, [2/5, 2/5, 2/5])

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.