{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-discrete-logistic-growth",
  "semantic_family_id": "sf-discrete-logistic-growth",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "離散 logistic 增量模型"
  },
  "summary": {
    "zh-Hant": "固定容量K與0≤r≤1時，本條目以x_(n+1)=x_n+r x_n(1-x_n/K)描述每步受限增量。"
  },
  "statement": {
    "zh-Hant": "本條目的離散 logistic 增量模型固定 K>0 與 0≤r≤1，以 x_(n+1)=x_n+r x_n(1-x_n/K) 更新。若 0≤x_n≤K，則下一步仍在 [0,K]；0 與 K 是固定點。這是指定離散模型，不是連續 logistic 方程的解析解。"
  },
  "formula": {
    "tex": "x(n+1)=G(x(n))",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>x</mi><mo>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mi>G</mi><mo>(</mo><mrow><mi>x</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        },
        "arguments": [
          {
            "type": "addition",
            "operands": [
              {
                "type": "symbol",
                "name": "n",
                "glyph": "n"
              },
              {
                "type": "number",
                "value": 1
              }
            ]
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        "arguments": [
          {
            "type": "function_call",
            "callee": {
              "type": "symbol",
              "name": "x",
              "glyph": "x"
            },
            "arguments": [
              {
                "type": "symbol",
                "name": "n",
                "glyph": "n"
              }
            ]
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "0eed7402e27a4e0bb8aa466347ce3e366d8637dfa98a961925a2299579811f49"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "時間步、K與r固定，初值在[0,K]，所有計算採精確有理數。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫最多展開短有限軌道；不由有限結果宣稱穩定、收斂或真實族群適配。"
    }
  ],
  "symbols": [
    {
      "token": "x",
      "role": "modeled_population",
      "role_zh": "模型狀態",
      "scope": "representative_formula"
    },
    {
      "token": "K",
      "role": "carrying_capacity",
      "role_zh": "固定容量",
      "scope": "representative_formula"
    },
    {
      "token": "r",
      "role": "increment_rate",
      "role_zh": "每步增量參數",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-discrete-exponential-growth",
      "title_zh": "離散指數成長模型",
      "reason_zh": "logistic增量在固定倍增上加入容量抑制因子。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "模型狀態由離散時間索引。"
    },
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "更新式使用精確加減乘除。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "當x接近0時容量因子接近1；當x=K時增量為0。",
      "0≤r≤1與0≤x≤K使一次更新保留區間，但這不自動證其他參數域或其他離散logistic定義。",
      "實際族群會受隨機性、時變容量、遷移與量測影響；固定模型的有限軌道不是經驗驗證。"
    ],
    "common_misconception_zh": "把不同形式的logistic map混為同一模型，或把短軌道趨勢當長期穩定性與實測預測。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪固定增量模型與參數域；不宣稱穩定性、收斂或生物適配。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "比較標準logistic map、Euler離散化、平衡點穩定性與資料校準。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-discrete-logistic-growth.md 整理；2026-10-13 脈衝 R07（使用者手動接續 2026-10-21 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}