# ⚛  L1 Principle — Lotka-Volterra Predator-Prey Dynamics

**ID:** `L1-407` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Computational Biology — *Population dynamics*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** LV parameter vector
> **📡 Carrier:** N/A · **🌫 Noise:** lognormal
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555218

---

## 🧠 1. Introduction

**Lotka-Volterra Predator-Prey Dynamics** is a **parameter-estimation problem** whose unknown lives in **LV parameter vector** space, within the **Population dynamics** sub-domain of **Computational Biology**.

Measurements consist of N/A via a **population census survey** sensing mechanism.

The forward operator applies, in order: time evolution of the state; O · nls · population fit operator; S · continuation · bifurcation LV operator.

Observations are corrupted by log-normal multiplicative noise. Existence of the recovered LV_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); stochastic_demographic_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Time derivative → O · nls · population fit → S · continuation · bifurcation LV → **y** (detector).

```
y = `S.continuation.bifurcation_LV` `O.nls.population_fit` ∂_t x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.time` | Time evolution of the state |
| `O.nls.population_fit` | O · nls · population fit operator |
| `S.continuation.bifurcation_LV` | S · continuation · bifurcation lv operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Computational Biology |
| Sub domain | Population dynamics |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | LV_parameter_vector |
| Noise model | lognormal |
| Integration axis | time |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered LV_parameter_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 20); stochastic_demographic_noise dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | population_trajectory_RMSE_log |
| Secondary | period_amplitude_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `population_census_survey`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Period yr | — | 5 |
| N observations | — | 50 |
| Amplitude h max | — | 1000 |
| Observation noise log sigma | — | 0.1 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Period yr | — | 0.5 – 50 |
| N observations | — | 10 – 300 |
| Amplitude h max | — | 10 – 100000 |
| Observation noise log sigma | — | 0.01 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.10 population_trajectory_RMSE_log

| Metric | Range |
|---|---|
| Population trajectory rmse log | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **population_trajectory_RMSE_log**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x2ddd2c04866e2278ad4c71739cb7b04448bd5d90fd7d2e384f238157a7bfd067`
- **Chain tx hash:** `0xfa70a63fa21039d7210901a3a16e7df7bfa2d224f561aac8c85797e637a10da7`
- **Chain block:** `41555218`

---

## File Mapping

This bundle consists of: `L1-407.md`, `L1-407.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-407.md` | Source of truth — edit this | Human or LLM |
| `L1-407.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-407`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._