Empirical Parameter Scaling & Calibration Strategy

This document details the scientific strategy for bridging raw empirical database entries (TRY, PanTHERIA, GloBI, Pherobase, ToxValDB) and discrete simulation parameters within the Plant-Herbivore Interaction & Defense Simulator (PHIDS). It establishes a non-dimensionalization framework ensuring Design Space Exploration (DSE) searches yield biologically authentic ecosystems rather than unphysical mathematical artifacts.

1. The Core Calibration Challenge

1. Unconstrained DSE Fallacy: Search algorithms easily locate parameter sets that achieve Lotka-Volterra limit cycles or stable equilibria. However, without empirical bounds, these equilibria frequently rely on absurd biological ratios—such as a single aphid consuming \(500\text{ kcal/sec}\) or a grass blade synthesizing \(10\text{ MJ}\) per tick.

2. Heterogeneous Unit Mismatch: Open empirical databases store quantitative traits in SI or scientific units:

   • TRY: Specific Leaf Area (\(\text{mm}^2/\text{mg}\)), Canopy Height (\(\text{m}\)), Seed Dry Mass (\(\text{mg}\)), Tensile Strength (\(\text{N/mm}^2\)).
   • PanTHERIA: Basal Metabolic Rate (\(\text{mL O}_2/\text{hr}\)), Adult Body Mass (\(\text{g}\)), Weaning Age (\(\text{days}\)).
   • Pherobase / NIST: Volatile Diffusion Coefficients (\(\text{cm}^2/\text{s}\)), Henry's Law Constants (\(\text{mol/m}^3\cdot\text{Pa}\)).
   • ToxValDB: \(\text{LD}_{50}\) (\(\text{mg/kg}\)).

PHIDS operates within a discrete spatial-temporal grid where spatial coordinates are cells (\(1\text{ cell} = \Delta L\)), time progresses in ticks (\(1\text{ tick} = \Delta \tau\)), and energy is represented as bounded floating-point scalars (\(E_i \in [0, E_{\text{max}}]\)).

2. Five-Tier Empirical Calibration Pipeline

flowchart TD
    RawDB["1. Raw Empirical Trait Ingestion<br><i>TRY, PanTHERIA, GloBI, Pherobase</i>"] --> NonDim["2. Non-Dimensionalization (Buckingham Π)<br><i>Spatiotemporal Grid Anchoring (ΔL, Δτ, ΔE)</i>"]
    NonDim --> Allometric["3. Allometric & Biophysical Scaling<br><i>Kleiber's Law (BMR ∝ M^0.75) & MTE</i>"]
    Allometric --> Invariants["4. Thermodynamic & Mass-Balance Invariants<br><i>Primary Production ≥ Metabolic Upkeep</i>"]
    Invariants --> BoundedDSE["5. Empirically Bounded DSE Search<br><i>Search Hyper-Cubes [μ - 2σ, μ + 2σ]</i>"]

Tier 1: Spatiotemporal Dimensional Anchoring (\(\Pi\)-Groups)

All physical traits are non-dimensionalized using explicit grid anchors:

• Length Scale: \(L_0 = \Delta L\) (meters per cell).
• Time Scale: \(T_0 = \Delta \tau\) (seconds per tick).
• Energy Scale: \(E_0 = \Delta E\) (Joules per unit energy).

Key Dimensionless Groups:

• Metabolic Upkeep Ratio: \(\Pi_M = \frac{C_{\text{max}} \cdot \eta_{\text{dig}}}{m_i} > 1.0\) (guarantees herbivores can physically survive on ingested food).
• Diffusion Courant Number: \(\Pi_D = \frac{D_{\text{physical}} \cdot \Delta \tau}{\Delta L^2} \le 0.25\) (guarantees Numba 2D stencil numerical stability).

Tier 2: Allometric & Biophysical Scaling (Kleiber's Law & Arrhenius Kinetics)

Parameters scale automatically across 6 orders of magnitude in organism mass (\(M_{\text{body}}\)) and account for ambient micro-climate temperature (\(T\)):

• Metabolic Upkeep (Kleiber-Arrhenius Law):

Where \(E_a \approx 0.65\,\text{eV}\) is the metabolic activation energy (Metabolic Theory of Ecology standard), and \(k_B\) is Boltzmann's constant.

• Consumption Limit:

• Foraging Velocity & Grid Resolution Scaling:

• Cell Carrying Capacity: Cell energy bounds scale quadratically with grid resolution: \(E_{\text{max, cell}} = E_{\text{ref, cell}} \cdot \left(\frac{\Delta L}{\Delta L_{\text{ref}}}\right)^2\).

Tier 3: Thermodynamic Invariants & Hard Biological Safeguards

• Photosynthetic Cap: Plant daily growth \(r_{\text{growth}}\) bounded by incident solar irradiance per cell surface area: \(r_{\text{growth}} \le I_{\text{solar}} \cdot \Delta L^2 \cdot \Delta \tau\).
• Digestibility Ceiling: \(\eta_{\text{digest}} \le 1.0\); Handling Time \(T_h \ge \frac{1}{\text{max\_bite\_rate}}\).

Tier 4: Empirically Bounded DSE (Log-Normal Hyper-Cubes & Taxonomic Imputation)

1. Log-Normal Space Transformation: Because ecological traits (body mass, seed mass, SLA, metabolic rates) span multiple orders of magnitude following log-normal distributions, DSE hyper-cubes are constructed in log-transformed parameter space to prevent unphysical negative lower bounds or linear skew:

   \[p_k \in \left[ 10^{\mu_{\log, k} - 2\sigma_{\log, k}}, \, 10^{\mu_{\log, k} + 2\sigma_{\log, k}} \right]\]

2. Hierarchical Taxonomic Imputation: When empirical database entries are missing for a specific species, the ETL pipeline imputes trait parameters using geometric means following a strict phylogenetic hierarchy:

   \[\text{Species} \to \text{Genus} \to \text{Family} \to \text{Functional Group}\]

3. Multi-Objective Cost Function:

   \[J_{\text{eco}} = w_1 \cdot S_{\text{LV}} + w_2 \cdot \sum_{k=1}^K \left( \frac{\log_{10}(p_k) - \mu_{\log, k}}{\sigma_{\log, k}} \right)^2 + w_3 \cdot P_{\text{thermo}}\]
   • Rewards stable Lotka-Volterra limit cycles (\(S_{\text{LV}}\) via FFT spectral analysis).
   • Penalizes Mahalanobis distance from empirical log-space biological distributions (\(D_{\text{bio}}\)).
   • Imposes step penalties for thermodynamic violations (\(P_{\text{thermo}}\)).

3. Parameter Mapping Matrix