Morphological Defenses and Dynamic Resource Reallocation

This document provides the formal mathematical and biological formulation for plant morphological (constitutive) defenses and dynamic resource reallocation (apparent nutrition withdrawal) in the PHIDS ecosystem simulation model.

1. Biological and Ecological Context

Plants possess two distinct evolutionary defense paradigms against herbivory:

Constitutive (Morphological) Defenses: Permanent structural barriers such as trichomes, thorns, spines, and indigestible cell-wall compounds (lignin and silica). These defenses are metabolically fixed and directly impair herbivore feeding mechanics or reduce nutrient assimilation efficiency during every trophic attack.
Inducible Defenses & Dynamic Allocation: Physiological responses catalyzed by herbivore damage or volatile organic compound (VOC) warning signals. Upon perception, plants actively withdraw mobile nutrients and carbohydrates from vulnerable tissues (leaves and shoots) via vascular phloem transport into below-ground root sinks, temporarily lowering their nutritional attractiveness to foraging herbivores.

2. Mathematical Formulation

2.1 Constitutive Mechanical Attrition

Mechanical defenses (thorns and spines) inflict physical damage on grazing mouthparts during feeding events. Let \(m_{\text{bite}} \ge 0\) represent the species-specific mechanical damage coefficient per bite, and \(\rho_{\text{morph}} \in [0, 1]\) represent the herbivore's morphological resistance. The integer headcount reduction \(\Delta n\) for a swarm cohort of size \(n(t)\) feeding on plant tissue is given by:

\[\Delta n = \left\lfloor m_{\text{bite}} \cdot (1 - \rho_{\text{morph}}) \right\rfloor\]

\[n(t + \Delta t) = \max\left(0, n(t) - \Delta n\right)\]

Where \(\lfloor \cdot \rfloor\) denotes the floor function, enforcing discrete integer mortality within the herbivore population cohort.

Dual Perspectives: Mechanical Attrition

Biological Perspective: Thorns and trichomes do not alter plant nutrient chemistry; instead, they inflict physical trauma on herbivore mouthparts. This enforces an immediate population penalty per bite, favoring herbivores with specialized morphological mouthpart adaptations (\(\rho_{\text{morph}}\)).
Computer Science & Mathematical Rationale: Modeled as an in-place \(O(1)\) integer subtraction during the feeding pass inside feeding.py. Enforcing \(\lfloor \cdot \rfloor\) prevents floating-point "fractional individuals" from cluttering the ECS entity array, maintaining clean headcount integers and strict zero-allocation JIT execution.

2.2 Caloric Attenuation & Digestibility Discounting

Structural cell-wall barriers (lignin and silica) reduce the fraction of ingested plant biomass that herbivores can digest and metabolize into reproductive energy reserves. Let \(\mu_{\text{digest}} \in [0, 1]\) represent the plant's digestibility modifier and \(\delta_{\text{eff}} \ge 0\) represent the herbivore's digestive adaptation efficiency.

Given total plant energy consumed \(E_{\text{consumed}}\), the net metabolized energy \(E_{\text{metabolized}}\) added to the herbivore energy pool is:

\[\eta_{\text{net}} = \min\left(1.0, \max\left(0.0, \mu_{\text{digest}} \cdot \delta_{\text{eff}}\right)\right)\]

\[E_{\text{metabolized}} = E_{\text{consumed}} \cdot \eta_{\text{net}}\]

Dual Perspectives: Caloric Attenuation

Biological Perspective: High-lignin or silica-heavy foliage reduces the net energetic yield per gram consumed. Herbivores ingest biomass but fail to assimilate calories, slowing their reproductive rate and mitigating population explosions without killing swarms outright.
Computer Science & Mathematical Rationale: Implemented as a scalar multiplication \(\eta_{\text{net}} = \min(1.0, \max(0.0, \mu_{\text{digest}} \cdot \delta_{\text{eff}}))\) during consumption accumulation. Decouples raw biomass consumption from energy transfer in a single arithmetic operation, preserving \(O(1)\) per-entity state updates.

2.3 Rate-Limited Phloem Translocation Kinetics

When a plant activates a resource_withdrawal defense action, it commits to reducing its apparent nutritional factor \(N(t)\) toward a target factor \(N_{\text{target}} \in [0, 1]\) (where lower values represent lower nutritional attraction).

Because carbohydrate transport through vascular phloem sieve tubes is limited by hydrostatic pressure gradients and fluid viscosity, the transition is governed by a first-order rate-limited differential equation with translocation rate \(k_{\text{trans}} \in (0, 1]\):

\[\frac{d N(t)}{dt} = -k_{\text{trans}} \cdot \left(N(t) - N_{\text{target}}\right)\]

In discrete simulation time with tick step \(\Delta t = 1\):

Active Withdrawal Phase (\(\tau_{\text{withdrawal}} > 0\)):

\[N^{t+1} = N^t - k_{\text{trans}} \cdot \left(N^t - N_{\text{target}}\right)\]

Recovery Phase (\(\tau_{\text{withdrawal}} = 0\)):

\[N^{t+1} = N^t + k_{\text{trans}} \cdot \left(1.0 - N^t\right)\]

Dual Perspectives: Rate-Limited Phloem Translocation

Biological Perspective: Real plants cannot instantaneously empty their leaves of sugars. Vascular transport takes time (\(k_{\text{trans}}\)), creating an empirical "window of vulnerability" during which grazing herbivores continue feeding while translocation is underway.
Computer Science & Mathematical Rationale: Approximated via an exponential relaxation recurrence step \(N^{t+1} = N^t + k (N_{\text{target}} - N^t)\) evaluated in lifecycle.py. Prevents discontinuous step-function jumps in the flow-field potential matrix, ensuring smooth spatial gradient transitions for herbivore navigation.

2.4 Attractant Field Gradient Scaling

The spatial attractiveness of a plant cell \((x, y)\) to foraging herbivores in the Numba-accelerated flow-field solver is scaled directly by the plant's current apparent nutrition factor \(N(x, y)\):

\[F(x, y) = \alpha \cdot \left(E_{\text{plant}}(x, y) \cdot N(x, y)\right) - \beta \cdot \sum_{k} T_k(x, y)\]

Where \(E_{\text{plant}}(x, y)\) is the total plant energy, \(T_k(x, y)\) represents localized repellent toxin concentrations, and \(\alpha, \beta\) are flow-field weighting coefficients. Down-regulating \(N(x, y)\) flattens the local attractant gradient, causing spatial flow-field navigation to steer approaching herbivore swarms away from the defended plant patch.

Dual Perspectives: Attractant Field Scaling

Biological Perspective: By translocating nutrients to root sinks, plants mask their nutritional value from sensory-seeking herbivores (trophic camouflage), forcing grazers to disperse toward other patches.
Computer Science & Mathematical Rationale: Multiplied directly into the 2D energy tensor \(E(x,y) \cdot N(x,y)\) prior to JIT flow-field generation. Modifies the global potential surface without requiring separate pathfinding re-computations or dynamic graph re-routing.

Appendix: Engine Implementation & Schema Mappings

For software engineers and data interface developers, this section maps the mathematical formulation above to concrete data schemas and system modules in the PHIDS codebase.

Schema Definitions (`src/phids/api/schemas/species.py` & `triggers.py`)

```python
class PassiveDefensesSchema(StrictBaseModel):
    """Constitutive morphological defenses of a flora species."""

    mechanical_damage_per_bite: float = Field(default=0.0, ge=0.0)
    digestibility_modifier: float = Field(default=1.0, ge=0.0, le=1.0)


class ResourceWithdrawalAction(StrictBaseModel):
    """Action configuring rate-limited phloem nutrient translocation."""

    type: Literal["resource_withdrawal"] = "resource_withdrawal"
    apparent_nutrition_factor: float = Field(default=0.1, ge=0.0, le=1.0)
    withdrawal_duration: int = Field(default=5, ge=1)
```

Component State (`src/phids/engine/components/plant.py`)

```python
@dataclass(slots=True)
class PlantComponent:
    apparent_nutrition_factor: float = 1.0
    target_nutrition_factor: float = 1.0
    translocation_rate: float = 0.2
    withdrawal_ticks_remaining: int = 0
```

System Execution Pipeline (`src/phids/engine/systems/lifecycle.py`)

```python
# Phloem translocation update during lifecycle tick
if plant.withdrawal_ticks_remaining > 0:
    plant.withdrawal_ticks_remaining -= 1
    plant.apparent_nutrition_factor += (
        plant.target_nutrition_factor - plant.apparent_nutrition_factor
    ) * plant.translocation_rate
else:
    plant.apparent_nutrition_factor += (
        1.0 - plant.apparent_nutrition_factor
    ) * plant.translocation_rate
```