Chemotaxis & Flow Fields

Herbivore swarms navigate the PHIDS biotope via a unified scalar guidance field, simulating a sensory-driven process called chemotaxis.

Biological and Physical Context

Chemotaxis is the phenomenon whereby somatic cells, bacteria, and other single-cell or multicellular organisms direct their movements according to certain chemicals in their environment.

In nature, organisms do not possess a top-down, global map of the world. They cannot calculate the most efficient Euclidean path to a food source three miles away while simultaneously avoiding a predator blocking a narrow mountain pass. Instead, they sense local chemical gradients—moving towards higher concentrations of attractants (food, mating pheromones) and away from repellents (toxins, predators).

By modeling chemotaxis, PHIDS ensures swarm navigation is inherently local, imperfect, and biologically plausible. Furthermore, real-world chemical plumes are subjected to wind, thermal eddies, and turbulent diffusion, making scent trails noisy and chaotic rather than perfectly smooth geometric cones.

The Unified Flow Field

Rather than solving a separate navigation problem for every individual herbivore swarm (e.g., \(N\) swarms calculating paths across an \(M \times M\) grid), PHIDS constructs a single Flow Field \(F_t(x, y)\) at the beginning of each tick \(t\).

This scalar lattice is a spatial superposition of two primary potentials:

1. Attractants (\(E_t\)): The aggregate caloric energy of all flora species.
2. Repellents (\(T_t\)): The aggregate concentration of localized defensive toxins emitted by flora.

Mathematical Formulation

The baseline gradient at cell \((x, y)\) before propagation is computed as:

Where:

• \(E_t(x,y)\) is the total plant energy available at the coordinate.
• \(T_{k,t}(x,y)\) is the concentration of the \(k\)-th toxin channel.
• \(\alpha, \beta\) are non-negative weighting constants.

To create an "influence map" that swarms can detect from a short distance away, this baseline gradient undergoes a rapid, one-pass local propagation (spreading) with a steep decay coefficient \(\delta\) (e.g., \(0.5\)).

The Gradient Ascent (Stochastic Taxis)

A swarm located at \((x, y)\) determines its next position by evaluating the Flow Field \(F_t\) in its immediate Von-Neumann Neighborhood \(\mathcal{V}(x,y)\) (the current cell plus its 4 orthogonal adjacent cells: North, South, East, and West).

Rather than deterministically selecting the absolute highest gradient (strict gradient ascent), the engine applies probability-weighted sampling. The probability \(P(u,v)\) of transitioning to a neighbor \((u,v) \in \mathcal{V}(x,y)\) is strictly proportional to its normalized flow-field magnitude relative to the neighborhood minimum.

This stochastic approach mathematically models biological sensory noise, receptor saturation, and the physical turbulence of volatile organic compounds in a real ecosystem.

Numerical Example

Imagine a swarm at center coordinate (1, 1) evaluating its Von-Neumann neighborhood in a simplified subset of the Flow Field \(F_t\):

Flow Field Segment (Orthogonal Only)

• North (1, 0): 1.2
• South (1, 2): 0.4
• East (2, 1): 1.5
• West (0, 1): 0.1

• Center (1, 1): 0.8

• The swarm extracts the 5 relevant scalars.

• It shifts the values relative to the local minimum (0.1) and applies a small epsilon (1e-6) to ensure non-zero weights for all valid moves.
• The adjusted weights formulate a discrete probability distribution. The East cell (1.5) holds the highest statistical probability of being selected, but the North cell (1.2) and Center cell (0.8) remain highly viable options.
• A random sample is drawn from the distribution, determining the transition.

If the center cell (1,1) had an overwhelmingly dominant value (e.g., the swarm is currently situated on a high-energy plant), the probability distribution collapses heavily onto the center cell, an act defined as Anchoring.

Alternatives Considered & Architectural Decisions

During the engineering of the PHIDS interaction system, several navigation paradigms were evaluated and explicitly rejected to maintain ecological fidelity and simulation invariants:

1. Deterministic Convergence (arg max) vs. Probabilistic Taxis

• The Rejected Model: Using a strict \(\operatorname*{arg\,max}\) mathematical function forces the swarm to perfectly select the steepest gradient every single tick.
• Why we rejected it: In a discrete grid, absolute determinism forces all swarms on a gradient slope to merge into the exact same optimal trajectory, forming unnatural, single-file "conga lines." By embracing a Probabilistic Taxis model, the engine natively introduces lateral dispersion. Swarms fan out organically as they approach a target, faithfully recreating the chaotic search patterns and klinokinesis observed in real biological foragers.

2. Moore Neighborhood vs. Von-Neumann Neighborhood

• The Rejected Model: The Moore Neighborhood (evaluating 8 directions including diagonals) allows for smoother visual pathing without jagged edges.
• Why we rejected it: Switching to Moore on a discrete grid introduces the Euclidean Distance Exploit. A diagonal step covers a physical distance of \(\sqrt{2} \approx 1.414\) units. If not explicitly mathematically penalized in the hot-path kernel, swarms traveling diagonally outpace orthogonal swarms by 41%, corrupting the fundamental velocity constants of the ecosystem. The Von-Neumann Neighborhood strictly preserves a 1:1 ratio between ticks and physical distance traversed, allowing the Numba @njit kernels to remain incredibly lean while guaranteeing absolute kinematic consistency.
• Future Alternative Consideration (The Sensory Penalty): Should the visual "blockiness" of orthogonal movement ever become undesirable, an 8-way Moore search could be safely implemented by applying a mathematical penalty to the diagonal cells during probability weighting. By dividing the flow gradient score of the four diagonal cells by \(\approx 1.414\) before inserting them into the probability distribution, the engine would naturally suppress the appeal of diagonal routes, effectively balancing the Euclidean speed exploit with a proportional sensory handicap.

3. A* (A-Star) or Dijkstra Pathfinding

• The Rejected Model: Calculating optimal, obstacle-avoidant paths from every swarm to the nearest food source.
• Why we rejected it: Classic pathfinding scales poorly. Calculating paths for hundreds of swarms across a dynamic grid per tick would create a computational bottleneck of \(O(N \cdot M^2)\). Furthermore, swarms lack "global knowledge" of the map. Our \(O(1)\) unified Flow Field sampling perfectly mimics biological sensory constraints while maintaining extreme computational efficiency.

Zero-Gradient Navigation (The Isotropic Search)

A critical edge case in spatial ecology occurs when an organism is entirely outside the sensory horizon of any resource or predator. Mathematically, this happens when the entire neighborhood evaluates to a flat zero-gradient:

If a swarm relied strictly on gradient ascent, a zero-gradient would result in indefinite paralysis. In biological systems, when an organism loses a scent trail, it transitions from directed movement (taxis) to undirected, exploratory movement (kinesis).

Algorithmic Resolution

When PHIDS evaluates a zero-gradient (flat) neighborhood, the swarm enters a Random Walk state. If the swarm has existing momentum (inertia), it heavily weights its probability distribution toward its previous heading (last_dx, last_dy), simulating orthokinesis. If no inertia exists, it selects an orthogonal neighbor from a uniform random distribution, effectively performing an isotropic search until it re-enters an active Flow Field.

Impact of Resource Reallocation on Chemotaxis

When a plant triggers a resource_withdrawal action, its apparent_nutrition_factor scalar drops below 1.0. Inside the Numba JIT Chemotaxis Flow Field resolution loop (flow_field.py), the base attractant landscape matrix is scaled before diffusion: