The model

This page describes the model that pyLEAFS v1 implements: a replenishing resource field and a single greedy forager living on it. It is the base of the layered architecture; sections marked (later layer) name the extensions that attach to it and are not yet implemented. The notation follows the papers listed on the pyLEAFS.

The resource field

Resources are energy quanta scattered through a \(D\)-dimensional toroidal box and treated as a Poisson point process. Within each region of the grid, one timestep of duration \(dt\) births and kills resources stochastically:

\[\begin{split}k_\mathrm{grow} &\sim \mathrm{Poisson}\!\left(\frac{\Gamma\,L^{D}\,dt}{\epsilon}\right) \\ k_\mathrm{decay} &\sim \mathrm{Binomial}\!\left(N,\; 1 - e^{-\gamma\,dt}\right)\end{split}\]

where \(\Gamma\) is the energy influx per unit volume, \(\gamma\) the resource decay rate, \(\epsilon\) the energy carried by one resource, and \(L\) the region side length. Births place a resource uniformly in the region; deaths remove uniformly chosen ones. Birth and death counts for every region are drawn in two vectorized calls per step, and the grow/decay order is randomized per region to avoid bias — the ResourceField is the struct-of-arrays descendant of the MATLAB Region/Environment pair.

In the absence of agents the field relaxes to an equilibrium count per region

\[N_\mathrm{eq} = \frac{\Gamma\,L^{D}}{\epsilon\,\gamma},\]

which ResourceField exposes as N_eq.

Homogeneity

Rather than set \(\epsilon\) directly, the model is parameterized by a single dimensionless homogeneity \(\Xi\), the ratio of the agent length scale \(\ell_A\) to the mean resource spacing \(\ell_E\):

\[\Xi = \frac{\ell_A}{\ell_E}, \qquad \ell_E = \left(\frac{\epsilon}{\Gamma\,\tau_E}\right)^{1/D}, \qquad \tau_E = 1/\gamma .\]

With the agent length scale set by the collection radius (\(\ell_A = r_\mathrm{col}\) in the 2d applet), inverting for the energy per resource gives the relation the forager() factory uses:

\[\epsilon = \frac{\Gamma}{\,(\Xi / r_\mathrm{col})^{D}\,\gamma\,}.\]

Small \(\Xi\) is a heterogeneous world — few, energy-rich resources, widely spaced. Large \(\Xi\) is a homogeneous world — many, lean resources, densely packed. The homogeneity is the central control variable of the whole research program: the semantic-information threshold, the foraging phenotypes, and the punctuated evolutionary dynamics are all studied as functions of \(\Xi\).

The agent

An agent is a body with a position \(\mathbf{x}\), a unit heading \(\hat{\mathbf{n}}\), and a fuel reserve \(s\). Each timestep it senses, steers, moves, harvests, and metabolizes.

Sensing and steering. The v1 controller is greedy: the agent looks for the nearest resource within a sensing radius \(R_\mathrm{sense}\) and turns to face it. If nothing is in range, its heading diffuses,

\[\hat{\mathbf{n}} \leftarrow R\!\left(\frac{\sigma}{\sqrt{\tau_A}}\sqrt{dt}\;\xi\right)\hat{\mathbf{n}}, \qquad \xi \sim \mathcal{N}(0, 1),\]

a rotation by Gaussian angular noise of strength \(\sigma/\sqrt{\tau_A}\) (realized in 2d as a heading-angle perturbation, in 3d as a renormalized tangential kick). This greedy rule is the perfect-fidelity limit of the sensor model; replacing it with a learned controller is a (later layer).

Motion. The agent advances ballistically and wraps on the torus:

\[\mathbf{x} \leftarrow \big(\mathbf{x} + v\,\hat{\mathbf{n}}\,dt\big) \bmod \mathbf{extent},\]

with \(v\) the translational speed.

Harvesting and metabolism. Every resource within the collection radius \(r_\mathrm{col}\) is consumed, each returning energy \(\epsilon\). Fuel follows

\[\frac{ds}{dt} = -\mu_0 + h,\]

where \(\mu_0\) is the basal metabolic rate and \(h\) the harvest rate, capped at \(s \le s_\mathrm{max}\). In the present version metabolism is a single basal term; the (later layer) in which sensors and actions modulate \(\mu_0\) — sensors as behaviors that cost energy — is the subject of the energy-harvesting notes.

Reproduction and death. When fuel reaches the threshold \(s_\mathrm{rep} = 0.8\,s_\mathrm{max}\) the agent buds: it spawns a daughter a short distance away and the two split the fuel equally. An agent dies when \(s \le 0\). In v1 the daughter is an exact clone; mutation of a genome is a (later layer).

Population dynamics

With agents drawing down the field, the resource and agent counts trace a predator–prey-like relation. Two carrying capacities frame it:

\[N_R^{\max} = \frac{\Gamma\,\tau_E\,V}{\epsilon}, \qquad N_A^{\max} = \frac{\Gamma\,V}{\mu_0},\]

with \(V\) the domain volume. \(N_R^{\max}\) is the no-agent resource equilibrium and varies with \(\Xi\); \(N_A^{\max}\) is the agent carrying capacity and is independent of it. At equilibrium the two normalized counts lie on a line,

\[\frac{N_R}{N_R^{\max}} + \frac{N_A}{N_A^{\max}} = 1,\]

the diagonal of the phase plane. Starting from the empty-agent corner, a seeded population overshoots, grazes the field down, and settles into a sustained oscillation about this line rather than going extinct — the behavior the v1 test suite pins. Plotting the trajectory in the \((N_R/N_R^{\max},\,N_A/N_A^{\max})\) plane, and detecting the transition onto the equilibrium line, is the first of the (later) observables.

Semantic information

The reason the model is built around energy and sensing is to measure the information an agent needs to stay alive. Following the Kolchinsky–Wolpert framework, the semantic information is the share of the agent–environment mutual information that is actually necessary for viability (the expected lifetime). Scrambling the sensor past a critical noise level \(\eta_c\) collapses viability; below it, noise barely matters. The threshold depends only on the agent’s geometry,

\[\sin(\pi\,\eta_c) = \frac{r}{R},\]

with \(r\) the collection radius and \(R\) the sensing radius. Computing the mutual information, transfer entropy, and the viability curve from logged trajectories is a planned analysis (later layer); v1 provides the dynamics they are measured on.

Roadmap of layers

The pieces above marked (later layer) attach to the v1 core through the extension seams described in The core and its seams:

a pheromone field (reaction–diffusion) as a second entry in the field list;
predators and trophic levels as additional populations in the step loop;
a heterogeneous environment with spatially varying \(\Gamma\);
neuroevolution: an evolvable recurrent controller and sensor array replacing the greedy rule, with mutation and selection;
a chemoton / internal metabolism replacing the single basal term;
information-theoretic observables: semantic information, mutual information, transfer entropy.