SP-Hamiltonian in Foraging Models

• SP-Hamiltonian is an energy function that encodes forager movement dynamics by quantifying resource intake, depletion, and recovery.
• It models a deterministic foraging process where a walker depletes resource sites and forms cyclic home ranges based on parameters like bite size and recovery time.
• The framework bridges microscopic movement decisions with macroscopic spatial patterns, offering insights into optimization strategies and attractor landscapes in foraging systems.

The SP-Hamiltonian, in the context of foraging models, refers to the effective energy function that governs the movement and resource-consumption dynamics of a forager interacting with a renewable, spatially disordered substrate. The relevant paradigmatic formulation is introduced in "The movement of a forager: Strategies for the efficient use of resources" (Kazimierski et al., 2016), where a deterministic, locally-constrained walker depletes fixed bites from patches (plants) and the substrate recovers according to a relaxation law. The mathematical structure underlying the resource-mediated movement dynamics, the recurrent home-range formation, and the emergent attractor landscape is naturally encoded by the SP-Hamiltonian.

1. System Definition and Resource Dynamics

The modeled ecosystem consists of $N$ discrete resource sites (plants, patches) at randomly chosen positions in a bounded domain (e.g., the unit square). Each site $i$ carries a time-dependent crop value $f_i(t)$, initialized within $(0,f_i(0))$. The key ingredients of the model are:

• Resource depletion: When a walker visits site $i$ at time $t_v$, it reduces $f_i$ by a fixed bite size $b$ (Eq. 1):

$$f_i(t_v^+) = f_i(t_v^-) - b, \quad f_i(t_v^-) \geq b.$$

• Resource recovery: Each removed bite $b$ is replenished exactly $\tau$ time steps after removal, unless the site is already saturated at $f_i(0)$ (Eq. 2):

$$f_i(t) = \min\left\{f_i(t-1) + b \sum_{t_v}\delta_{t, t_v + \tau}, f_i(0)\right\}.$$

The substrate's state thus evolves as a piecewise-deterministic Markov process, affected by the forager's visitation history.

2. Forager Dynamics and Deterministic Movement Rule

At each discrete time step $t$, the forager resides at some site $i_t$. It selects its next move as follows (Eq. 3):

• Local availability constraint: Only considers candidate sites $j \neq i_t$ such that $f_j(t) \geq b$.
• Nearest-neighbor search: Among available candidates, chooses $j^* = \arg\min_{j: f_j(t) \geq b} d_{i_t, j}$.

This rule produces a deterministic, zero-temperature movement regime: the forager always moves to the nearest still-sufficiently-resourced neighbor, never exhausting a site's resource below the bite threshold, and otherwise ignores all other potential sites (Kazimierski et al., 2016).

3. Energetics and the SP-Hamiltonian Framework

The energetic state of the forager is updated as a balance of intake and locomotor cost (Eq. 4):

$E(t) = E(t-1) + g(b) - h(d_{i_{t-1}, i_t}),$

with

• $g(b) = \beta(b) b$, where $\beta(b) \leq 1$ (accounts for possible bite size penalties such as processing time or predation risk).
• $h(d) = \alpha d$, with $\alpha > 0$ a per-unit-distance cost.

Over a trajectory of $T$ steps, the time-averaged energetic efficiency is (Eq. 6):

$$\bar{E} = \frac{1}{T} \sum_{t=1}^T \left[\beta(b) b - \alpha d_{i_{t-1}, i_t}\right].$$

This functional, which serves as an SP-Hamiltonian in the sense of encoding the energy dynamics over state space, forms the basis for analyzing optimal foraging strategies and phase behavior of the walker-substrate system.

4. Home Range, Cyclic Attractors, and Dynamical Landscape

The feedback between the forager's movement, substrate depletion, and recovery leads to nontrivial periodic trajectories after a transient phase:

• Home range: The walker falls into a cyclic sequence of patch visits, identified as the emergent home range.
• Cycle period ($T$): Grows linearly with resource recovery time $\tau$ and decreases with bite size $b$.
• Fraction of space used ($S$): Increases with both $\tau$ and $b$.

Transient duration and the onset of the recurrent cycle also depend strongly on $b$; larger bites lead to shorter transients (Kazimierski et al., 2016).

Crucially, the number of distinct cyclic attractors (i.e., unique home-range cycles accessible from different initializations) changes non-monotonically with model parameters:

• For fixed $N$, number of attractors decreases with $b$ and $\tau$.
• Small $b, \tau$ lead to a rugged, multi-attractor landscape; large $b, \tau$ produce a coarse landscape, often with a unique attractor.

Thus, the SP-Hamiltonian landscape determines not only local movement, but also the global structure of possible steady-state behaviors and their partitioning of resource space.

5. Efficiency Optimization, Morphological Constraints, and Biological Implications

The energy-balance phase diagrams $\bar{E}(b,\alpha)$ reveal:

• If $\beta \equiv 1$ (no penalty), the optimal strategy is always "greedy": larger $b$ leads to greater efficiency.
• With realistic penalization factors ($\beta(b)$ decreasing in $b$), an interior maximum emerges: moderate $b$ achieves the best balance between net intake and cost (as shown in $\bar{E}(b,\alpha)$ contours).

Biologically, this reflects the trade-off between maximizing instantaneous gain and reducing processing costs or risk: morphological and physiological constraints affecting $b$ therefore co-evolve with environmental renewal rates ($\tau$), with direct impact on forager fitness (Kazimierski et al., 2016).

6. Model Extensions, Multi-Forager Use, and Attractor Structure

Although the core SP-Hamiltonian model is single-forager, the attractor landscape analysis generalizes: if multiple foragers act with independent or weakly coupled trajectories on the same substrate, the structure of home-range tilings and their degree of overlap are set by the same underlying Hamiltonian dynamics—modulated by the dependence of attractor count on $b$ and $\tau$. This has direct consequences for resource partitioning, contest competition, and coexistence strategies in animal populations (Kazimierski et al., 2016).

7. Synthesis and Relevance to Foraging Theory

The SP-Hamiltonian formalism, as instantiated in the energy-balanced walker-substrate model, provides:

• A mechanistic link between optimal foraging theory and the spontaneous emergence of home-range patterns from local rules.
• A parameter-dependent landscape that predicts the empirically observed phenomena of periodic movement cycles, bounded spatial use, and phase transitions in efficiency and spatial organization.
• An explicit bridge between microscopic movement decisions and macroscopic behavioral patterning in foraging systems.

This framework thereby offers an analytically tractable, ecologically interpretable foundation for understanding resource use, movement ecology, and evolutionary constraints on forager morphology in heterogeneous, renewable landscapes (Kazimierski et al., 2016).