MVT Explorer — Marginal Value Theorem

Optimal Patch Departure — Single Patch Type

g′(h^*) = R^* = g(h^*) s + h^*

Leave when instantaneous gain rate equals the long-term habitat rate · h = Time in Patch

R^* acts as an opportunity cost of time: each additional moment in the patch must justify itself against the habitat's long-run return rate. This is structurally identical to a temporal discount rate in sequential decision-making — foragers implicitly discount future patch opportunities by R^*.

Marginal Value Theorem: Maximize R^* = g(h^*) / (s + h^*) by equating marginal gain to opportunity cost.

↔ drag ● on graph · adjust s
↕ drag curve · adjust G_max · ↔ drag curve · adjust τ

Inter-patch Search Time

s = 5.0

Average time between patch encounters

Statistics

Opt. Residence h^*

—

Habitat Rate R^*

—

Gain g(h^*)

—

Slope g′(h^*)

—

Patch Quality G_max

12.0

Rise Time τ

6.0

Multi-Patch Equimarginal Condition

g′₁(h₁^*) = g′₂(h₂^*) = ⋯ = g′_m(h_m^*) = R^*

Optimal times h_i^* equalize marginal returns across all patch types · h = Time in Patch

The equimarginal R^* is the shared opportunity cost of time across all patch types. No patch can offer a better marginal return at the optimum — analogous to a common discount rate at which all options become equally attractive at the margin, mirroring value-function convergence in reinforcement learning.

Equimarginal Principle: Richer patch → longer h_i^*. Sparser habitat (↑ s) → all h_i^* increase, R^* decreases. Rarer patch type (↓ p_i) → R^* shifts toward other patches.

↕ drag any curve · adjust G_max
↔ drag any curve · adjust rise time τ

Inter-patch Search Time

s = 5.0

Common search time across all patches

Portfolio Statistics

Habitat Rate R^* (equimarginal)

—

Residence h₁^*

—

g₁(h₁^*)= —

Residence h₂^*

—

g₂(h₂^*)= —

Residence h₃^*

—

g₃(h₃^*)= —

Spread h₁^*−h₃^*

—

Encounter Proportions

p₁=0.33 p₂=0.33 p₃=0.33

drag to adjust relative encounter rates

Patch Types

Richest G_max=14.5 τ=8.0

OK G_max=11.0 τ=5.0

Poor G_max=8.5 τ=3.0

Tangent slope = R^* (all equal)

τ = depletion time constant · small τ → step-like (fast reward); large τ → ramp-like (slow depletion)