POPULATION DYNAMICS¶

This section provides a comprehensive specification of how population measures evolve through the stage structure. The transition functions (gₐᵥ, gᵥₑ, gₑₐ₊, gᵥₐ₊) defined in the Recursive Problem Definition induce a corresponding evolution of population measures at each step.

Within-Stage Perch Transitions¶

1. Arrival → Decision¶

Individual Level: - Transition: (xₐ, ζₐᵥ) ↦ xᵥ = gₐᵥ(xₐ, ζₐᵥ) - Properties: For each fixed shock ζₐᵥ, the map xₐ ↦ xᵥ is bijective, preserving dimensionality - When 𝒵ₐᵥ is trivial: xₐ ↦ xᵥ = gₐᵥ(xₐ) (deterministic)

Population Level: - Measure evolution: μᵥ(B) = E_μₐ[P({ζₐᵥ: gₐᵥ(xₐ,ζₐᵥ) ∈ B})] - When 𝒵ₐᵥ is trivial: μᵥ(B) = μₐ({xₐ: gₐᵥ(xₐ) ∈ B})

2. Decision → Continuation¶

Individual Level: - Transition: (xᵥ, 𝜋, ζᵥₑ) ↦ xₑ = gᵥₑ(xᵥ, 𝜋, ζᵥₑ) - Properties: For each fixed choice 𝜋 and shock ζᵥₑ, the map xᵥ ↦ xₑ is bijective, preserving dimensionality - When 𝒵ᵥₑ is trivial: (xᵥ, 𝜋) ↦ xₑ = gᵥₑ(xᵥ, 𝜋) (deterministic given policy)

Population Level: - With post-decision shocks: μₑ(C) = E_μᵥ[P({ζᵥₑ: gᵥₑ(xᵥ,𝜋(xᵥ),ζᵥₑ) ∈ C})] - When 𝒵ᵥₑ is trivial: μₑ(C) = E_μᵥ[1{gᵥₑ(xᵥ,𝜋(xᵥ)) ∈ C}]

Between-Stage Transitions¶

1. Sequential Stages¶

Individual Level: - Within period [i] → [i+1]: xₑ ↦ xₐ₊ = gₑₐ₊(xₑ) - Between periods [t,-1] → [t+1,0]: xₑ ↦ xₐ₊ = gₑₐ₊(xₑ) - Properties: Bijective, unique next stage

Population Level: - Measure evolution: μₐ₊(A) = E_μₑ[1{gₑₐ₊(xₑ) ∈ A}]

2. Branching Stages¶

Individual Level: - First: Draw j ∈ N₊ with probability p(j|xᵥ,𝜋) - Then: (xᵥ, 𝜋) ↦ xₐ₊ = gᵥₐ₊(xᵥ, 𝜋) - Properties: Bijective for each branch j ∈ N₊ - Same mechanism for within-period and between-period transitions

Population Level: - Branch-specific: μₐ₊ⱼ(A) = E_μᵥ[p(j|xᵥ,𝜋(xᵥ))1{gᵥₐ₊(xᵥ,𝜋(xᵥ)) ∈ A}] - Total: μₐ₊(A) = ∑_{j∈N₊} μₐ₊ⱼ(A)

3. Terminal Period¶

No transitions from final stage 𝕊₀[-1]
Terminal value ℰ₀(xₑ) specified directly

Conservation Properties¶

Mass Conservation¶

Within stages: μᵥ(𝓧ᵥ) = μₐ(𝓧ₐ) and μₑ(𝓧ₑ) = μᵥ(𝓧ᵥ)
Sequential transitions: μₐ₊(𝓧ₐ₊) = μₑ(𝓧ₑ)
Branching transitions: ∑_{j∈N₊} μₐ₊ⱼ(𝓧ₐ) = μᵥ(𝓧ᵥ)

Normalization¶

If input measure is normalized (E_μₐ[1] = 1), then:
All within-stage measures remain normalized
Sequential transition preserves normalization
Sum of branch-specific measures equals 1

Measure Properties¶

Non-negativity¶

All measures are non-negative: μ(A) ≥ 0 for all measurable A
Branch-specific measures inherit non-negativity

Additivity¶

Measures are countably additive
Total measure in branching case is sum over branches

Regularity¶

All measures are defined on Borel σ-algebras
Transition functions preserve measurability

Reconvergent Branch Combination¶

When multiple branches j ∈ 𝒥 produce measures μₐ₊ⱼ that arrive at a shared downstream perch (reconvergence), the combined measure is:

μₐ₊(A) = ∑_{j ∈ 𝒥} μₐ₊ⱼ(A)

Conservation: The total mass is preserved through the branching-reconvergence cycle:

μₐ₊(𝓧ₐ₊) = ∑_j μₐ₊ⱼ(𝓧ₐ₊) = μᵥ(𝓧ᵥ)

Multi-source perch: A perch with multiple incoming transitions (from different branches) is a multi-source perch. The combine operation is the summation of branch-specific measures. This is the symmetric counterpart to the branching split defined in "Between-Stage Transitions § Branching Stages" above.

Computational note: The combine operation is method-independent in its mathematical specification (summation of measures) but method-dependent in implementation: - Monte Carlo: concatenate agent lists from all branches; each agent carries its branch-specific weight (importance sampling) or unit weight (equal-weight particles) - Transition matrix: sum sub-distribution vectors; interpolate to a common grid if branches use different grids - Function approximation: mixture distribution with branch-mass weights: f_{a+}(x) = ∑j w_j f is the normalized density}(x) where w_j = μₐ₊ⱼ(𝓧) is the mass in branch j and f_{a+,j