Relative Allocation

Published

June 6, 2025

The RelativeAllocation module implements a proportional allocation strategy to distribute available biomass or nutrient supply among plant organs. This strategy is a general-purpose mechanism that underpins how APSIM handles partitioning of dry matter (DM) and nutrients (e.g. nitrogen) based on each organ’s demand.

This method is part of the broader plant allocation system and is invoked by arbitrator components such as BiomassTypeArbitrator.

Biological Motivation

Plants must allocate available assimilates or nutrients among organs in a coordinated way. The RelativeAllocation method mimics this by:

  • Prioritizing structural and metabolic demands (essential growth),
  • Then allocating any residual supply to storage pools (non-essential, luxury uptake).

This reflects physiological prioritization in plant growth.

Allocation Logic

The allocation occurs in two sequential passes:

  • First Pass – Structural and Metabolic Allocation
  • Second Pass – Storage (Luxury) Allocation

First Pass – Structural and Metabolic Allocation

Objective

Distribute available supply to fulfill structural and metabolic demands proportionally.

Definitions

Let:

  • \(S_{\text{total}}\): Total supply available (e.g. Dry biomass or nitrogen).
  • \(D_s^{(i)}\): Structural demand of organ \(i\)
  • \(D_m^{(i)}\): Metabolic demand of organ \(i\)
  • \(D_{s,\text{total}} = \sum_i D_s^{(i)}\): Total structural demand across all organs.
  • \(D_{m,\text{total}} = \sum_i D_m^{(i)}\): Total metabolic demand across all organs.

We define the structural fraction of supply as:

\[ f_s = \frac{D_{s,\text{total}}}{D_{s,\text{total}} + D_{m,\text{total}}} \]

Then, for each organ \(i\):

  • Structural allocation:

\[ A_s^{(i)} = \min\left(R_s^{(i)}, S_{\text{total}} \cdot f_s \cdot \frac{D_s^{(i)}}{D_{s,\text{total}}} \right) \]

  • Metabolic allocation:

\[ A_m^{(i)} = \min\left(R_m^{(i)}, S_{\text{total}} \cdot (1 - f_s) \cdot \frac{D_m^{(i)}}{D_{m,\text{total}}} \right) \]

where \(R_s^{(i)}\) and \(R_m^{(i)}\) are the remaining unmet demand for structural and metabolic pools, respectively.

Second Pass – Storage (Luxury) Allocation

Objective

Allocate any remaining supply to storage pools (non-essential or luxury growth) based on proportional demand.

Definitions

Let:

  • \(S_{\text{rem}}\): Remaining supply after first pass
  • \(D_t^{(i)}\): Storage demand of organ \(i\)
  • \(D_{t,\text{total}} = \sum_i D_t^{(i)}\): Total storage demand across all organs

Then, for each organ \(i\):

\[ A_t^{(i)} = \min\left(R_t^{(i)}, S_{\text{rem}} \cdot \frac{D_t^{(i)}}{D_{t,\text{total}}} \right) \]

where \(R_t^{(i)}\) is the remaining unmet storage demand for organ \(i\).

Final Accounting

After both passes, the module tracks:

  • Total allocation: Sum of structural, metabolic, and storage allocations across all organs.
  • Not allocated: Any supply that remains unallocated due to demand being fully satisfied or due to zero total demand (e.g., at early stages).

\[ S_{\text{not\_allocated}} = S_{\text{total}} - \sum_i \left( A_s^{(i)} + A_m^{(i)} + A_t^{(i)} \right) \]

This is useful for diagnostics, such as reporting sink limitations.

Key Features

  • Prioritizes essential growth (structural and metabolic) over luxury (storage).
  • Proportional to each organ’s share of total demand.
  • Applies generically to both biomass and nutrients.
  • Uses a two-pass method to avoid allocating luxury growth when essential needs are unmet.

Practical Example

Assume:

  • Leaf has a structural demand of 50 units and a metabolic demand of 20 units.
  • Stem has a structural demand of 60 units and a metabolic demand of 60 units.
  • Total supply is 120 units.

Then:

  • Total structural demand = 50 (leaf) + 60 (stem) = 110 units

  • Total metabolic demand = 20 (leaf) + 60 (stem) = 80 units

  • Structural fraction:
    \(f_s = \frac{110}{110 + 80} = 0.579\)

  • First pass allocates:

    • Structural: \(120 \times 0.579 = 69.48\) units, distributed as:
      • Leaf: \(69.48 \times \frac{50}{110} = 31.58\) units
      • Stem: \(69.48 \times \frac{60}{110} = 37.90\) units
    • Metabolic: \(120 \times 0.421 = 50.52\) units, distributed as:
      • Leaf: \(50.52 \times \frac{20}{80} = 12.63\) units
      • Stem: \(50.52 \times \frac{60}{80} = 37.89\) units

  • Second pass allocates any leftover proportionally to storage demands.

Applications

  • Biomass partitioning after photosynthesis, reallocation, and retranslocation.
  • Nitrogen allocation after uptake, reallocation, and retranslocation.
  • Any generic resource arbitration among competing sinks.

References