Cohen and King allocation (cohen_king_roughgarden)

Question. For an annual plant (one vegetative compartment, one reproductive compartment), how should allocation switch through a fixed season?

Here plant means that annual-plant model.

Formulation

Symbol	Role	Meaning
\(x_1(t)\)	state	vegetative biomass
\(x_2(t)\)	state	reproductive biomass
\(u(t)\in[0,1]\)	control	fraction of production kept vegetative

Dynamics (relative growth rate 1 in these units).

\[ \dot x_1 = u\,x_1, \qquad \dot x_2 = (1-u)\,x_1. \]

Objectives (same \(f\), two hypotheses).

\[ \text{Cohen:}\quad \max x_2(T); \qquad \text{King:}\quad \max\int_0^T \ln x_2(t)\,\mathrm{d}t. \]

Constraints. \(x_1(0)=X_{10}>0\), \(x_2(0)=0\), \(0\le u\le 1\). Horizon. Fixed season \(T\).

What was done

One collocation grid; \(u(t)\) free in \([0,1]\). Cohen uses a Mayer term \(t_f(x_2)\); King uses a running log payoff.

What we found

Cohen is bang–bang: \(u^*=1\) on \([0,T-1]\), then \(u^*=0\). For defaults \(T=5\), \(X_{10}=1\), switch at \(4\) and \(x_2(T)=e^{4}\). King is bang–singular–bang: a mixed-growth arc with \(x_1=(T-1-t)x_2\). The OCP recovers both closed forms.

Interpretation

The same plant ODE yields different schedules when \(J\) changes: a single terminal harvest (Cohen) versus a running reproductive payoff (King). The switch is not inserted in \(f\); it is implied by \(J\) and \(0\le u\le 1\).

Public API

Cohen terminal yield (examples/cohen_king_roughgarden/cohen_terminal_yield.py):

import eco_opt
from eco_opt import Namespace


def plant_model(t, x, u):
    veg, repro, alloc = x[0], x[1], u[0]
    y = Namespace(veg=veg, repro=repro, alloc=alloc)
    return [alloc * veg, (1.0 - alloc) * veg], y


sys = eco_opt.System(states=2, controls=1, model=plant_model)
ocp = eco_opt.Ocp()
ocp.max(eco_opt.t_f(sys.y.repro))
ocp.st(system=sys, constraints=[...])

Run and tests

cd examples/cohen_king_roughgarden
python cohen_terminal_yield.py
python king_singular_allocation.py
python plot_cohen_king.py
python -m pytest ../../tests/test_cohen_king_roughgarden.py -q

References

Cohen (1971), maximizing terminal seed yield. King and Roughgarden (1982), graded allocation with a log running payoff.

Not validated against published figures

Numerics are checked against the papers’ closed-form controls and yields, not against figures from those papers.