Lindh flowering (`lindh_2016_flowering`)¶

Question. When should an annual plant switch from vegetative growth to reproduction, and how does that timing change with productivity when growth is constrained?

Here plant means that phenology model.

Formulation¶

Symbol	Role	Meaning
\(V(t)\)	state	vegetative biomass
\(R(t)\)	state	reproductive biomass
\(u(t)\in[0,1]\)	control	fraction of production sent to \(V\)

Dynamics.

\[ \dot V = u\,F(V),\qquad \dot R=(1-u)\,F(V),\qquad F(V)=P\,U(V). \]

Three \(U\): exponential \(U=V\); logistic; WBE (size-constrained).

Objective. \(\max R(T)\). Horizon. Fixed season. \(V(0)=V_0\), \(R(0)=0\).

What was done¶

Relaxed \(u(t)\) in \([0,1]\) on one grid. Flowering time is read off as the switch of \(u^*\). An independent 1-D line search over a single switch time is the cross-check.

What we found¶

The OCP is bang–bang for all three \(U\). Under exponential growth, optimal flowering time increases with productivity \(P\). Under constrained growth it can decrease — the qualitative result of the source paper.

Flowering vs productivity

Interpretation¶

The same allocation ODE plus \(\max R(T)\) is not enough to fix the sign of \(\mathrm{d}t_F/\mathrm{d}P\). That sign is a property of \(U(V)\). Constrained growth is a hypothesis about production, not an extra phenology rule.

Public API¶

Full script: examples/lindh_2016_flowering/optimal_flowering.py.

import eco_opt
from eco_opt import Namespace


def model(t, x, u):
    V, R = x[0], x[1]
    F = P * U_fn(V)
    y = Namespace(V=V, R=R, F=F)
    return [u[0] * F, (1 - u[0]) * F], y


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

Run and tests¶

cd examples/lindh_2016_flowering
python optimal_flowering.py
python plot_figure2.py
python -m pytest ../../tests/test_lindh_2016_flowering.py -q

References¶

Lindh, Johansson, Bolmgren, Lundström, Brännström and Jonzén (2016), New Phytologist 209:1591–1599, doi:10.1111/nph.13706.

Not validated against the paper’s figures

The eco_opt trajectories are not checked against the numerical results in that paper. The example follows their formulation (exponential / logistic / WBE production, \(\max R(T)\)). Tests compare the OCP to an independent 1-D line search in this package, not to Lindh et al.’s plots.

Lindh flowering (lindh_2016_flowering)¶