Chemostat PDE example: trait-structured population model

[jc-macdonald]

Chemostat PDE example: trait-structured population model

Add a chemostat PDE example that demonstrates op_engine on a real reaction-diffusion system. This is the first PDE testbed for the package and serves as validation for the structured-observables manuscript.

Model

Trait-structured chemostat with resource coupling:

∂n/∂t = n·g(θ,S) − D·n + d·∂²n/∂θ²    (population density)
dS/dt  = D·(S_in − S) − ∫ n·g(θ,S) w(θ) dθ  (resource)

where g(θ,S) is the growth function, D is dilution rate, d is trait diffusion, w(θ) is uptake weight.

IMEX splitting

• Implicit (linear): diffusion d·∂²n/∂θ² — handled by op_engine's implicit solver with the non-uniform Laplacian (Non-uniform grid Laplacian operator in matrix_ops #45)
• Explicit (nonlinear): growth n·g(θ,S), dilution −D·n, resource dynamics

Implementation

• examples/chemostat_pde.py — complete working example
• Non-uniform trait grid via adaptive generation (Adaptive curvature-weighted grid generation #46)
• IMEX time-stepping via CoreSolver (imex-euler or imex-heun-tr)
• Observable computation: total biomass I, mean trait ρ, variance Σ, skewness κ, kurtosis ξ
• Comparison with explicit RK45 reference solution
• Docs-site guide with figures

Source code reference

• Thoery_1_Code.ipynb: simulate_model1_crank_nicolson — CN FV on non-uniform grid
• Chemostat_Simulator.ipynb: simulate_model2_semi_implicit — CN with positivity projection

Depends on

• Non-uniform grid Laplacian operator in matrix_ops #45 (non-uniform Laplacian)
• Adaptive curvature-weighted grid generation #46 (adaptive grid generation)