Detecting the most probable transition phenomenon of a nutrient–phytoplankton–zooplankton system

The paper defines the Onsager–Machlup (OM) functional with V=(σσ^T)^{-1}, b_i(ω)=\tilde b_i(ω)−(1/2)∑_{l,j}(V^{-1})_{lj}Γ^i_{lj}, and includes div b and the −R/6 curvature term; for σ=diag(εx,εy,εz) it computes Γ^1_{11}=−1/x, Γ^2_{22}=−1/y, Γ^3_{33}=−1/z, R=0, the modified drift b=(f+ε^2x/2, g+ε^2y/2, h+ε^2z/2), and div b=f_x+g_y+h_z−f/x−g/y−h/z, yielding the explicit Lagrangian and the Euler–Lagrange system (3.3) for the six-dimensional first-order form in (x,y,z,u,v,w) with u=ẋ, v=ẏ, w=ż. These match the candidate’s derivation step-for-step, including the cancellations and the final ODEs; see the paper’s definitions and computations of the OM functional, modified drift, divergence, R=0, and the displayed ODEs (3.3) . The probabilistic conclusion that minimizing S(ω) maximizes the leading-order tube probability is also stated in the paper . The model additionally notes the natural transversality condition at the terminal set Γ (not spelled out in the paper), but this is a standard consequence of free-endpoint variational problems and does not create a discrepancy with the paper’s results.