Figure 1. Transition precursors for the LE model, equation (4).

In (a) the steady homogeneous phosphorus concentration ψ (orange) and the numerically obtained spatial average (black, further averaged over R = 1000 temporal snapshots for each value of p) as a function of the control parameter p which is slowly increased from low to high values. In (b) , the spatial standard deviation (black) of ψ used in refs ¹³, ¹⁴, ¹⁵ as a transition precursor is displayed averaged over R = 1000 temporal snapshots. In orange, the clustering of the functional network built with threshold γ = 0.21 also used as a precursor in refs ²²,²³. In (c) the relative size of the giant component, S₁ (orange), and the average size of the leftover clusters (〈s²〉, black). (d) The probabilities c₂ (black), c₅ (orange), and c₉ (purple) are shown. In all the panels, the vertical green line marks the position of the observed abrupt transition (P_d = 0.658). The grey area indicates the anticipation in parameter space gained over previous methods by using the peak of c₂ as precursor of the percolation transition. All curves have been further averaged over 100 realizations of the random noise and initial conditions.