Figure 2
Numerical validation of theoretical results.
(a) A five-gene model for a DNB and an early-warning signal. The network model and detailed background are described in Supplementary Information B. The tipping point is at P = 0 in the theoretical model, at which the system undergoes a critical transition or a bifurcation detected by z1 and z2. (b)–(c) When the system approaches the tipping point (P = 0), z1 and z2 become closely correlated with increasingly strong deviations from P = 0.4 to P = 0.01. (d)–(e) Figures show the curves of SDs and PCCs for the variables against the parameter P, which clearly indicate the tendency of z1 and z2, i.e., their fluctuations (SD(z1) and SD(z2)) and correlation (|PCC(z1, z2)|) increase drastically whereas their correlations with other nodes (|PCC(z1, z3)|), |PCC(z1, z4)|, |PCC(z1, z5)|, |PCC(z2, z3)|, |PCC(z2, z4)| and |PCC(z2, z5)|) decrease drastically when the system approaches the tipping point, which satisfies all three criteria for the DNB. (f) The curve shows the clear tendency of the composite index near the tipping point for the DNB composed of (z1, z2), which can be used as the early-warning signal for predicting the imminent change in the concerned system.
