Fig. 3: Universal bounds for the critical scaling exponents at the transcritical bifurcation.

a Power-law decay of the speed of the growth rate (v_s, black solid line), the speed of the change in diversity (v_I, dark-gray solid line), the speed of the type index (v_b, light-gray solid line), and the speed limit (\({v}_{\lim }\), red dashed line) at the transcritical bifurcation point (r = r_c) of the evolutionary model with selection and mutation. Here, t is the time, r is the growth rate for type 1 relative to that for type 2 or 3, and r_c is the value of r at the transcritical bifurcation point. The asymptotic forms (\({v}_{\lim } \sim {t}^{-3/2}\) and v_b \(\sim\) t⁻²) are shown with dotted lines. b Time and parameter dependence of \({v}_{\lim }\) and c the corresponding scaling plot near the bifurcation point (0.999 ≤ r/r_c ≤ 1.001). The exponents at the transcritical (TC) point are given as \({\alpha }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}=3/2\) [see Eq. (4)] and \({\beta }_{\lim }^{{{{{{{{\rm{TC}}}}}}}}}=1\) [see Eq. (7)]. d Power-law decay of v_I (black solid line) and \({v}_{\lim }\) (red dashed line) at the transcritical bifurcation point of the susceptible-infected-recovered (SIR) model. The asymptotic form (\({v}_{\lim } \sim {t}^{-3/2}\)) is shown with a dotted line. For a–c, we use the same parameters as those for Fig. 2. See Supplementary Method 4 for the parameters used for d.