Fig. 4: Universal bounds for the critical scaling exponents at the supercritical Hopf bifurcation.
Limit-cycle oscillation of a the proportion of type i (density of type i divided by the total density), Pi, and b the speed of the growth rate (vs, black solid line), the speed of change in diversity (vI, gray solid line), and the speed limit (\({v}_{\lim }\), red dashed line) as a function of time t in the competitive Lotka–Volterra model. c Power-law decay of \({v}_{\lim }\) (red line), compared with vs (black line) at the Hopf bifurcation point. The asymptotic form of the amplitude relaxation (\({v}_{\lim } \sim {t}^{-1/2}\)) is shown with a dotted line. The curves are rattling since the number of plotted points is finite; similarly to b, vs oscillates between zero and nonzero values, while \({v}_{\lim }\) stays nonzero. d Scaling plot of the time and interaction dependence of \({v}_{\lim }\) near the bifurcation point (0.999 ≤ c21/cc ≤ 1.001). The limit cycle appears for c21 < cc, while the steady-state coexistence of three types appears for c21 ≥ cc, where c21 is the competitive interaction strength from type 1 to type 2, and cc is the value of c21 at the Hopf bifurcation point (Supplementary Method 6). The exponents are given as \({\alpha }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}=1/2\) and \({\beta }_{\lim }^{{{{{{{{\rm{Hopf}}}}}}}}}=1\). See Supplementary Method 6 for the parameters used.
