Stochastic Diversity Stabilization in Nonlinear Competitive Dynamics: Fluctuation Suppression and Relaxation Restructuring

Stochastic forcing can prevent diversity collapse in competitivedynamical systems, yet the quantitative conditions and thedynamical mechanism responsible remain poorly characterized.We address both questions in a class of nonlinear stochasticreplicator dynamics on the $N$-dimensional probability simplex,parameterized by a nonlinearity exponent $\gamma$ and additiveGaussian noise of intensity $\sigma$.Under the baseline interaction, the minimum noise requiredto sustain diversity decays as $\sigma_c \sim N^{-2.31}$;this scaling collapses under superlinear variants,identifying $\alpha \approx 2.31$ as a model-specificquantity rather than a universal index.Scanning the $(\gamma,\sigma)$ plane reveals a smoothdynamical crossover: the fluctuation susceptibility$\chi_H = N\,\mathrm{Var}(H)$ peaks at$\gamma \approx 0.96$ ($\chi_H \approx 0.74$) andis suppressed at high $\gamma$, with no divergencenear $\gamma \approx 1.4$;matched finite-size analysis finds no systematicdivergence of the peak with $N$,ruling out a genuine second-order phase transition.The entropy relaxation time $\tau_\mathrm{relax}(\gamma)$is non-monotonic and mirrors this three-regime landscape,with fast but broadly distributed relaxation at high$\gamma$ attributable to pathway multiplicity acrosstrajectories.These findings identify \emph{nonlinear fluctuationsuppression via pathway multiplicity}---rather thancritical organization---as the operative stabilizationmechanism.