Lyapunov instability threshold in stochastic Lotka–Volterra dynamics without horizontal gene transfer

Determine the threshold value of the Lyapunov function E(B,V) = (B − n_G^* log(B/n_G^*)) + (V − n_G^* log(V/n_G^*)) at which the prey–predator oscillator governed by the stochastic Lotka–Volterra model without horizontal gene transfer transitions from stable oscillations to instability leading to extinction. Specify the criterion in terms of the parameters s and n_G^* and the current state (B,V), and provide a rigorous relation that delineates stability versus instability in this finite-population, noise-driven setting.

Background

In the Supplementary Information, the authors analyze the stochastic Lotka–Volterra predator–prey dynamics in the absence of horizontal gene transfer to understand extinction driven by demographic noise. They use a Lyapunov function E(B,V) to characterize the distance from the steady state and show that in the deterministic limit E is conserved, while in the stochastic setting E grows, eventually leading to extinction.

While they provide an approximate scaling for the persistence time using a linear-noise approximation, they explicitly state that the threshold value of the Lyapunov function at which the oscillator becomes unstable is unknown. Establishing this threshold would provide a precise and predictive criterion for instability and extinction in the model, complementing their approximate persistence-time estimates.