Ergodic trichotomy for hybrid mass-conserving biological switching diffusions under multiplicative common noise

This paper establishes an asymptotic ergodic classification for mass-conserving biological switching diffusions subjected to multiplicative common noise and Markovian regime-switching. The inclusion of multiplicative common noise yields a positive predictable quadratic variation density on the spatial intersection manifold whenever the biological mass is active, precluding the application of classical pathwise spatial comparison theorems. To resolve this measure-theoretic bottleneck, we impose constant stoichiometry and Beddington–DeAngelis spatial interference to guarantee the uniform fractional integrability and derive algebraic bounds for the continuous spatial variance terms. The global dynamics are classified into a trichotomy governed by the algebraic sign of the principal continuous-time hybrid Lyapunov exponent, λ . For λ < 0 , we bypass intersection bounds using a continuous-time logarithmic discrepancy tracking sequence and continuous local martingale limit dichotomies to establish global exponential extinction. At the critical threshold λ = 0 , we prove biological extinction in expectation by applying the multidimensional generator to the boundary Poisson resolvent. Because the abiotic stochastic flow is pathwise affine, it preserves spatial concavity, yielding a non-positive spatial operator difference. For λ > 0 , we construct a separable Meyn–Tweedie Foster–Lyapunov mapping to secure positive Harris recurrence and exponential convergence in total variation distance. The theoretical results are illustrated numerically.