A note on nonautonomous logistic equation with random perturbation
Consider the nonautonomous extension of randomized logistic equation \[ dN(t) = N(t) [(a(t)-b(t) N(t)) dt + \alpha (t) dB(t)], \quad N(0)=N_0 > 0,\;t \geq 0, \] driven by a \(1\)-dimensional Brownian motion \(B\) on a probability space \((\Omega,{\mathcal F},({\mathcal F}_t)_{t \geq 0}, P)\). Suppose that the coefficients \(a, b, \alpha\) are continuous, \(T\)-periodic functions satisfying \[ a(t) > 0, \quad b(t) > 0, \quad \int^T_0 [a(s)-\alpha^2(s)] \,ds > 0 . \] This note shows that \(E [1/N(t)]\) has a unique positive \(T\)-periodic solution under these conditions as a natural extension of a well-known property of the underlying deterministic model.
scientific article; zbMATH DE number 2141748