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A note on nonautonomous logistic equation with random perturbation. (English) Zbl 1076.34062
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.

MSC:
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
37H10 Generation, random and stochastic difference and differential equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
92D25 Population dynamics (general)
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