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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 $$dN(t) = N(t) [(a(t)-b(t) N(t)) dt + \alpha (t) dB(t)], \quad N(0)=N_0 > 0,\ t \ge 0,$$ driven by a $1$-dimensional Brownian motion $B$ on a probability space $(\Omega,{\cal F},({\cal F}_t)_{t \ge 0}, P)$. 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 ODE with randomness 60H10 Stochastic ordinary differential equations 37H10 Generation, random and stochastic difference and differential equations 60H30 Applications of stochastic analysis 92D25 Population dynamics (general)
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##### References:
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