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Persistence and extinction in stochastic non-autonomous logistic systems. (English) Zbl 1214.34045
Beginning with a general discussion of the classical non-autonomous logistic equation
$dx(t)/dt = x(t)[r(t) -a(t)x(t)],\;\;x(0) =x_{0}>0,$
including definitions of persistence in both the deterministic and the stochastic sense, the authors examine the equations
$dx(t) = x(t)[r(t) -a(t)x(t)]dt+ \sigma(t)x^{2}(t)dB(t)$
and
$dx(t) = x(t)[r(t) -a(t)x(t)]dt+ \sigma(t)x(t)dB(t).$
Carrying out the survival analysis, they find sufficient conditions for extinction and examine the questions of persistence and permanence.

MSC:
 34F05 Ordinary differential equations and systems with randomness 92D25 Population dynamics (general) 34D05 Asymptotic properties of solutions to ordinary differential equations
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