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Zbl 1214.34045
Liu, Meng; Wang, Ke
Persistence and extinction in stochastic non-autonomous logistic systems. (English)
[J] J. Math. Anal. Appl. 375, No. 2, 443-457 (2011). ISSN 0022-247X

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.
[Andrew Dale (Durban)]

MSC 2000:: *34F05 ODE with randomness
92D25 Population dynamics
34D05 Asymptotic stability of ODE

Keywords: stochastic perturbation; non-autonomous logistic model; persistence

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