Persistence and extinction in stochastic non-autonomous logistic systems (Q615909)

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.