We do consider models of the population or opinion dynamics which result in non-linear stochastic differential equations (SDEs) exhibiting spurious long-range memory. In this context, the correspondence between the description of birth-death processes as continuous-time Markov chains and continuous SDEs is of high importance for the alternatives of modeling. We propose and generalize Bessel-like birth-death process having clear representation by SDEs. The new process helps to integrate

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... ves of description and to derive equations for the probability density function (PDF) of burst and inter-burst duration of the proposed continuous time birth-death processes. This PDF might be used to discriminate between spurious memory and true long-range memory in complex systems exhibiting continuing fluctuations.