Amendment to: populations in environments with a soft carrying capacity are eventually extinct

This sharpens the result in the paper Jagers and Zuyev (J Math Biol 81:845–851, 2020): consider a population changing at discrete (but arbitrary and possibly random) time points, the conditional expected change, given the complete past population history being negative, whenever population size exceeds a carrying capacity. Further assume that there is an [Formula: see text] such that the conditional probability of a population decrease at the next step, given the past, always exceeds [Formula: see text] if the population is not extinct but smaller than the carrying capacity. Then the population must die out.