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

Sergei Zuyev,Peter Jagers

doi:10.1007/s00285-021-01624-z

Sergei Zuyev, Peter Jagers

Open Access

https://doi.org/10.1007/s00285-021-01624-z

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Abstract

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 epsilon > 0 such that the conditional probability of a population decrease at the next step, given the past, always exceeds epsilon if the population is not extinct but smaller than the carrying capacity. Then the population must die out.

Highlights

Denote population sizes, starting at time τ0 = 0, by Z0, changing into Z1, Z2, . . . ∈ N at subsequent time points 0 < τ1 < τ2
Let Fn be the sigmaalgebra of all events up to and including the n-th change - i.e. really all events, population size changes - and introduce a carrying capacity K > 0, the population size where reproduction turns conditionally subcritical
Assumption 2 There is no resurrection or immigration but, otherwise, a change is a change in population size: Zn = 0 ⇒ Zn+1 = 0, (2)

Summary

Three assumptions and one result

Denote population sizes, starting at time τ0 = 0, by Z0, changing into Z1, Z2, . ∈ N at subsequent time points 0 < τ1 < τ2. N is the set of non-negative integers, and we make no assumptions about the times between changes. Let Fn be the sigmaalgebra of all events up to and including the n-th change - i.e. really all events, population size changes - and introduce a carrying capacity K > 0, the population size where reproduction turns conditionally subcritical.

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The proof