1 Three assumptions and one result

Denote population sizes, starting at time \(\tau _0=0\), by \(Z_0\), changing into \(Z_1, Z_2, \ldots \in {\mathbb {N}}\) at subsequent time points \(0< \tau _1 < \tau _2 \ldots \). Here \({\mathbb {N}}\) is the set of non-negative integers, and we make no assumptions about the times between changes. Let \({\mathscr {F}}_n\) be the sigma-algebra of all events up to and including the n-th change - i.e. really all events, not only population size changes - and introduce a carrying capacity \(K>0\), the population size where reproduction turns conditionally subcritical. More precisely:

Assumption 1

$$\begin{aligned} {{\,\mathrm{{{\mathbb {E}}}}\,}}[Z_{n+1}|{\mathscr {F}}_n] \le Z_n, \quad \mathrm {if}\ \ Z_n \ge K. \end{aligned}$$
(1)

Further,

Assumption 2

There is no resurrection or immigration but, otherwise, a change is a change in population size:

$$\begin{aligned}&Z_n = 0 \Rightarrow Z_{n+1} =0, \end{aligned}$$
(2)
$$\begin{aligned}&Z_n > 0 \Rightarrow Z_{n+1} \ne Z_n. \end{aligned}$$
(3)

Assumption 3

Non-extinct populations, smaller than the carrying capacity, run a definite risk of decreasing:

$$\begin{aligned} \exists \epsilon > 0; \forall n\in {\mathbb {N}}, 0< Z_n< K \Rightarrow {\mathbb {P}}(0\le Z_{n+1} < Z_n|{\mathscr {F}}_n] \ge \epsilon . \end{aligned}$$
(4)

Then:

Theorem 1

Under the three assumptions given, the population must die out: with probability 1, \(Z_n = 0\) eventually.

The original paper (Jagers and Zuyev 2020) had a stronger third assumption, viz. that, whatever the population history, there must be a definite, strictly positive risk that the population size decreases by exactly one unit at the next change. This is not unnatural and can be interpreted as a possibility that a change involves no reproduction but merely the death of one individual. But it turns out to be unnecessary.

2 The proof

Like the original proof, this starts from stopping times \(\nu _1, \nu _2, \ldots \) and \(\mu _1, \mu _2, \ldots \), the former denoting the times of successive visits to the integer interval [0, K), the latter the subsequent first hittings of levels \(\ge K\). More precisely,

$$\begin{aligned} \nu _1 := \inf \{n\in {\mathbb {N}}; Z_n < K\}, \end{aligned}$$

and for \(k=1,2, \ldots \) ,

$$\begin{aligned} \mu _k := \inf \{n\in {\mathbb {N}}; n> \nu _k \text{ and } Z_n \ge K\}, \nu _{k+1} := \inf \{n\in {\mathbb {N}}; n > \mu _k \text{ and } Z_n < K\}. \end{aligned}$$

As was noted, \(\nu _1<\infty \), whereas the \(\mu _k\) constitute an increasing sequence, possibly hitting infinity. Clearly, \(\nu _k< \infty , \mu _k=\infty \) means that the population dies out at or after \(\nu _k\), without ever reaching K again. Also for any k, \(\mu _k<\infty \Rightarrow \nu _{k+1} < \infty \). Proceeding like in the original paper, note that

$$\begin{aligned} Z_n \rightarrow 0 \Leftrightarrow \exists n \in {\mathbb {N}}; Z_n=0 \Leftrightarrow \exists k; \mu _k = \infty , \end{aligned}$$

and

$$\begin{aligned} {\mathbb {P}}( \exists k; \mu _k = \infty ) = \lim _{k \rightarrow \infty }{\mathbb {P}}(\mu _k = \infty ) = 1 - \lim _{k \rightarrow \infty }{\mathbb {P}}(\mu _k < \infty ). \end{aligned}$$

But

$$\begin{aligned} {\mathbb {P}}(\mu _k< \infty ) = {\mathbb {P}}(\mu _k< \infty , \nu _k< \infty ) = {{\,\mathrm{{{\mathbb {E}}}}\,}}[{\mathbb {P}}(\mu _k< \infty | {\mathscr {F}}_{\nu _k}) ; \nu _k < \infty .] \end{aligned}$$

For short, write

$$\begin{aligned} D_n := \{Z_n \le (Z_{n-1} -1)^+ \} \end{aligned}$$

for the event that the n-th change is a decrease, provided \(Z_{n-1} >0\) (and of course the population remains extinct if \(Z_{n-1} =0\)). By Assumption 3, \(Z_n < K\) implies that

$$\begin{aligned} {\mathbb {P}}(\cap _{j=1}^K D_{n+j}| {\mathscr {F}}_n)= & {} {{\,\mathrm{{{\mathbb {E}}}}\,}}[{\mathbb {P}}(D_{n+K} |{\mathscr {F}}_{n+K-1} ; \cap _{j=1}^{K-1} D_{n+j} | {\mathscr {F}}_n] \\\ge & {} \epsilon {\mathbb {P}}(\cap _{j=1}^{K-1} D_{n+j}| {\mathscr {F}}_n) \ge \ldots \ge \epsilon ^K .\end{aligned}$$

Since \(Z_n < K\) implies that \(Z_{n+K} =0\) on the set

$$\begin{aligned} \cap _{j=1}^K D_{n+j} , \end{aligned}$$

and the population size never crosses the carrying capacity, we can conclude that

$$\begin{aligned} {\mathbb {P}}(\mu _k = \infty )= & {} 1- {\mathbb {P}}(\mu _k< \infty ) \\\ge & {} 1- (1-\epsilon ^K) {\mathbb {P}}(\mu _{k-1} <\infty ) \ge \ldots \ge 1-(1-\epsilon ^K)^k \rightarrow 1. \end{aligned}$$

The theorem follows.