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

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

Three assumptions and one result

Denote population sizes, starting at time ${\tau }_0=0$, by $Z_0$, changing into $Z_1,Z_2,\dots \in \mathbb{N}$ at subsequent time points $0<{\tau }_1<{\tau }_2\dots$. Here $\mathbb{N}$ is the set of non-negative integers, and we make no assumptions about the times between changes. Let ${\mathcal{F}}_n$ 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

$$\begin{array}{c}\hfill \mathbb{E}[Z_{n+1}|{\mathcal{F}}_n]\le Z_n,\mathrm{if}{Z}_n\ge K.\end{array}$$ 1

Further,

Assumption 2

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

$$\begin{array}{cc}& Z_n=0\Rightarrow Z_{n+1}=0,\hfill \end{array}$$ 2

$$\begin{array}{cc}& Z_n>0\Rightarrow Z_{n+1}\ne Z_n.\hfill \end{array}$$ 3

Assumption 3

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

$$\begin{array}{c}\hfill \exists ϵ>0;\forall n\in \mathbb{N},0<Z_n<K\Rightarrow \mathbb{P}(0\le Z_{n+1}<Z_n|{\mathcal{F}}_n]\ge ϵ.\end{array}$$ 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.

The proof

Like the original proof, this starts from stopping times ${\nu }_1,{\nu }_2,\dots$ and ${\mu }_1,{\mu }_2,\dots$, 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,

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

and for $k=1,2,\dots$ ,

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

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

$$\begin{array}{c}\hfill Z_n\to 0\iff \exists n\in \mathbb{N};Z_n=0\iff \exists k;{\mu }_k=\infty ,\end{array}$$

and

$$\begin{array}{c}\hfill \mathbb{P}(\exists k;{\mu }_k=\infty )=\underset{k\to \infty }{lim}\mathbb{P}({\mu }_k=\infty )=1-\underset{k\to \infty }{lim}\mathbb{P}({\mu }_k<\infty ).\end{array}$$

But

$$\begin{array}{c}\hfill \mathbb{P}({\mu }_k<\infty )=\mathbb{P}({\mu }_k<\infty ,{\nu }_k<\infty )=\mathbb{E}[\mathbb{P}({\mu }_k<\infty |{\mathcal{F}}_{{\nu }_k});{\nu }_k<\infty .]\end{array}$$

For short, write

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

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

$$\begin{array}{cc}\hfill \mathbb{P}({\cap }_{j=1}^K{D}_{n+j}|{\mathcal{F}}_n)=& \mathbb{E}[\mathbb{P}(D_{n+K}|{\mathcal{F}}_{n+K-1};{\cap }_{j=1}^{K-1}{D}_{n+j}|{\mathcal{F}}_n]\hfill \\ \hfill \ge & ϵ\mathbb{P}({\cap }_{j=1}^{K-1}{D}_{n+j}|{\mathcal{F}}_n)\ge \dots \ge {ϵ}^K.\hfill \end{array}$$

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

$$\begin{array}{c}\hfill {\cap }_{j=1}^K{D}_{n+j},\end{array}$$

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

$$\begin{array}{cc}\hfill \mathbb{P}({\mu }_k=\infty )=& 1-\mathbb{P}({\mu }_k<\infty )\hfill \\ \hfill \ge & 1-(1-{ϵ}^K)\mathbb{P}({\mu }_{k-1}<\infty )\ge \dots \ge 1-{(1-{ϵ}^K)}^k\to 1.\hfill \end{array}$$

The theorem follows.

Funding

Open access funding provided by Chalmers University of Technology.