Impact statement
Conservation decisions are often taken as though a species’ fate were settled by its average trend: stable numbers mean safety, falling numbers mean something is driving them down. Both readings can mislead. A small population can be lost while its average tendency is stable, and a population that appears to be declining may be in the trough of a fluctuation that will reverse. Whether an extinction is called "stochastic" therefore carries practical weight, because it decides whether managers look for a cause to remove or for numbers to rebuild.The difficulty is that the phrase is used for several different things: noise in births and deaths, variation in the environment, the erosion of genetic variability, and, in palaeontology, extinction that simply does not sort species by their traits. This Explainer separates those meanings and sets out the four things a careful use of the term must specify. It is written for readers who meet the phrase from outside extinction research (journalists, policymakers, students and practitioners) and who need to know what is, and is not, being claimed.
The idea
Most extinctions are frequently pictured as deterministic: a population undergoes a sustained, directional decline (habitat loss, overharvest, a changing climate) until it reaches zero. Stochastic extinction is the complement. It is the recognition that even a population with, on average, a positive or stable growth rate can disappear through the accumulation of chance events. Births and deaths, good and bad years, the shuffling and loss of genes and rare severe years are all probabilistic. A population that is only holding steady or slowly declining is, over a long enough interval, certain to be extinguished this way; even one growing on average is not immune: it retains a non-zero probability of being driven to zero by a run of bad luck (Feller, Reference Feller1939). Feller’s result, which already contains the whole idea, is stated in full in Supplementary Section S1.
It is tempting to picture stochastic extinction as the last individual dying at random, the final shot in a field of bullets. That image captures a real but different sense of the word, used mainly in palaeontology (see below). In population biology the important point is earlier and more practical: chance dominates the fate of a lineage long before the last individual, once the population becomes small enough that random variation outweighs its average tendency to grow. The expected time to extinction lengthens steeply with population size, so small populations are far more vulnerable than large ones (MacArthur and Wilson, Reference MacArthur and Wilson1967) (Figure 1).
Stochastic extinction as a chance-driven loss of viability. Simulated trajectories of a small population with, on average, a stable growth rate. Both panels use the same random shocks, so the difference between them is the biology, not the noise. (A) With chance alone the trajectories wander; numbers can sit low for a long time and still recover, and the expected path of a population that has entered the shaded region (heavy dashed line) returns. (B) When growth itself weakens as numbers fall, because mate-finding failure, sex-ratio failure and lost genetic variability reinforce one another, that expected path steepens instead: the extinction vortex. Neither panel contains a line that gets crossed. The shading marks a risk that deepens as numbers fall, at a rate neither constant nor predictable across densities. The vertical axis is qualitative (few to many).
Figure 1 Long description
The figure has two panels side by side, labelled A and B. Both share the same axes. The vertical axis is labelled Population size and is qualitative, running from few at the bottom to many at the top. The horizontal axis is labelled Time and has no units. Both panels are driven by the same sequence of random shocks, so the only difference between them is the biology.
In each panel a band of red shading rises from the baseline and fades out towards the middle of the panel. The shading deepens continuously towards zero and has no boundary line at any height.
A legend in panel A identifies three line types. Pale blue grey lines are populations that persist. Red lines are populations that are lost. A heavy black dashed line is the expected path of a population that has entered the shaded region, and a small open circle marks where that expected path begins.
Panel A is headed Chance alone, with the subheading no feedback: the expected path returns. Most trajectories are pale blue and fluctuate irregularly across the upper part of the panel from start to finish. One red trajectory descends through the shaded region, reaches zero part way along the time axis, and then runs flat along the baseline. The heavy dashed expected path starts low inside the shaded region and rises towards the right. An annotation in the upper right reads: numbers can sit low for a long time and still recover.
Panel B is headed With positive feedback: the extinction vortex, with the subheading growth itself weakens as numbers fall. The same shocks give a different outcome. Three red trajectories descend into the shaded region and reach zero, at three different times. The heavy dashed expected path starts at the same open circle position as in panel A, but falls to the baseline and stays there. An annotation in the upper right reads: same shocks, same zone, but the expected path now steepens.
The visual comparison is between the two heavy dashed paths. From the same starting position, that path rises in panel A and falls to zero in panel B. Neither panel contains a threshold line.
An umbrella, not one concept
Conservation biology inherited a partition of the sources of randomness, named explicitly by Shaffer (Reference Shaffer1981) and formalised by Lande (Reference Lande1993). Two of its divisions are better collapsed, leaving what is external to the population and what is internal to it:
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• Environmental fluctuation: year-to-year variation in vital rates driven by a fluctuating environment, which can threaten even large populations (Lande, Reference Lande1993). Rare, severe events belong here too. Setting “catastrophes” apart as a class of their own begs the question of what counts as one, and there is no scale-free answer: a population’s measured variability keeps rising with the length of the census, so extremes appear at every scale of observation (Inchausti and Halley, Reference Inchausti and Halley2001; Pimm and Redfearn, Reference Pimm and Redfearn1988), which is also why recovery from a disturbance is hard to delimit and hard to measure (Pimm et al., Reference Pimm, Donohue, Montoya and Loreau2019). A catastrophe is the tail of the environmental distribution, not a different kind of thing.
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• Demographic accidents: the unavoidable consequences of finite numbers, in three forms: the vagaries of death (who dies, when individuals die independently), the vagaries of birth (how many young appear, and of which sex) and the loss of genetic variability through genetic drift and inbreeding, at a rate set by the effective population size. They matter most in small populations (Melbourne and Hastings, Reference Melbourne and Hastings2008). The first two can be modelled exactly; the third cannot, because how a given loss of variation translates into survival and reproduction is still poorly known and has to be established case by case (Pimm et al., Reference Pimm, Dollar and Bass2006). Analyses that model only the first two therefore understate risk (Kong et al., Reference Kong, Xu, Xiao, Pimm, Shi and Ouyang2021). The three are not independent, which is why they belong together: as numbers fall each strengthens the others, so the growth rate itself declines, producing the extinction vortex (Fagan and Holmes, Reference Fagan and Holmes2006; Gilpin and Soulé, Reference Gilpin and Soulé1986).
A third, conceptually different usage comes from palaeontology. There “stochastic” extinction usually means extinction that is non-selective with respect to traits, Raup’s field of bullets, in which who dies is unrelated to morphology, ecology or body size (Raup, Reference Raup1991). Crucially, this describes a pattern and is compatible with a perfectly deterministic cause: a bolide impact is not random, yet it can kill clades at random with respect to their traits. The two classes above, by contrast, describe a process. Conflating the two, pattern versus process, is a principal source of confusion in how the term is used (Figure 2).
One term, several meanings. The two population-level classes of randomness (environmental fluctuation, external to the population, and demographic accidents, internal to it) act on the process of population change and together set the chance that a self-replacing population is no longer attainable. The palaeontological usage, extinction non-selective with respect to traits (Raup’s “field of bullets”), describes an observed pattern and can be produced even by a deterministic cause. Clarity requires saying which meaning, at which scale, over which horizon, is intended.
Figure 2 Long description
The figure is a diagram divided by a vertical dashed line into a wide left portion and a narrower right portion. The left portion is headed PROCESS: where the randomness comes from. The right portion is headed PATTERN.
The left portion contains two blue outlined boxes, one above the other.
The upper box is labelled EXTERNAL and then, in bold, Environmental fluctuation, with the gloss good years and bad: weather, disease, resources, predators. A note inside the box reads: catastrophe has no clean, scale free definition, it is the tail of this same distribution, not a separate class.
The lower box is labelled INTERNAL and then, in bold, Demographic accidents, with the gloss unavoidable in a finite population. Inside it are three white sub boxes, one above the other. The first reads the vagaries of death, glossed who dies, in a finite population. The second reads the vagaries of birth, glossed how many, and of which sex. The third reads loss of genetic variability, glossed drift and inbreeding, set by Ne. A curved double headed arrow runs down the right hand side of the three sub boxes, and a note beneath them reads: these three feed back on one another, the extinction vortex.
One arrow leaves each of the two blue boxes. Both arrows converge on a red outlined box at the centre of the figure, which reads: The chance that a self replacing population becomes unattainable. A note inside that box reads: a probability, not a line that gets crossed, and not consistent or predictable across densities.
To the right of the dashed line stands a single gold outlined box headed Palaeontological usage. It reads: extinction non selective with respect to traits, Raup’s field of bullets. It continues: This describes who died, not why. A perfectly deterministic cause, such as a bolide, can produce it. In bold at the foot of that box: PATTERN, not PROCESS. No arrow joins this box to anything on the left of the dashed line.
A single line of text runs across the foot of the whole figure: Clarity requires saying which meaning is intended, at which scale, over which time horizon.
A population-biology reading: When viable populations are not attainable
For a working population biologist, the most useful reading is operational and scale-explicit. A lineage is effectively lost not at the death of its last individual but when the probability, over a stated horizon, that a self-replacing population completing its full life cycle in the wild remains attainable has become small (Figure 3). Standing adults that cannot recruit, because mates, pollinators, or symbiotic partners have become too rare, are already on a one-way trajectory, even when census numbers look reassuring. This is the regime in which stochastic processes become decisive: Allee effects, of which mate-finding failure is one mechanism, amplify demographic accidents. Viewed this way, stochastic extinction is a chance-driven loss of viability rather than the crossing of a line, and it sits naturally alongside the concept of functional extinction (Jarić, Reference Jarić2015).
Stochastic extinction as failure to complete the life cycle. A taxon-neutral, stage-structured life cycle (read each stage for your organism: seed, spore, egg, larva, or planula). (A) When every transition needed to close the cycle is intact, the population self-replaces. (B) When recruitment collapses (establishment falls to near zero because mates, pollinators, or symbiotic partners are lost), adults persist and still produce propagules, but the cycle cannot close and the population can no longer self-replace. Crucially, the presence of all life-cycle stages does not guarantee viability: a population can contain every stage, adults included, yet still be unable to replace itself. Stochastic extinction, in this view, is the chance-driven weakening of any transition needed to close the cycle, and the limiting process can act at any stage or transition. Establishment is only the example shown here; survival, maturation, or fecundity can equally be the failing step. Read each stage for your organism: a propagule is a seed (plants), spore (fungi), egg or larva (animals and fish), or planula (corals); propagule dormancy/stasis applies chiefly to seed and spore banks, and many propagules (eggs, larvae, planulae) lack a persistent stage.
Figure 3 Long description
The figure has two panels, one above the other. Each panel shows the same life cycle diagram, built from three rounded pale blue boxes arranged left to right and joined by arrows.
The left box reads Propagule, with the gloss seed, spore, egg, larva, planula. The middle box reads Juvenile, with the gloss seedling, immature. The right box reads Adult, with the gloss reproductive.
Above each box is a curved arrow that loops back to the same box. The loop above Propagule is labelled dormancy or stasis. The loops above Juvenile and above Adult are both labelled survival.
A straight arrow runs from Propagule to Juvenile, labelled establishment. A second straight arrow runs from Juvenile to Adult, labelled maturation. A long curved arrow runs back from Adult to Propagule beneath the three boxes, labelled fecundity. Together these close the cycle.
The upper panel is labelled A and headed Intact cycle: the population self replaces. Every arrow in it is solid.
The lower panel is labelled B and headed, in red, Recruitment collapses: adults persist, no replacement. It is identical to panel A with one change. The establishment arrow from Propagule to Juvenile is now red and dashed, and its label, also in red, reads establishment approximately zero, recruitment fails. Every other arrow stays solid, including adult survival and fecundity, so adults are still present and still producing propagules while the loop cannot close.
This reading is deliberately conceptual rather than tied to any single regulatory definition, and it is not without tension. Reserving the word “extinct” for the disappearance of the last individual is more conservative and more easily audited; treating a lineage as effectively extinct once viability is unattainable is more biologically honest about commitment to loss but harder to apply at the moment of classification. The disagreement is real and worth stating plainly: it is, at root, a disagreement about scale and time horizon, not about whether chance matters.
Why precision matters
Because the boundary between “stochastic” and “deterministic” shifts with scale and time horizon, it is often a methodological choice rather than a property of nature. A drought is an environmental driver (deterministic) to a demographer modelling one population and a random shock (stochastic) to a metapopulation model that does not resolve weather. Neither is wrong; they describe different levels of the same system.
Clear use of stochastic extinction therefore means specifying, every time: (1) which source of randomness is meant; (2) the biological scale (individual, population, metapopulation, or clade); (3) the time horizon over which the claim holds and (4) whether the statement is about an underlying process or an observed pattern. With those four coordinates fixed, the term is precise and powerful. Without them, it is one of the most ambiguous phrases in extinction research.
The historical development of the concept, from Feller’s birth-and-death processes to MacArthur and Wilson, Shaffer, Lande, and the palaeontological tradition, together with the supporting literature and a fuller treatment of scale, is given in the Supplementary Material.
Open peer review
To view the open peer review materials for this article, please visit http://doi.org/10.1017/ext.2026.10023.
Supplementary material
The supplementary material for this article can be found at http://doi.org/10.1017/ext.2026.10023.
Data availability statement
No data were generated or analysed in this article. Figure 1 illustrates a simulated example and has no underlying dataset; Figures 2 and 3 are schematic.
Acknowledgements
Generative AI (Claude, Anthropic) was used to assist with spelling and grammar checking and with formatting the text. All scientific content was written, checked and verified by the author.
Author contribution
R.L.T. is the sole author and is responsible for the conception, writing, figures and revision of this article.
Funding statement
This research received no specific grant from any funding agency, or from commercial or not-for-profit sectors.
Competing interests
The author declares no competing interests.
Ethics and approvals
Ethical approval was not required: this article is a conceptual Perspective and involved no human participants, no animal subjects, and no collection of biological material.
Open access
Comments
Dear Dr Roberts,
Please find enclosed my invited contribution to the Explainer mini-series in Cambridge Prisms: Extinction, “Explainer: Stochastic extinction,” submitted as a Perspective. I am grateful to Jessica Jones for the invitation of 14 May 2026, and to you and the Editors-in-Chief, Professor Brook and Associate Professor Alroy, for the opportunity to contribute to this series and its accompanying reference glossary.
The Explainer treats “stochastic extinction” as what it has become in practice: an umbrella term rather than a single concept. It distinguishes the four population-level sources of randomness — demographic, environmental, genetic, and catastrophic — from the separate palaeontological usage, in which “stochastic” means extinction non-selective with respect to traits, and it stresses the distinction between process and pattern that these usages so often conflate. Its central, practical reading is that stochastic extinction is best understood at the population level as the chance-driven crossing of a viability threshold: even a population with, on average, a stable or positive growth rate can be lost to chance once it becomes small. The piece closes with the four coordinates — source, scale, time horizon, and process versus pattern — that make the term precise. The manuscript is original, is not under consideration elsewhere, and its claims have been checked against the primary literature. I have no competing interests to declare, and the article processing fee is waived under the terms of the invitation.
I would be glad to make any revisions the editors feel would strengthen the piece, and I thank you for your consideration.
Yours sincerely,
Raymond Tremblay
Department of Biology, University of Puerto Rico, San Juan, Puerto Rico
raymond.tremblay@gmail.com