New research finds that the fertility rate needed to sustain a population is much higher than once thought, especially when sex ratios or mortality rates shift. This raises important questions for human societies and endangered species.
Study: Threshold fertility for the avoidance of extinction under critical conditions. Image Credit: sogane / Shutterstock
In a recent article published in the journal PLoS ONE, researchers investigated the minimum fertility rate required to prevent the extinction of a sexually reproducing population, accounting for the concept of demographic stochasticity, which is defined as random fluctuations in survival and reproduction.
Their findings indicate that the extinction threshold fertility is significantly higher than the commonly accepted replacement level, usually estimated as 2.1. Under model conditions assuming no mortality and a balanced sex ratio, women may need to have an average of 2.7 children each to avoid extinction. However, this threshold is reduced where the population shows a sex ratio biased towards female children, a factor that may help explain observed increases in female births under stressful conditions.
Background
With many developed countries experiencing fertility crises, total fertility rates (TFRs) are dropping well below the conventional replacement level fertility (RLF) of 2.1 children per woman.
Currently, about two-thirds of the global population lives in regions with sub-replacement fertility, with countries like Japan and South Korea showing extreme declines. In Japan, for example, the population may shrink by 31% per generation if current fertility rates persist.
The RLF is based on low death rates, a balanced sex ratio at birth, and large population sizes, where chance variations in individual births and deaths (demographic stochasticity) are negligible.
However, in small or declining populations, such stochasticity becomes critical. Events such as skewed sex ratios or childhood mortality can significantly affect population sustainability, potentially increasing the true RLF beyond 2.1.
Earlier ecological and demographic studies suggest that random fluctuations can raise the threshold for population survival. Furthermore, a mismatch in sex ratios can lead to fewer mating pairs, exacerbating the decline.
For populations with low survival rates or unbalanced sex ratios, an RLF much higher than 2.1 – sometimes above 3.0 – may be necessary to avoid extinction. Thus, traditional RLF estimates may underestimate the actual fertility required to maintain population stability in modern, low-fertility societies.
About the Study
The researchers modeled a sexually reproducing population with non-overlapping generations, focusing on the extinction probability of a lineage originating from a single female.
Four key components were incorporated into the model. The first was the fertility rate; the number of children per female was assumed to follow a Poisson distribution, representing a random but biologically plausible variation in births. This allowed for a realistic spread in offspring numbers, including a non-trivial chance of having no children.
To model the sex ratio, offspring sex was assigned through a binomial distribution, where each offspring had a probability r of being male and (1–r) of being female. This reflected the natural variation in sex at birth and allowed the study to explore the impact of skewed sex ratios.
To model mortality rates, researchers assumed that male and female children have separate probabilities of dying before reaching reproductive age, accounting for differing mortality rates.
Lineage tracking and extinction evaluation were also included in the model. A branching process was used to simulate reproduction across generations. If any generation fails to produce both sexes, the lineage is considered extinct. This process was iterated many times using simulations, and extinction probabilities were also derived analytically through recurrence relations.
No real-life biological samples were used; the study was purely theoretical and computational, relying on established probability distributions and stochastic processes to model population dynamics under demographic uncertainty.
Findings
The study analyzed how the risk of population extinction changes with fertility rates, mortality rates, and sex ratios. It found that a female-biased sex ratio significantly reduces the probability of extinction.
The critical fertility rate (RLF)—the threshold below which extinction is certain—was greater than the usual RLF of 2.1. For example, with an equal sex ratio and no mortality, the critical fertility was about 2.7.
Simulations showed that populations with subcritical fertility (below the critical value) almost always went extinct within 20 generations, though a few rare populations survived and continued growing.
As generations progressed, the proportion of extinct populations approached 100%, especially for lower fertility rates. The risk of extinction was slightly lower for female-biased populations.
Histograms of survival duration confirmed that most populations died out quickly, typically within five generations, although some rare cases lasted longer, especially when fertility was near the critical value. These findings highlight the high extinction risk for small populations, even if average fertility is above the RLF, due to random fluctuations in birth numbers.
Conclusions
This study examined how demographic randomness affects the survival of small populations. It simplified assumptions like non-overlapping generations and a constant fertility rate using a Poisson distribution.
Despite these, the key insight remains robust: populations can go extinct even if they meet the standard RLF of 2.1, especially when small in size. The results show that extinction is almost certain for subcritical fertility levels, with only rare exceptions surviving. This highlights a risk for small or endangered populations and suggests that many family lineages may be statistically likely to go extinct over time.
A female-biased sex ratio can help mitigate this risk, suggesting that such biases seen under stress may serve an adaptive role in enhancing survival. This has been observed in both humans and other mammals
Journal reference:
- Threshold fertility for the avoidance of extinction under critical conditions. Cuaresma, D.C.N., Ito, H., Arima, H., Yoshimura, J., Morita, S., Okabe, T. PLoS ONE (2025). DOI: 10.1371/journal.pone.0322174, https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0322174