In many species, small populations shrink on their own. Mates are harder to find, groups defend themselves less well, and cooperative foraging breaks down. Ecologists call this the Allee effect. When it is "strong", a population has a tipping point: below a critical abundance it declines to extinction, even though a larger population in the same environment would persist.
What happens when such a population is also harvested by hunting, fishing or logging, and the removal targets individuals of a particular age?
In a new paper in Computational Mathematics and Modeling (Springer, 2026), we study this with the McKendrick–von Foerster model. In this transport equation, age "flows" at one year per year. Natural mortality and harvesting act as losses along the way, and births enter through a boundary condition. The recruitment law captures both extremes: an Allee effect at low abundance and crowding at high abundance.
The main result is an explicit threshold formula:
K_cr = A / (√R₀ − 1)²
Here A is the Allee scale, K is the density-limitation scale (the "capacity" of the environment), and R₀ is the reproductive number. R₀ carries all the age information: who reproduces at what age, who survives to that age, and whom harvesting removes. If K exceeds K_cr, the population has two equilibria. The lower one is unstable and is the extinction threshold; the upper one is stable. At K = K_cr the two merge, and below it only extinction remains. Mathematicians call this merger a saddle-node bifurcation. We prove that the lower equilibrium is unstable. We check the upper one numerically for the chosen parameters, without a general theorem.
An example with illustrative parameters (not fitted to any species): without harvesting, the threshold is K_cr ≈ 4,850. With harvesting centred on 35-year-olds at an intensity of 0.06 per year, it rises to about 14,500, and at 0.10 per year to about 38,500. That means an environment about eight times "richer" is needed for the population to persist at all. Age matters too. Removal before or during the main reproductive years shifts the threshold far more than the same removal of old individuals.
Simulations show the threshold as well. At K = 20,000, the lower equilibrium is about 1,000 individuals and the upper about 5,000. A population starting 1% below the threshold dies out, while one starting 1% above it settles at the stable level. For the computations, we built a positivity-preserving characteristic scheme. The computations check the proofs rather than replace them.
The main takeaway: selective harvesting does not just lower abundance, it moves the tipping point itself. In the same environment, removing young individuals can leave the population no way to persist, while removing old individuals does not.
Kudayeva F.Kh., Kaygermazov A.A. (Kabardino-Balkarian State University, Nalchik), Znamenskiy V.S. (BMCC, City University of New York). Explicit saddle-node extinction thresholds in a continuous age-structured population model with strong Allee recruitment and selective harvesting. Computational Mathematics and Modeling, 2026.
https://orcid.org/0000-0002-3428-7180
