Predator extinction and total extinction in a non-autonomous predator–prey system with Allee effect and feedback control

Qin Yue; Fengde Chen; Qin Yue; Fengde Chen

doi:10.3934/era.2026344

Electronic Research Archive

2026, Volume 34, Issue 11: 8069-8100. doi: 10.3934/era.2026344

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Predator extinction and total extinction in a non-autonomous predator–prey system with Allee effect and feedback control

Qin Yue ¹,
Fengde Chen ^{2
,
,}

1.
School of Finance and Mathematics, West Anhui University, Lu'an 237000, China
2.
School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, China

Received: 08 July 2026 Revised: 01 September 2026 Accepted: 08 September 2026 Published: 22 September 2026

Lai et al. ^[1] recently studied a non-autonomous predator–prey system with an additive Allee effect, prey refuge, antipredator behavior, and indirect feedback control. Their Theorem 5.1 gives a sufficient condition for predator extinction and prey persistence. We show that its hypotheses involve unknown solution components and therefore cannot be checked a priori; additionally, we identify a sign error in the proof. Then, we replace that result with conditions stated entirely in terms of the prescribed coefficients and parameters. The first condition yields predator extinction while the prey remains uniformly persistent. Its main ingredient is an optimization bound for the predator's net benefit from the prey, which avoids the sign issue in the original argument. Additionally, we give three sufficient mechanisms for total extinction: strong intraspecific competition, uniformly bounded intrinsic prey growth combined with strong competition, and a quantitative integral condition involving the prey dissipativity bound. These conditions are not intended to be necessary, but they are directly checkable. For the autonomous model, we derive an explicit critical refuge level for predator invasion and discuss invasion numbers in autonomous and periodic settings. Numerical examples illustrate both the autonomous and non-autonomous cases.
- predator–prey system,
- Allee effect,
- non-autonomous system,
- extinction,
- persistence,
- feedback control,
- anti-predator behavior
Citation: Qin Yue, Fengde Chen. Predator extinction and total extinction in a non-autonomous predator–prey system with Allee effect and feedback control[J]. Electronic Research Archive, 2026, 34(11): 8069-8100. doi: 10.3934/era.2026344

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Abstract

Lai et al. ^[1] recently studied a non-autonomous predator–prey system with an additive Allee effect, prey refuge, antipredator behavior, and indirect feedback control. Their Theorem 5.1 gives a sufficient condition for predator extinction and prey persistence. We show that its hypotheses involve unknown solution components and therefore cannot be checked a priori; additionally, we identify a sign error in the proof. Then, we replace that result with conditions stated entirely in terms of the prescribed coefficients and parameters. The first condition yields predator extinction while the prey remains uniformly persistent. Its main ingredient is an optimization bound for the predator's net benefit from the prey, which avoids the sign issue in the original argument. Additionally, we give three sufficient mechanisms for total extinction: strong intraspecific competition, uniformly bounded intrinsic prey growth combined with strong competition, and a quantitative integral condition involving the prey dissipativity bound. These conditions are not intended to be necessary, but they are directly checkable. For the autonomous model, we derive an explicit critical refuge level for predator invasion and discuss invasion numbers in autonomous and periodic settings. Numerical examples illustrate both the autonomous and non-autonomous cases.

References

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