Delayed predator-prey dynamics with Holling type Ⅳ functional response and rational nonlinear harvesting

Wei Liu; Wei Liu

doi:10.3934/era.2026029

Electronic Research Archive

2026, Volume 34, Issue 1: 627-656. doi: 10.3934/era.2026029

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Delayed predator-prey dynamics with Holling type Ⅳ functional response and rational nonlinear harvesting

Wei Liu ^,

School of Information Management and Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China

Received: 21 November 2025 Revised: 28 December 2025 Accepted: 06 January 2026 Published: 20 January 2026

A Holling type Ⅳ predator-prey system with a rational nonlinear harvesting rate and gestation delay of prey species is studied, which is formulated by delayed differential-algebra equations. Its dynamical behaviors are investigated in terms of differential-algebra system theory, bifurcation theory, and center manifold theorem. By choosing the gestation delay as a bifurcation parameter, we first show that Hopf bifurcations can occur as the delay increases through a sequence of threshold values. Second, we derive an explicit algorithm for determining the stability and direction of the Hopf bifurcations. Last, some numerical simulations are performed to illustrate the analytical results, and their biological significances are explained.
- Holling type Ⅳ,
- nonlinear harvesting,
- stability switches,
- Hopf bifurcation,
- periodic orbits
Citation: Wei Liu. Delayed predator-prey dynamics with Holling type Ⅳ functional response and rational nonlinear harvesting[J]. Electronic Research Archive, 2026, 34(1): 627-656. doi: 10.3934/era.2026029

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Abstract

A Holling type Ⅳ predator-prey system with a rational nonlinear harvesting rate and gestation delay of prey species is studied, which is formulated by delayed differential-algebra equations. Its dynamical behaviors are investigated in terms of differential-algebra system theory, bifurcation theory, and center manifold theorem. By choosing the gestation delay as a bifurcation parameter, we first show that Hopf bifurcations can occur as the delay increases through a sequence of threshold values. Second, we derive an explicit algorithm for determining the stability and direction of the Hopf bifurcations. Last, some numerical simulations are performed to illustrate the analytical results, and their biological significances are explained.

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