Turing bifurcation and linear Hopf threshold in a diffusive predator–prey model with Smith growth and ratio-dependent Holling type Ⅲ functional response

Jiahong Li; Wenjie Li; Wanqin Wu; Xuewen Tan; Xinzhi Liu; Jiahong Li; Wenjie Li; Wanqin Wu; Xuewen Tan; Xinzhi Liu

doi:10.3934/era.2026323

Electronic Research Archive

2026, Volume 34, Issue 10: 7480-7499. doi: 10.3934/era.2026323

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Turing bifurcation and linear Hopf threshold in a diffusive predator–prey model with Smith growth and ratio-dependent Holling type Ⅲ functional response

1.
School of Mathematics and Computer Science, Yunnan Minzu University, Yunnan 650500, China
2.
Department of Mathematics, Kunming University of Science and Technology, Yunnan 650500, China
3.
Department of Applied Mathematics, University of Waterloo, Waterloo N2L 3G1, Canada

Received: 02 June 2026 Revised: 23 August 2026 Accepted: 24 August 2026 Published: 28 August 2026

This paper studies a diffusive predator–prey model with Smith growth, a ratio-dependent Holling type Ⅲ functional response, and Leslie–Gower predator dynamics. By analyzing the characteristic equations, we derive conditions for Turing instability and a linear stability threshold that signals the onset of Hopf-type oscillations. For the Turing branch, we prove the local existence of non-constant steady states bifurcating from the positive equilibrium, treating simple cases using the Crandall–Rabinowitz theorem. Numerical simulations are then carried out to illustrate these theoretical results and to explore the system's behavior near the intersection of the Turing and Hopf curves, where long-lived transient spatial patterns, spatially inhomogeneous periodic solutions, and complex spatio-temporal oscillatory patterns are observed. Overall, this work offers a mathematical account of how nonlinear growth, ratio-dependent predation, and spatial diffusion together shape the dynamics of this predator–prey system.
- Turing bifurcation,
- linear Hopf threshold,
- diffusive predator–prey model,
- ratio-dependent Holling type Ⅲ functional response,
- Smith growth
Citation: Jiahong Li, Wenjie Li, Wanqin Wu, Xuewen Tan, Xinzhi Liu. Turing bifurcation and linear Hopf threshold in a diffusive predator–prey model with Smith growth and ratio-dependent Holling type Ⅲ functional response[J]. Electronic Research Archive, 2026, 34(10): 7480-7499. doi: 10.3934/era.2026323

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Abstract

This paper studies a diffusive predator–prey model with Smith growth, a ratio-dependent Holling type Ⅲ functional response, and Leslie–Gower predator dynamics. By analyzing the characteristic equations, we derive conditions for Turing instability and a linear stability threshold that signals the onset of Hopf-type oscillations. For the Turing branch, we prove the local existence of non-constant steady states bifurcating from the positive equilibrium, treating simple cases using the Crandall–Rabinowitz theorem. Numerical simulations are then carried out to illustrate these theoretical results and to explore the system's behavior near the intersection of the Turing and Hopf curves, where long-lived transient spatial patterns, spatially inhomogeneous periodic solutions, and complex spatio-temporal oscillatory patterns are observed. Overall, this work offers a mathematical account of how nonlinear growth, ratio-dependent predation, and spatial diffusion together shape the dynamics of this predator–prey system.

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