Turing–Hopf bifurcation in a predator–prey model with cross-diffusion and herd behavior

Junci He; Dan Jin; Ruizhi Yang; Junci He; Dan Jin; Ruizhi Yang

doi:10.3934/era.2026243

Electronic Research Archive

2026, Volume 34, Issue 8: 5428-5451. doi: 10.3934/era.2026243

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Research article

Turing–Hopf bifurcation in a predator–prey model with cross-diffusion and herd behavior

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received: 08 June 2026 Revised: 22 June 2026 Accepted: 23 June 2026 Published: 29 June 2026

We studied a predator–prey reaction–diffusion system with cross-diffusion and a saturated herd-type functional response. The saturation effect introduces an extra ecological parameter that shifts both the Hopf and Turing–Hopf bifurcation thresholds away from those of the classical square-root model. The positive equilibrium was obtained in closed form, and linearization at this equilibrium yields characteristic equations for every spatial mode. Sharp conditions for local stability, Turing instability, and the codimension-two Turing–Hopf bifurcation were derived from these equations. A complete spectral description was provided so that the center spectrum at the bifurcation point consists of one pair of simple purely imaginary eigenvalues together with one simple zero eigenvalue. Using center manifold reduction and normal form theory for reaction–diffusion systems with cross-diffusion, we derived the third-order normal form on the center manifold. Numerical simulations verified the normal-form classification and displayed the stable spatially homogeneous and inhomogeneous steady states and periodic solutions that emerge near the bifurcation point.
- predator–prey model,
- herd behavior,
- cross-diffusion,
- Turing instability,
- Turing–Hopf bifurcation,
- normal form
Citation: Junci He, Dan Jin, Ruizhi Yang. Turing–Hopf bifurcation in a predator–prey model with cross-diffusion and herd behavior[J]. Electronic Research Archive, 2026, 34(8): 5428-5451. doi: 10.3934/era.2026243

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Abstract

We studied a predator–prey reaction–diffusion system with cross-diffusion and a saturated herd-type functional response. The saturation effect introduces an extra ecological parameter that shifts both the Hopf and Turing–Hopf bifurcation thresholds away from those of the classical square-root model. The positive equilibrium was obtained in closed form, and linearization at this equilibrium yields characteristic equations for every spatial mode. Sharp conditions for local stability, Turing instability, and the codimension-two Turing–Hopf bifurcation were derived from these equations. A complete spectral description was provided so that the center spectrum at the bifurcation point consists of one pair of simple purely imaginary eigenvalues together with one simple zero eigenvalue. Using center manifold reduction and normal form theory for reaction–diffusion systems with cross-diffusion, we derived the third-order normal form on the center manifold. Numerical simulations verified the normal-form classification and displayed the stable spatially homogeneous and inhomogeneous steady states and periodic solutions that emerge near the bifurcation point.

References

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