Journal of Guangxi Normal University(Natural Science Edition) ›› 2026, Vol. 44 ›› Issue (6): 122-133.doi: 10.16088/j.issn.1001-6600.2026031303

• Mathematics • Previous Articles     Next Articles

Periodic solutions of a predator system with cross-diffusion

Yuan Hailong*, Dang Nan, Liu Shushu   

  1. School of Mathematics & Data Science, Shaanxi University of Science & Technology, Xi’an Shaanxi 710021, China
  • Received:2025-11-18 Revised:2026-03-08 Online:2026-11-05 Published:2026-11-10

Abstract: The dynamical behavior of a predator system with Smith growth and cross-diffusion is investigated. Firstly, the direction of Hopf bifurcation and the stability of periodic solutions for the ODE are analyzed using normal form theory and center manifold theorem. Secondly, an analytical criterion for cross-diffusion inducing the Turing instability of the periodic solutions is derived through Floquet theory and eigenvalue perturbation method. The results show that when cross-diffusion coefficients exceed a critical value, the originally stable spatially homogeneous periodic solutions lose stability under spatial perturbations, leading to spatial pattern formation. Finally, numerical simulations are performed using Matlab to verify the theoretical results.

Key words: predator system, cross-diffusion, periodic solution, Turing instability, Hopf bifurcation

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