JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (10): 32-42, 53.doi: 10.6040/j.issn.1671-9352.0.2023.279

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Hopf bifurcation of a vegetation-water reaction-diffusion model with time delay

Gaihui GUO(),Jingjing WANG,Wangrui LI   

  1. School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, Shaanxi, China
  • Received:2023-06-27 Online:2023-10-20 Published:2023-10-17

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Abstract:

Taking the time delay τ as the bifurcation parameter, the effect of time delay on the stability of the positive steady state point and the existence of Hopf bifurcation are given by analyzing the characteristic equation. The criteria for the direction of Hopf bifurcation and the stability of periodic solutions are obtained by the normal form theory and the center manifold theorem. Finally, the theoretical results are verified by numerical simulations.

Key words: vegetation-water model, time delay, Hopf bifurcation, stability

CLC Number: 

  • O175.26

Fig.1

Parameter a=14.74>a0, positive equilibrium point (u*, v*) of system (3) is locally asymptotically stable without time delay and diffusion"

Fig.2

Parameter a=14.240, system (3) produces stable periodic closed orbits without time delay and diffusion"

Fig.3

Parameter τ=0.46 < τ00, the positive equilibrium point (u*, v*) of system (3) is locally asymptotically stable"

Fig.4

Parameter τ=0.57>τ00, system (3) produces stable bifurcating periodic solutions"

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