Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions

Weiyu Li; Hongyan Wang; Weiyu Li; Hongyan Wang

doi:10.3934/era.2023211

Electronic Research Archive

2023, Volume 31, Issue 7: 4139-4154. doi: 10.3934/era.2023211

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Research article

Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions

Weiyu Li ,
Hongyan Wang ^,

College of Science, Heilongjiang Bayi Agricultural University, Daqing 163319, China

Received: 04 April 2023 Revised: 10 May 2023 Accepted: 22 May 2023 Published: 29 May 2023

In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.
- schnakenberg model,
- cross-diffusion,
- spatially homogeneous periodic solutions,
- Turing instability
Citation: Weiyu Li, Hongyan Wang. Dynamics of a three-molecule autocatalytic Schnakenberg model with cross-diffusion: Turing patterns of spatially homogeneous Hopf bifurcating periodic solutions[J]. Electronic Research Archive, 2023, 31(7): 4139-4154. doi: 10.3934/era.2023211

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Abstract

In this paper, a three-molecule autocatalytic Schnakenberg model with cross-diffusion is established, the instability of bifurcating periodic solutions caused by diffusion is studied, that is, diffusion can destabilize the stable periodic solutions of the ordinary differential equation (ODE) system. First, utilizing the local Hopf bifurcation theory, the central manifold theory, the normal form method and the regular perturbation theory of the infinite dimensional dynamical system, the stability of periodic solutions for the ODE system is discussed. Second, for this model, according to the implicit function existence theorem and Floquet theory, the Turing instability of spatially homogeneous Hopf bifurcating periodic solutions is studied. It is proved that the otherwise stable Hopf bifurcating periodic solutions in the ODE system produces Turing instability in the Schnakenberg model with cross-diffusion. Finally, through numerical simulations, it is verified that Turing instability of periodic solutions is determined by cross-diffusion rates.

References

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