Spatiotemporal patterns and multiple bifurcations of a reaction- diffusion model for hair follicle spacing

Zhili Zhang; Aying Wan; Hongyan Lin; Zhili Zhang; Aying Wan; Hongyan Lin

doi:10.3934/era.2023099

Electronic Research Archive

2023, Volume 31, Issue 4: 1922-1947. doi: 10.3934/era.2023099

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Spatiotemporal patterns and multiple bifurcations of a reaction- diffusion model for hair follicle spacing

1.
School of Mathematics and Statistics, Hulunbuir University, Hailar, Inner Mongolia 021008, China
2.
Department of Science and Technology, Hulunbuir University, Hailar, Inner Mongolia 021008, China

Academic Editor: Fengqi Yi

Received: 29 December 2022 Revised: 28 January 2023 Accepted: 02 February 2023 Published: 13 February 2023

In this paper, the dynamical behaviors of a 2-component coupled diffusive system modeling hair follicle spacing is considered. For the corresponding ODEs, we not only consider the stability and instability of the unique positive equilibrium solutions, but also show the existence of unstable Hopf bifurcating periodic solutions. For the reaction-diffusion equations, we are mainly interested in the Turing instability of the positive equilibrium solution, as well as Hopf bifurcations and steady-state bifurcations. Our results showed that, under certain conditions, the reaction-diffusion system not only has Hopf bifurcating periodic solutions (both spatially homogeneous and non-homogeneous, all unstable), but it also has non-constant positive bifurcating equilibrium solutions. This allows for a clearer understanding of the mechanism for the spatiotemporal patterns of this particular system.
- reaction-diffusion model,
- Hopf bifurcation,
- steady-state bifurcation,
- Turing instability
Citation: Zhili Zhang, Aying Wan, Hongyan Lin. Spatiotemporal patterns and multiple bifurcations of a reaction- diffusion model for hair follicle spacing[J]. Electronic Research Archive, 2023, 31(4): 1922-1947. doi: 10.3934/era.2023099

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Abstract

In this paper, the dynamical behaviors of a 2-component coupled diffusive system modeling hair follicle spacing is considered. For the corresponding ODEs, we not only consider the stability and instability of the unique positive equilibrium solutions, but also show the existence of unstable Hopf bifurcating periodic solutions. For the reaction-diffusion equations, we are mainly interested in the Turing instability of the positive equilibrium solution, as well as Hopf bifurcations and steady-state bifurcations. Our results showed that, under certain conditions, the reaction-diffusion system not only has Hopf bifurcating periodic solutions (both spatially homogeneous and non-homogeneous, all unstable), but it also has non-constant positive bifurcating equilibrium solutions. This allows for a clearer understanding of the mechanism for the spatiotemporal patterns of this particular system.

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