CPAA
Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis
Fengqi Yi Hua Zhang Alhaji Cherif Wenying Zhang
In this paper, a homogeneous reaction-diffusion model describing the control growth of mammalian hair is investigated. We provide some global analyses of the model depending upon some parametric thresholds/constraints. We find that when one of the dimensionless parameter is less than one, then the unique positive equilibrium is globally asymptotically stable. On the contrary, when this threshold is greater than one, the existence of both steady-state and Hopf bifurcations can be observed under further parametric constraints. In addition, we find that both spatially homogeneous and heterogeneous oscillatory solutions can be seen for some spatially independent parameters provided that some conditions are met. Under these conditions, the direction and stability of these oscillatory behaviors, global stability of the unique constant steady state and the local orbital asymptotic stability of the spatially homogeneous periodic orbits are also investigated.
keywords: Lyapunov function. spatiotemporal patterns steady state and Hopf bifurcations Diffusive hair growth model globally asymptotically stable
DCDS-B
On spatiotemporal pattern formation in a diffusive bimolecular model
Rui Peng Fengqi Yi
This paper continues the analysis on a bimolecular autocatalytic reaction-diffusion model with saturation law. An improved result of steady state bifurcation is derived and the effect of various parameters on spatiotemporal patterns is discussed. Our analysis provides a better understanding on the rich spatiotemporal patterns. Some numerical simulations are performed to support the theoretical conclusions.
keywords: Diffusive bimolecular model steady state bifurcation. spatiotemporal pattern autocatalysis and saturation law
DCDS-B
The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system
Fengqi Yi Eamonn A. Gaffney Sungrim Seirin-Lee

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

keywords: Schnakenberg model delay and diffusion normal form method Hopf bifurcation diffusion driven instability delay-diffusion driven instability

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