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January  2014, 13(1): 347-369. doi: 10.3934/cpaa.2014.13.347

## Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis

 1 Center for Partial Differential Equations, East China Normal University, Minhang, 200241, Shanghai, China 2 Department of Applied Mathematics, Harbin Engineering University, Harbin, 150001, Heilongjiang, China, China 3 Centre for Mathematical Biology, Mathematical Institute, University of Oxford, St Giles 24-29, OX1 3LB, United Kingdom

Received  March 2013 Revised  April 2013 Published  July 2013

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.
Citation: Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347
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