• Zhou, Jun (School of Mathematics and Statistics Southwest University)
  • Received : 2016.03.23
  • Published : 2017.03.31


In this paper, we study the pattern formation to a general Degn-Harrison reaction model. We show Turing instability happens by analyzing the stability of the unique positive equilibrium with respect to the PDE model and the corresponding ODE model, which indicate the existence of the non-constant steady state solutions. We also show the existence periodic solutions of the PDE model and the ODE model by using Hopf bifurcation theory. Numerical simulations are presented to verify and illustrate the theoretical results.


Supported by : CQC-STC, Central Universities, NSFC, China Postdoctoral Science Foundation


  1. H. Degn and D. E. F. Harrisson, Theory of oscillations of respiration rate in continuous culture of klebsiella aerogenes, J. Theoret. Biol. 22 (1969), no. 22, 238-248.
  2. B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981.
  3. S. Li, J. Wu, and Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations 259 (2015), no. 5, 1990-2029.
  4. R. Peng, F. Q. Yi, and X. Q. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations 254 (2013), no. 6, 2465-2498.
  5. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), no. 641, 37-72.
  6. J. Wang, J. Shi, and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations 251 (2011), no. 4-5, 1276-1304.
  7. S. Wiggins and M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.
  8. F. Yi, J. Wei, and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl. 9 (2008), no. 3, 1038-1051.
  9. F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations 246 (2009), no. 5, 1944-1977.