Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme
The authors treat the reaction-diffusion model proposed in the article [\textit{H. Degn} and \textit{D. E. F. Harrison}, ``Theory of oscillations of respiration rate in continuous culture of Klebsiella aerogenes'', J. Theor. Biol. 22, 238--248 (1969)] to explain the observed oscillatory behavior of respiration rate in the continuous cultures of the bacteria Klebsiella aerogenes, described by the system in dimensionless parameters \[ u_{t}-\Delta u=a-u-\frac{uv}{1+ku^{2}} \] \[ v_{t}-d\Delta v=b-\frac{uv}{1+ku^{2}} \] in \(\Omega\subset \mathbf{R}^{n},\) \(n\geq 1\) with homogeneous Neumann boundary conditions \(\partial_{v}u=\partial_{v}v=0\) on \(\partial\Omega\) and nonnegative continuous initial values \(u(x,0)=u_{0}(x)\geq 0,\) \(v(x,0)=v_{0}(x)\geq 0\) in \(\Omega.\) The formation of spatial and temporal patterns in this system are investigated with special attention to Turing patterns. Following Turing's idea of ``diffusion driven instability'' the relevant steady state problem is investigated. The obtained results include the global attracting of the constant steady state, the existence and nonexistence conditions of positive nonconstant steady states, the Hopf and steady state bifurcations, generation of spatiotemporal patterns.
inhomogeneous periodic oscillatory pattern
scientific article; zbMATH DE number 6140825