Hopf bifurcations in a reaction-diffusion population model with delay effect
The authors consider the following reaction diffusion system with time delay \[ {{\partial u(x,t)}\over{\partial t}} = d{{\partial^2 u(x,t)}\over{\partial x^2}} + \lambda u(x,t)f(u(x,t-\tau)), \quad x \in (0, 1), \] with Dirichlet boundary conditions \[ u(0, t) = u(l, t) = 0. \] The growth rate function \(f(.)\) satisfies \(f(0)=1\) and \(f'(0)<0\). Also the case is considered, that \(f(u)\) increases for small \(u\). For this system the authors first study the stationary case, when the delay vanishes and postulate the primary bifurcating solution branch. Then they investigate the influence of the delay on the stability of this branch and claim, that only Hopf bifurcations are possible for increasing delays. They also derive formulas for the bifurcating periodic solutions. Unfortunately the presentation of the material leaves out many details and doesn't seem to be completely correct: The solution of the system without delay also contains higher harmonics, not only the simple \(\sin(x\pi/l)\) contribution. Also the calculation of the shape of the critical eigenfunctions is hard to find in the article. The presentation is quite technical without sufficient explanation of the corresponding steps. So it seems impossible to judge from this presentation, whether the necessary steps for the derivation of the bifurcation behaviour is performed correctly. I cannot find the steps to correctly take the contributions by the stable modes into account, as that is accomplished by center manifold theory, for example.
scientific article; zbMATH DE number 5577112
scientific article