Stability and bifurcation analysis of reaction-diffusion systems with delays

Hu, Rui (2009) Stability and bifurcation analysis of reaction-diffusion systems with delays. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

The work focuses on the stability of steady state and local bifurcation analysis in partial differential equations with different delays. Especially, a neural network model with discrete delay and diffusion is proposed in the first part; a diffusive competition model with uniformly distributed delay is studied in part 2. An extended reaction-diffusion system with general distributed delay is treated in part 3. In the last part, a Nicholson's blowflies model with nonlocal delay and diffusion is discussed. -- For a diffusive neural network model with discrete delay, by analyzing the distributions of the eigenvalues of the system and applying the center manifold theory and normal form computation, we show that, regarding the connection coefficients as the perturbation parameter, the system, with different boundary conditions, undergoes some bifurcations including transcritical bifurcation, Hopf bifurcation and Hopf-zerobifurcation. The normal forms are given to determine the stabilities of the bifurcated solutions. -- In some cases, the model with distributed delay is more accurate than that with discrete delay. We study a competition diffusion system with uniformly distributed delay. The complete analysis of the characteristic equation is given. And via the analysis, the stability of the constructed positive spatially non-homogeneous steady state solution is obtained. Moreover, the occurrence of Hopf bifurcation near the steady state solution is proved by using the implicit function theorem with time delay as the bifurcation parameter. Finally, the formula determining the stability of the periodic solutions is given. -- The uniformly distributed kernel is only one of the widely used time kernel. It is natural to discuss more general time kernels. We consider a class of reaction-diffusion system with general kernel functions. The stability of the constructed positive spatially non-homogeneous steady state solution is obtained under general kernels by using the similar method in part 2. Moreover, taking minimal time delay as the bifurcation parameter, we can not only show the existence of Hopf bifurcations near the steady state solution, but also prove that the Hopf bifurcation is forward and the bifurcated periodic solutions are stable under certain condition. The general results are applied to competitive and cooperative systems with weak kernel function. -- In many application models, if individuals move, it is more reasonable to model delay and diffusion simultaneously, which induces nonlocal delay by employing Britton's random walk method. We study the stability of the uniform steady states and Hopf bifurcation of diffusive Nicholson's blowflies equation with nonlocal delay. By using the upper- and lower- solutions method, we have obtained the global stability conditions at the constant steady states, and discussed the local stability. Moreover, for a special kernel, we have proved the occurrence of Hopf bifurcation near the steady state solution and given formula in determining stability of bifurcated periodic solutions.

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