We study the global (in time) existence of nonnegative solutions of the Gierer-Meinhardt system with mixed boundary conditions. In the research, the Robin boundary and Neumann boundary conditions were used on the acti...We study the global (in time) existence of nonnegative solutions of the Gierer-Meinhardt system with mixed boundary conditions. In the research, the Robin boundary and Neumann boundary conditions were used on the activator and the inhibitor conditions respectively. Based on the priori estimates of solutions, the considerable results were obtained.展开更多
In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric an...In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric and asymmetric twospike solutions using a Lyapunov-Schmidt reduction.The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain 2×2 matrix.Our rigorous results are complemented by formal calculations of N-spike solutions using the method of matched asymptotic expansions.In addition,we explicitly consider examples of one-and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds.By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities.Furthermore,when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity.However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.展开更多
Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simul...Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization.We numerically show the convergence of our algorithm.Pattern transitions in irregular domains are shown.We also provide various parameter settings on some irregular domains for different patterns appeared in nature.To further simulate patterns in reality,we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.展开更多
The Gierer-Meinhardt reaction-diffusion model is analyzed using a spectralcollocation method. This reaction-diffusion system is governed by activator and inhibitorconcentrations. Initially, the system is considered in...The Gierer-Meinhardt reaction-diffusion model is analyzed using a spectralcollocation method. This reaction-diffusion system is governed by activator and inhibitorconcentrations. Initially, the system is considered in one dimension and thenin two dimensions;numerical results are presented for both cases. The algorithmiccomplexity and accuracy are compared to those of a moving finite element method.Finally, observations are made concerning when to use the proposed spectral methodas opposed to the established moving mesh method.展开更多
文摘We study the global (in time) existence of nonnegative solutions of the Gierer-Meinhardt system with mixed boundary conditions. In the research, the Robin boundary and Neumann boundary conditions were used on the activator and the inhibitor conditions respectively. Based on the priori estimates of solutions, the considerable results were obtained.
文摘In this paper we consider the existence and stability of multi-spike solutions to the fractional Gierer-Meinhardt model with periodic boundary conditions.In particular we rigorously prove the existence of symmetric and asymmetric twospike solutions using a Lyapunov-Schmidt reduction.The linear stability of these two-spike solutions is then rigorously analyzed and found to be determined by the eigenvalues of a certain 2×2 matrix.Our rigorous results are complemented by formal calculations of N-spike solutions using the method of matched asymptotic expansions.In addition,we explicitly consider examples of one-and two-spike solutions for which we numerically calculate their relevant existence and stability thresholds.By considering a one-spike solution we determine that the introduction of fractional diffusion for the activator or inhibitor will respectively destabilize or stabilize a single spike solution with respect to oscillatory instabilities.Furthermore,when considering two-spike solutions we find that the range of parameter values for which asymmetric two-spike solutions exist and for which symmetric two-spike solutions are stable with respect to competition instabilities is expanded with the introduction of fractional inhibitor diffusivity.However our calculations indicate that asymmetric two-spike solutions are always linearly unstable.
基金supported by a Hong Kong Research Grant Council GRF Grant,and a Hong Kong Baptist University FRG Grant.
文摘Pattern formations by Gierer-Meinhardt(GM)activator-inhibitor model are considered in this paper.By linear analysis,critical value of bifurcation parameter can be evaluated to ensure Turing instability.Numerical simulations are tested by using second order semi-implicit backward difference methods for time discretization and the meshless Kansa method for spatially discretization.We numerically show the convergence of our algorithm.Pattern transitions in irregular domains are shown.We also provide various parameter settings on some irregular domains for different patterns appeared in nature.To further simulate patterns in reality,we construct different kinds of animal type domains and obtain desired patterns by applying proposed parameter settings.
基金the NSF Grant DMS-0453600.Special thanks are due to the Hong Kong Baptist University for their generous accommodations,specifically Dr.Tao Tang for his time and support.Also we would like to thank Dr.Graeme Fairweather for his guidance throughout this project,and Dr.Greg Fasshauer for his suggestions and contributions.
文摘The Gierer-Meinhardt reaction-diffusion model is analyzed using a spectralcollocation method. This reaction-diffusion system is governed by activator and inhibitorconcentrations. Initially, the system is considered in one dimension and thenin two dimensions;numerical results are presented for both cases. The algorithmiccomplexity and accuracy are compared to those of a moving finite element method.Finally, observations are made concerning when to use the proposed spectral methodas opposed to the established moving mesh method.
