Bifurcation of traveling spot patterns in 2-dimensional domain

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Bifurcation problem of traveling spot patterns in 2-dimensional domain is studied. Our model equations are 2- component reaction-diffusion systems including a non-local term. . Numerically they exhibit traveling spot patterns which is bifurcated from standing spot patterns stably. Theoretically we know that 2-component reaction-diffusion systems without a non-local term do not exhibit such phenomena. Then first we considered that such phenomena came from the non-local term. But later we could succeed in rewriting the 2-component systems with the non-local term into the 3-component reaction-diffusion systems without a non-local term. We conclude that such phenomena come from the delicate balance of the ratio of diffusions, the ratio of reactions and the nonlinear terms. This enables us to analyze them mathematically. In 1-dimensional case, we showed the existence and the stability of traveling front and back solutions under the assumption of bistability. Now we try to construct standing pulse solutions connecting these traveling front and back solutions. For the stability property, we will show that standing pulse solutions destabilize under the out-of-phase (asymmetric) modes first and the in-phase (symmetric) modes secondly when some parameter is changed. At this time, the informatin of the stability of traveling front and back solutions and the connecting manner of these two solutions will help us to analyze the stability property. These new bifurcated solutions are stable traveling pulse solutions. In 2-dimensional case, it is confirmed numerically that these systems have stable 2-dim traveling spot patterns. For this case, we will show the existence of radial symmetric standing pulse solutions and then catch traveling spot patterns as the destabilization of radial symmetric standing pulse solutions.