Mathematical studies on spatial and temporal patterns in reaction-diffusion systems

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Concerning the above research project, we obtained the following results.(1) We carried out the study on skew-gradient systems, which are a generalized activator-inhibitor systems. When a steady state is characterized as a min-max point of an energy functional, we showed that the steady state is stable regardless of parameter values. We also make clear the relation with the so-called Turing instability.(2) We carried out numerical simulations on a model equation which describes a spatial pattern formation observed In a colony of some bacteria. It is shown that such spatial patterns appear as a history of the process.(3) Two-species competition system is a main problem in the theory of mathematical ecology. We gave a mathematically rigorous verification for the fact that the system reduces to a Stephen -type free boundary value problem in a singular limit where the competition rate is infinitely large.(4) The Gierer-Meinhardt system is a mathematical model for the morphogenesis in Mathematical biology. We studied stability of spiky stationary patterns, and gave some criteria for the stability and Instability.