Concentration Phenomena in Diffusion and Cross-Diffusion Systems

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Proposal DMS-0400452Title: Concentration phenomena in diffusion and cross-diffusion systemsPI: Wei-Ming Ni (University of Minnesota- Twin Cities)(I) Technical Description:Professor Ni plans to continue his research in understandingmathematically the effects of various diffusion-related mechanisms. Usingvariational approach and techniques in classical analysis, Professor Niand his collaborators have established the spike-layer steady states foran activator-inhibitor system in morphogenesis proposed by Gierer andMeinhardt based on the celebrated idea - diffusion-driven instability - ofTuring in 1952. The method exploits the gap between the diffusion ratesfor the two chemical substances, and stability results in one spacedimension have also been obtained. Very recently, progress has been madefor concentration phenomena on multi-dimensional subsets. However, thecomplete dynamics is still far from being fully understood. It has beennoted that the gap between the diffusion rates alone is insufficient tocreate patterns. Thus, the notion of "cross-diffusion," introduced bytheoretical biologists in 1979 in modeling segregation phenomena inpopulation dynamics, deserves systematic studies. Cross-diffusion systems,which have also been used in recent years to model singular phenomenaincluding dendritic growth of bacteria colonies, are both nonlinear andstrongly-coupled in highest order terms, thus are mathematically verychallenging. Professor Ni and his collaborators propose to study theeffects of cross-diffusion by first obtain necessary and sufficientconditions for cross-diffusion to help create patterns, and then toinvestigate their qualitative behavior as well as their stabilityproperties.(II) Non-technical Descriptions:Professor Ni plans to continue his research in understanding, in amathematically rigorous manner, the phenomena and effects of variousdiffusion-related mechanisms and hopefully thereby to have some impact inboth improving our ability in modeling more complicated and/or realisticphenomena in applied sciences, as well as in creating new and significantmathematics. In this proposal, from the viewpoint of pattern formation,Professor Ni intends to investigate the various "concentration phenomena"in diffusion and/or cross-diffusion systems. These, in particular, includeTuring patterns in chemical reactions (e.g. the CIMA reaction),Gierer-Meinhardt's activator-inhibitor systems in modeling theregeneration phenomena of hydra in morphogenesis, a nonlinear diffusionsystem modeling dendritic growth of bacteria colonies, and theLotka-Volterra competition systems with inter-specific populationpressures taken into considerations."