The concentration and localization phenomena play an important role in natural science and engineering. Various concentration effects can be observed in population biology, ecology, mechanics, and in engineering technological processes. For example, study ing when populations of bacteria or viruses are localized is very important in medicine. The models used are described with the help of linear or nonlinear differential equations and corresponding spectral problems, i.e the problems of finding the natural frequencies and modes.
The main objectives of the present project is to study analytically and numerically the localization effects arising in spectral problems of mathematical physics, and to use the obtained results for prediction of various localizat ion phenomena. We study a particular case when spectral problem is considered in a locally periodic perforated domain, where the size of the "holes" varies. The typical size of the perforation is h, h being a small parameter, is much smaller than the size of the domain.
It is expected that the eigenfunctions are asymptotically localized in the vicinity of the "biggest hole".
The homogenization theory and asymptotic analysis of singularly perturbed equations are relevant and very suitable technical tool s for studying this problem due to the presence of the small parameter h. Classical methods have to be adapted to our problem because of the localization effect and locally periodic geometry.