Asymptotic analysis of regularly and singularly perturbed problems and their applications in biology

Abstract

absolutely and uniformly converges in some norm as |e| 0 before the derivative in system (1) and means the possibility of the appearance of so-called boundary layers in some neighborhood of t = 0 with certain limitations on the spectrum of the matrix A(t, 0). The singular perturbation theory, which significantly differs from the regular perturbation theory (see [4, 5, 10, 13]), in particular, was developed by A. N. Tihonov, A. B. Vasilyeva, V. F. Butuzov, S. A. Lomov, and others mathematicians, who made a great contribution to the study of singularly perturbed linear (see [11]) and nonlinear (see [15]) Cauchy and boundary-value problems. It was shown that the structure of boundary layers is defined by the structure of the limit (e = 0) operator of the linearized system. The questions on the behavior of singularly perturbed Cauchy problems on the semi-axis have not been quite elucidated in the literature. In well-known monographs (see, e.g., [11, 15]), singularly perturbed problems such that the definition of the limiting operator spectrum is rather simple were considered. The study of more general problems of the form (1) on the semi-axis and the analysis of stable solutions are connected to the search of eigenvalues λj(t, e) and eigenvectors sj(t, e) of matrix (2). This is a nontrivial problem of the regular perturbation theory, which requires quite cumbersome methods for its analysis (see [4, 5, 10, 13]). New results in this direction were obtained by Konyaev [7], who proposed a new algorithm (cf. [4, 5, 10, 13]) for the calculation of eigenvalues λj(e) and eigenvectors sj(e) of the regularly perturbed matrix