The book examines the properties of developing systems, formalized mapping and the study of which requires the development of existing models of Dynamical Systems. A new class of dynamic models is introduced, based on nonlinear integro-functional equations with prehistory. Proven theorems about the existence and uniqueness of solutions , qualitative and numerical studies of the properties of solutions , Various optimization tasks have been solved , The problems of modeling the distribution of resources between the branches of macroeconomics of groups A and B , modeling of some mechanisms of regulation of cell redistribution in immunocompetent and hematopoietic tissues , modeling the process of sugar accumulation in sugar beet and a number of other tasks. Various modeling tools of developing systems. are considered