Deterministic Multivariate Evolution

Abstract

The discussions of deterministic evolution in Chap. 7 were focused on the modeling of the evolution of one variable (as heat, mass, the position of any body, or a population density). The consideration of such relatively simple problems is helpful for a basic understanding of the structure and the range of applicability of equations for typical problems. However, only a narrow range of problems can be described in this way: the analysis of most real problems requires the consideration of the multivariate evolution of several variables. The latter is required, for example, regarding the interaction of biological species and motions of bodies or fluids in three-dimensional space. To deal with such cases we extend here the concepts used for the modeling of mechanical and population ecology processes in Chap. 7 to the modeling of the joint evolution of several variables. We will continue with the consideration of global properties that change in time but not in space, i.e., partial differential equations that describe the evolution of processes in space will be not considered. The mathematics of models for the evolution of such processes can be formulated in terms of linear and nonlinear systems of coupled ordinary differential equations.