Modeling and Analysis of Compositional Data

9.5 Elementary differential equations

Differential equations involve unknown functions and their derivatives and, if they have solution(s), they can be considered as models of evolution in time. When the unknown function is an sv-function and the derivative is simplicial, they are called simplicial differential equations. Herein, simplicial differential equations are viewed as a way of obtaining models of compositional processes with special emphasis in processes that come from simple equations. Some of those models are also simple when formulated directly on the sv-function itself, but there are cases in which, although the sv-function is quite complex, the corresponding simplicial differential equation is easily interpretable.

A simplicial differential equation only gives relative information between the parts involved, and it does not convey any information about absolute abundances or masses. This fact, although obvious from the principles of compositional analysis, may cause misinterpretations and surprise to the analyst used to interpret models in terms of absolute abundances.

First-order differential equations can be expressed as implicit expressions, which are out of the scope of this introduction. Attention is paid to explicit first-order simplicial differential equations, that is, where the derivative is isolated,

9.12

with . When does not depend on explicitly, that ...