Deterministic Changes

Abstract

The models discussed in the previous chapters were based on observations: the goal was to develop models that agree with any given observations. A different modeling approach is the use of mathematical concepts that explain changes of variables (e.g., by means of a formula that provides the change of any population over a certain time). The comparison of predictions implied by such concepts with observations is usually needed to show the validity of the ideas applied, but the model development can be performed without the use of observations. Usually, such model developments are performed on the basis of difference equations or differential equations. Differential equations are applied for postulating general laws (e.g., the laws of mechanics or the laws of population ecology). Characteristic features of such equations for the deterministic evolution of single variables and several variables will be described in Chaps. 7 and 9, respectively. Difference equations are useful for the modeling of processes that require the modeling of changes over a specific time period (e.g., the change of any population per year). Apart from that, difference equations provide the basis for the mathematical formulation, numerical solution, and in some cases even the analytical solution of differential equations. Therefore, characteristic features of deterministic difference equations will be discussed in this chapter in order to prepare the discussion of differential equations in Chaps. 7 and 9.