  Extended systems governed by partial differential equations can, under
suitable conditions, be approximated by means of sets of ordinary differential
equations for global quantities capturing the essential features of the systems
dynamics. Here we obtain a small number of effective equations describing the
dynamics of single-front and localized solutions of Fisher-Kolmogorov type
equations. These solutions are parametrized by means of a minimal set of
time-dependent quantities for which ordinary differential equations ruling
their dynamics are found. A comparison of the finite dimensional equations and
the dynamics of the full partial differential equation is made showing a very
good quantitative agreement with the dynamics of the partial differential
equation. We also discuss some implications of our findings for the
understanding of the growth progression of certain types of primary brain
tumors and discuss possible extensions of our results to related equations
arising in different modelling scenarios.
