  In this paper we study a reduced continuous model describing the local
evolution of high grade gliomas - a lethal type of primary brain tumor -
through the interplay of different cellular phenotypes. We show how hypoxic
events, even sporadic and/or limited in space may have a crucial role on the
acceleration of the growth speed of high grade gliomas. Our modeling approach
is based on two cellular phenotypes one of them being more migratory and the
second one more proliferative with transitions between them being driven by the
local oxygen values, assumed in this simple model to be uniform. Surprisingly
even acute hypoxia events (i.e. very localized in time) leading to the
appearance of migratory populations have the potential of accelerating the
invasion speed of the proliferative phenotype up to speeds close to those of
the migratory phenotype. The high invasion speed of the tumor persists for
times much longer than the lifetime of the hypoxic event and the phenomenon is
observed both when the migratory cells form a persistent wave of cells located
on the invasion front and when they form a evanecent wave dissapearing after a
short time by decay into the more proliferative phenotype.
  Our findings are obtained through numerical simulations of the model
equations. We also provide a deeper mathematical analysis of some aspects of
the problem such as the conditions for the existence of persistent waves of
cells with a more migratory phenotype.
