  This paper provides the phase transition analysis of a reaction diffusion
equations system modeling dynamic instability of microtubules. For this purpose
we have generalized the macroscopic model studied by Mour\~ao et all [MSS].
This model investigates the interaction between the microtubule nucleation,
essential dynamics parameters and extinction and their impact on the stability
of the system. The considered framework encompasses a system of partial
differential equations for the elongation and shortening of microtubules, where
the rates of elongation as well as the lifetimes of the elongating shortening
phases are linear functions of GTP-tubulin concentration. In a novel way, this
paper investigates the stability analysis and provides a bifurcation analysis
for the dynamic instability of microtubules in the presence of diffusion and
all of the fundamental dynamics parameters. Our stability analysis introduces
the phase transition method as a new mathematical tool in the study of
microtubule dynamics. The mathematical tools introduced to handle the problem
should be of general use.
