From this expression, $\beta_{G}$, $\beta_{E}$, $\beta_{G*E}$ can be seen as coefficients which allow quantifying the participation of each factors (Genotype, Environment and interaction Genotype/Environment).
Then,
In mathematical term, it leads to a linear expression such:
$$
P = \beta_{G}*G + \beta_{E}*E + \beta_{G*E}*G.E + \beta_{0}
$$
In order to estimate these coefficients, a Generalized Linear Model (GLM) can be used.