  We consider the linear growth and fragmentation equation with general
coefficients. Under suitable conditions, the first eigenvalue represents the
asymptotic growth rate of solutions, also called \emph{fitness} or
\emph{Malthus coefficient} in population dynamics ; it is of crucial importance
to understand the long-time behaviour of the population. We investigate the
dependency of the dominant eigenvalue and the corresponding eigenvector on the
transport and fragmentation coefficients. We show how it behaves asymptotically
as transport dominates fragmentation or \emph{vice versa}. For this purpose we
perform suitable blow-up analysis of the eigenvalue problem in the limit of
small/large growth coefficient (resp. fragmentation coefficient). We exhibit
possible non-monotonic dependency on the parameters, conversely to what would
have been conjectured on the basis of some simple cases.
