Self-similarity in a General Aggregation-Fragmentation Problem ; Application to Fitness Analysis

We consider the linear growth and fragmentation equation ∂ ∂t u(x, t) + ∂ ∂x ( τ(x)u ) + β(x)u = 2 ∫ ∞ x β(y)κ(x, y)u(y, t) dy, with general coefficients τ, β and κ. Under suitable conditions (see [17]), the first eigenvalue represents the asymptotic growth rate of solutions, also called the fitness or Malthus coefficient in population dynamics. This value is of crucial importance in understanding the long-time behavior of the population. We investigate the dependence of the dominant eigenvalue and the corresponding eigenvector on the transport and fragmentation coefficients. We show how it behaves asymptotically depending on whether transport dominates fragmentation or vice versa. For this purpose we perform a suitable blow-up analysis of the eigenvalue problem in the limit of a small/large growth coefficient (resp. fragmentation coefficient). We exhibit a possible non-monotonic dependence on the parameters, in contrast to what would have been conjectured on the basis of some simple cases.