  We present a thorough study of the impact of a migrating planet on a
planetesimal disk, by exploring a broad range of masses and eccentricities for
the planet. We discuss the sensitivity of the structures generated in debris
disks to the basic planet parameters. We perform many N-body numerical
simulations, using the symplectic integrator SWIFT, taking into account the
gravitational influence of the star and the planet on massless test particles.
A constant migration rate is assumed for the planet. The effect of planetary
migration on the trapping of particles in mean motion resonances is found to be
very sensitive to the initial eccentricity of the planet and of the
planetesimals. A planetary eccentricity as low as 0.05 is enough to smear out
all the resonant structures, except for the most massive planets. The
planetesimals also initially have to be on orbits with a mean eccentricity of
less than than 0.1 in order to keep the resonant clumps visible. This numerical
work extends previous analytical studies and provides a collection of disk
images that may help in interpreting the observations of structures in debris
disks. Overall, it shows that stringent conditions must be fulfilled to obtain
observable resonant structures in debris disks. Theoretical models of the
origin of planetary migration will therefore have to explain how planetary
systems remain in a suitable configuration to reproduce the observed
structures.
