In this paper we formally prove that invading carnivores in thediscrete food-chain derived and preliminary analyzed inLindström (2002) always makes thesystem less stable and thus, limit the food-chain length in thecorresponding system. Hence, invading unsaturated carnivores arenot able to stabilize oscillatory dynamics.

What we prove constitutes a significant difference betweendiscrete and continuous food-chains. Actually, Freedman andWaltman (1977) related the stabilizingproperties of an invading carnivore in continuous food-chains to absence of saturation: An unsaturated carnivore keeps at least oneinterior equilibrium - if one exists - locally stable.

One consequence is that the dynamics of unsaturated discretefood-chains display similarities with saturated continuousfood-chains. Indeed, discrete dynamics seem to have a similardestabilizing impact on the dynamics as saturation has.