On the Stability-complexity Relation for Unsaturated Semelpareous Discrete Food-chains

Abstract

In this paper we formally prove that invading carnivores in the discrete food-chain derived and preliminary analyzed in [2] always makes the system less stable and thus, limit the food-chain length in the corresponding system. Hence, invading unsaturated carnivores are not able to stabilize oscillatory dynamics. What we prove constitutes a significant difference between discrete and continuous food-chains. Actually, Freedman and Waltman [3] related the stabilizing properties of an invading carnivore in continuous food-chains to absence of saturation: An unsaturated carnivore keeps at least one interior equilibrium - if one exists - locally stable. One consequence is that the dynamics of unsaturated discrete food-chains display similarities with saturated continuous food-chains. Indeed, discrete dynamics seem to have a similar destabilizing impact on the dynamics as saturation has. Key Words: Stability-complexity relation; Discrete food-chains