Effect of Parity on Boundedness of Orbits in Lotka-Volterra Food Chains

Abstract

Hairston, Slobodkin, and Smith (HSS) conjectured that top-down forces act on food chains with three trophic levels, which opposed the previously accepted theory that bottom-up forces dictate the dynamics of populations. HSS argued that plant life is abundant because carnivores prey upon herbivores, which prevents plants from being depleted. From this hypothesis Fretwell inferred that HSS could be applied to food chains with greater or fewer number of trophic levels. The exploitation ecosystem hypothesis (EHH) extends HSS to a food chain of any length. According to EHH in food chains with an odd number of trophic levels plants are resource limited, which indicates bottom-up forces control the dynamics of the food chain. Alternatively, in food chains with an even number of species plants are limited by the grazing of herbivores. Thus, for an even number of species top-down forces act on the food chain. This implies that the plant population can only increase in an odd level food chain. Volterra extended his original model to include any number of species interacting. This simple model contained only birth/death terms and interaction terms. Using this model, Volterra proved that for an even number of species in a food chain, all species persist and remain bounded. The ecological hypotheses are supported by Volterra's work on species interaction. We will use invariant surfaces to prove Volterra's work, and also extend it to show that for an odd number of species in a food chain, unbounded orbits exist. Thus, we mathematically validate the long-debated HSS conjecture.