Aggregation methods in food chains with nutrient recycling

Abstract

This paper is devoted to the study of food chain models under batch and chemostat conditions where nutrient recycling is taken into account. The food chain is formed by a nutrient and two populations, prey and predator (producers and consumers). Species at both trophic levels digest their food source only partly. The unusable part of the food is ejected in the reactor as faeces together with metabolic products. The excreted material together with death material, detritus, is decomposed and this gives the recycling of the nutrient. In closed (batch-type environment) systems the elemental matter needed by producers must be provided through recycling where light energy from the environment supplies the necessary energy that fuels the life processes. In open (chemostat-type environment) systems this energy is added to the system via the chemical energy stored in the organic compounds in the inflow. Bifurcation analysis is used to study the effects of material recycling on the long-term dynamic behaviour of these simple food chains. An aggregation method is developed for situations in which each trophic level is characterized by differing time scales. This allows us to reduce the dimension of the model which gives good approximations after the fast transient. We will show that first-order approximations are needed in order to get the same qualitative long-term dynamics for both the full and the reduced model.