Logistic Theory of Food Web Dynamics

Abstract

Classical food—web theory arises from Lotka—Volterra models. As an alternative, we develop a model from the logistic concept of demand and supply. We first extend the logistic to an arbitrary species in a trophic chain or stack by developing a simple equation for any population X i : 1/X i dX i /dt = a i — b i X i — X i /c i X i — 1 — d i X i + 1 /X i , which includes the effects of intra—specific competition for fixed resources (the term b i X i ), intra—specific competition for renewable resources in the lower trophic level (the term X i /c i X i — 1 ), and consumers in the upper trophic level (the term d i X i + 1 /X i ). This equation emerges from the basic logistic concept of demand and supply, as captured by the consumer/resource ratios, and fulfills all the requirements for a plausible food—chain equation. We then generalize the equation to any population in a food web of arbitrary complexity 1/X i dX i /dt = a i b i X i — X i /s j c i j X j F r j ( i ) — s k d i k X k F k c ( i ) /X i , where F r j ( i ) is the fraction of population X j that is a resource for i, and F k c ( i ) is the fraction of population X k that consumes i. This equation meets all the requirements for a general food web model. Some properties of the model are discussed.