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 Xi: 1/Xi dXi/dt = ai — biXi — Xi/ciXi—1 — diXi+1/Xi, which includes the effects of intra—specific competition for fixed resources (the term biXi), intra—specific competition for renewable resources in the lower trophic level (the term Xi/ciXi—1), and consumers in the upper trophic level (the term diXi+1/Xi). 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/Xi dXi/dt = ai biXi — Xi/sj cijXjFrj(i) — sk dikXkFkc(i)/Xi, where Frj(i) is the fraction of population Xj that is a resource for i, and Fkc(i) is the fraction of population Xk that consumes i. This equation meets all the requirements for a general food web model. Some properties of the model are discussed.