Bifurcation structure of a three-species food chain model
Since the time of Lotka (1925) and Volterra (1926), the use of ordinary differential equations in consumer-resource population modelling has been a source of great contention. These simple and general models have offered insights into community structure and other population phenomena. However, biologists have criticized the approach for its lack of biological rigor. Parameters are abstractly defined and do little to bridge the gap between theorists and empiricists. Recently, \textit{P. Yodzis} and \textit{S. Innes} [Am. Nat. 139, 1151-1175 (1992)] have begun to cross this void by clarifying the derivation of these parameters using well-known allometric and energetic relations. We will extend their model from a two-dimensional context to the dynamically more interesting three-dimensional simple food chain model. The ability to utilize body size relationships among consumers and resources in parameter estimation enables this increase in dimension -- and the corresponding increase in parameters -- to be executed judiciously. Any jump in dimension within a modelling context is dangerous as informational error may be multiplied. With the aid of Yodzis and Innes' estimation technique we now have a tool with which to measure the biological ``plausibility'' of any parametric setting by calculating consumer-resources mass ratios and comparing them to empirically derived relationships. Our analysis is undertaken with the primary motivation of gaining insight into trophodynamic behaviour of three-dimensional systems. Population models have traditionally been concerned primarily with local stability structure in qualifying dynamic behaviour. This paper emphasizes a more global perspective and finds that simple local structures can have a relatively complicated set of global phase spaces.
three-dimensional simple food chain model
scientific article; zbMATH DE number 820077