Bifurcation structure of a three-species food chain model
type: Item
instance of: scholarly article
title: Bifurcation structure of a three-species food chain model
zbMATH Open document ID: 0854.92022
author: Peter Yodzis
author: Kevin McCann
published in: Theoretical Population Biology
publication date: 1997-01-19T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 92D40
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37N99
zbMATH DE Number: 820077
zbMATH Keywords: type II functional responses
zbMATH Keywords: allometric relationships
zbMATH Keywords: equilibria structure
zbMATH Keywords: local bifurcation analysis
zbMATH Keywords: domains of attraction
zbMATH Keywords: quasi-periodicity
zbMATH Keywords: chaos
zbMATH Keywords: homoclinic events
zbMATH Keywords: transient chaos
zbMATH Keywords: consumer-resource population modelling
zbMATH Keywords: three-dimensional simple food chain model
zbMATH Keywords: body size relationships
zbMATH Keywords: trophodynamic behaviour
zbMATH Keywords: global phase spaces
MaRDI profile type: Publication
full work available at URL: tpbi.1995.1023
OpenAlex ID: W2042626082
DOI: 10.1006/TPBI.1995.1023
description: scientific article; zbMATH DE number 820077