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Possibly Longest Food Chain: Analysis of a Mathematical Model

Published online by Cambridge University Press:  04 December 2008

T. Matsuoka  and
H. Seno
T. Matsuoka*
Affiliation:
Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, Kagamiyama 1-3-1, Higashi-hiroshima, 739-8526 Japan
H. Seno
Affiliation:
Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, Kagamiyama 1-3-1, Higashi-hiroshima, 739-8526 Japan
*
seno@math.sci.hiroshima-u.ac.jp
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Abstract

We consider the number of trophic levels in a food chain given by the equilibrium state for a simple mathematical model with ordinary differential equations which govern the temporal variation of the energy reserve in each trophic level. When a new trophic level invades over the top of the chain, the chain could lengthen by one trophic level. We can derive the condition that such lengthening could occur, and prove that the possibly longest chain is globally stable. In some specific cases, we find that the possibly longest chain is such that the lower trophic level has a greater energy reserve than the higher has, so that the distribution of energy reserves can be regarded to have a pyramid shape, whereas, if any of its trophic levels is removed, the pyramid shape cannot be maintained. Further, we find the condition that arbitrary long chain can be established. In such unbounded case, we prove that any chain could not have the pyramid shape of energy reserve distribution.

Keywords

food chainenergy reservetrophic levelmathematical model
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 3 , Issue 4: Patterns and waves in ecology and evolution (2) , 2008 , pp. 131 - 160
Copyright
© EDP Sciences, 2008

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