Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1609–1626 | Cite as

Effect of parity on productivity and sustainability of Lotka–Volterra food chains

Bounded orbits in food chains
  • Nicole Massarelli
  • Kathleen HoffmanEmail author
  • Joseph P. Previte


Hairston, Slobodkin, and Smith conjectured that top down forces act on food chains, which opposed the previously accepted theory that bottom up forces exclusively dictate the dynamics of populations. We model food chains using the Lotka–Volterra predation model and derive sustainability constants which determine which species will persist or go extinct. Further, we show that the productivity of a sustainable food chain with even trophic levels is predator regulated, or top down, while a sustainable food chain with odd trophic levels is resource limited, which is bottom up, which is consistent with current ecological theory.


Sustainability Lotka–Volterra predator–prey equations Hairston, Slobodkin, and Smith conjecture Parity of food chains 

Mathematics Subject Classification (2000)

92D25 37N25 34C11 



We thank the reviewers for their insightful comments and helpful suggestions. J.P.P. acknowledges support from NSF-DMS-9987594, NSF-DMS-0236637, and NSF-DMS-0552148 for foundational work that led to the formulation of this problem.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nicole Massarelli
    • 1
  • Kathleen Hoffman
    • 1
    Email author
  • Joseph P. Previte
    • 2
  1. 1.University of Maryland Baltimore CountyBaltimoreUSA
  2. 2.Penn State ErieThe Behrend CollegeErieUSA

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