Modeling the dynamics of woody plant-herbivore interactions with age-dependent toxicity

Rongsong Liu, Stephen A. Gourley, Donald L. DeAngelis, John P. Bryant

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


In this paper we study the effects that woody plant chemical defenses may have on interactions between boreal hares that in winter feed almost entirely on twigs. We focus particularly on the fact that toxin concentration often varies with the age of twig segments. The model incorporates the fact that the woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that these hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modeled using age-structured model equations which are reduced to a system of delay differential equations involving multiple delays in the woody plant-hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. This showed that age dependence in plant chemical defenses has the capacity to cause hare-plant population cycles, a new result.

Original languageEnglish (US)
Pages (from-to)521-552
Number of pages32
JournalJournal of Mathematical Biology
Issue number3
StatePublished - Sep 1 2012


  • Age-structured model
  • Boreal ecosystem
  • Delay-differential equations
  • Functional response
  • Oscillations
  • Plant chemical defense
  • Plant-herbivore interactions
  • Time lag
  • Woody plants

ASJC Scopus subject areas

  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics
  • Modeling and Simulation


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