Abstract

Studies of the dynamics of plant-herbivore intera ctions that explicitly address model robustness are important in assessing uncertainty. Hence, identifying conditions that guarantee the global stability of plant-herbivore systems can be used to assess the rationale involved in, for example, the selection of management and/or control decisions. The model used here to illustrate these issues is naturally capable of supporting complex dynamics; the result of the explicit incorporation of plant toxicity in the functional response. Unlike the traditional Holling Type 2 functional response, the selected toxin-determined functional response loses its monotonicity at high levels of plant toxicity. Systems with nonmonotone functional responses are capable of supporting multiple interior equilibria and bistable attractors. Therefore, identifying conditions that guarantee global stability is not only mathematically challenging but important to scientists. We are able to find necessary and sufficient condition on the nonexistence of a closed orbit via the transformation of the model to a new equivalent system. The Poincaré-Bendixson theorem is used to show that the existence of a unique interior equilibrium point guarantees its global asymptotical stability whenever it is locally asymptotically stable. It is shown that, whenever there are multiple interior equilibria, the local stability of the "first interior equilibrium" implies model bistability. In other words, the phase space is characterized by two subregions: the basins of attraction of two stable equilibria, the interior and the boundary equilibria.

Original languageEnglish (US)
Pages (from-to)1002-1020
Number of pages19
JournalSIAM Journal on Applied Mathematics
Volume72
Issue number4
DOIs
StatePublished - 2012

Fingerprint

Functional Response
Global Dynamics
Interior
Toxicity
Global Stability
Model Robustness
Global Asymptotical Stability
Closed Orbit
Model
Bistability
Basin of Attraction
Interior Point
Local Stability
Complex Dynamics
Asymptotically Stable
Orbits
Equilibrium Point
Nonexistence
Monotonicity
Attractor

Keywords

  • Global stabili bistability
  • Plant-herbivore interaction
  • Toxin-determined functional response

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Global dynamics of a plant-herbivore model with toxin-determined functional response. / Castillo-Chavez, Carlos; Feng, Zhilan; Huang, Wenzhang.

In: SIAM Journal on Applied Mathematics, Vol. 72, No. 4, 2012, p. 1002-1020.

Research output: Contribution to journalArticle

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