Taiwanese Journal of Mathematics

Mathematical Analysis on a Droop Model with Intraguild Predation

Sze-Bi Hsu, Yi-hui Ho, and Feng-Bin Wang

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In this paper, we analyze a predator-prey chemostat system with internal storage, in which the predator not only competes for a single inorganic nutrient with the prey species but also consumes the prey for growth. The outcome for the corresponding model without intraguild predation is that the competitive exclusion holds, that is, the superior species will win the competition, and coexistence will not happen. When the mechanism of intraguild predation is added into the system, our analysis indicates that coexistence can be possible.

Article information

Taiwanese J. Math., Volume 23, Number 2 (2019), 351-373.

Received: 26 June 2018
Revised: 5 October 2018
Accepted: 29 October 2018
First available in Project Euclid: 8 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C12: Monotone systems 34D20: Stability 92D25: Population dynamics (general)

Droop's model internal storage competition coexistence intraguild predation


Hsu, Sze-Bi; Ho, Yi-hui; Wang, Feng-Bin. Mathematical Analysis on a Droop Model with Intraguild Predation. Taiwanese J. Math. 23 (2019), no. 2, 351--373. doi:10.11650/tjm/181011. https://projecteuclid.org/euclid.twjm/1541667764

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  • M. Arim and P. A. Marquet, Intraguild predation: a widespread interaction related to species biology, Ecol. Lett. 7 (2004), no. 7, 557–564.
  • A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theoret. Biol. 84 (1980), no. 2, 189–203.
  • ––––, Transients and Oscillations in Continuous Culture, Mathematics in Microbiology, Academic Press, New York, 1983.
  • M. R. Droop, Some thoughts on nutrient limitation in algae, J. Phycol. 9 (1973), no. 3, 264–272.
  • K. J. Flynn, D. K. Stoecker, A. Mitra, J. A. Raven, P. M. Glibert, P. J. Hansen, E. Granéli and J. M. Burkholder, Misuse of the phytoplankton-zooplankton dichotomy: the need to assign organisms as mixotrophs within plankton functional types, J. Plankton Res. 35 (2013), no. 1, 3–11.
  • J. P. Grover, Resource competition in a variable environment: phytoplankton growing according to the variable-internal-stores model, Amer. Natur. 138 (1991), no. 4, 811–835.
  • ––––, Constant- and variable-yield models of population growth: Responses to environmental variability and implications for competition, J. Theoret. Biol. 158 (1992), no. 4, 409–428.
  • ––––, Resource Competition, Chapman & Hall, London, 1997.
  • ––––, Resource storage and competition with spatial and temporal variation in resource availability, Amer. Natur. 178 (2011), no. 5, E124–E148.
  • M. Hartmann, C. Grob, G. A. Tarran, A. P. Martin, P. H. Burkill, D. J. Scanlan and M. V. Zubkov, Mixotrophic basis of Atlantic oligotrophic ecosystems, Proc. Natl. Acad. Sci. 109 (2012), no. 15, 5756–5760.
  • R. D. Holt and G. A. Polis, A theoretical framework for intraguild predation, Amer. Natur. 149 (1997), no. 4, 745–764.
  • S.-B. Hsu, Ordinary Differential Equations with Applications, Series on Applied mathematics 16, World Scientific, Singapore, 2006.
  • F. M. M. Morel, Kinetics of nutrient uptake and growth in phytoplankton, J. Phycol. 23 (1987), no. 2, 137–150.
  • G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: potential competitiors that eat each other, Annu. Rew. Ecol. Syst. 20 (1989), no. 1, 297–330.
  • H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs 41, American Mathematical Society, Providence, RI, 1995.
  • H. L. Smith and P. Waltman, Competition for a single limiting resouce in continuous culture: the variable-yield model, SIAM J. Appl. Math. 54 (1994), no. 4, 1113–1131.
  • ––––, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995.
  • H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (1992), no. 7, 755–763.
  • T. F. Thingstad, H. Havskum, K. Garde and B. Riemann, On the strategy of “eating your competitor": a mathematical analysis of algal mixotrophy, Ecology 77 (1996), no. 7, 2108–2118.
  • S. Wilken, J. M. H. Verspagen, S. Naus-Wiezer, E. V. Donk and J. Huisman, Comparison of predator-prey interactions with and without intraguild predation by manipulation of the nitrogen source, Oikos 123 (2013), no. 4, 423–432.
  • X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
  • M. V. Zubkov and G. A. Tarran, High bacterivory by the smallest phytoplankton in the North Atlantic Ocean, Nature 455 (2008), no. 7210, 224–226.