Journal of Applied Mathematics

Stability Analysis of Competing Insect Species for a Single Resource

Sizah Mwalusepo, Henri E. Z. Tonnang, Estomih S. Massawe, Tino Johansson, and Bruno Pierre Le Ru

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The models explore the effects of resource and temperature on competition between insect species. A system of differential equations is proposed and analysed qualitatively using stability theory. A local study of the models is performed around axial, planar, and interior equilibrium points to successively estimate the effect of (i) one species interacting with a resource, (ii) two competing species for a single resource, and (iii) three competing species for a single resource. The local stability analysis of the equilibrium is discussed using Routh-Hurwitz criteria. Numerical simulation of the models is performed to investigate the sensitivity of certain key parameters. The models are used to predict population dynamics in the selected cases studied. The results show that when a single species interacts with a resource, the species will be able to establish and sustain a stable population. However, in competing situation, it is observed that the combinations of three parameters (half-saturation, growth rate, and mortality rate) determine which species wins for any given resource. Moreover, our results indicate that each species is the superior competitor for the resource for the range of temperature for which it has the lowest equilibrium resource.

Article information

J. Appl. Math., Volume 2014 (2014), Article ID 285350, 14 pages.

First available in Project Euclid: 2 March 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Mwalusepo, Sizah; Tonnang, Henri E. Z.; Massawe, Estomih S.; Johansson, Tino; Le Ru, Bruno Pierre. Stability Analysis of Competing Insect Species for a Single Resource. J. Appl. Math. 2014 (2014), Article ID 285350, 14 pages. doi:10.1155/2014/285350.

Export citation


  • D. H. Robert, Predation, Apparent Competition, and the Structure of Prey Communities, Academic Press, New York, NY, USA, 1977.
  • I. Kaplan and R. F. Denno, “Interspecific interactions in phy-tophagous insects revisited: a quantitative assessment of competition theory,” Ecology Letters, vol. 10, no. 10, pp. 977–994, 2007.
  • G. S. K. Wolkowicz and Z. Lu, “Global dynamics of a mathematical model of competition in the chemostat: general response functions and differential death rates,” SIAM Journal on Applied Mathematics, vol. 52, no. 1, pp. 222–233, 1992.
  • A. J. Lotka, “The growth of mixed populations: two species competing for a common food supply,” Journal of Washington Academic of Science, vol. 22, no. 16, pp. 461–469, 1932.
  • G. F. Gause, The Struggle for Existence, Williams and Wilkins, Baltimore, Md, USA, 1934.
  • T. Jermy, “Is there competition between phytophagous insects?” Journal of Zoological Systematic and Evolutionary Research, vol. 23, no. 4, pp. 275–285, 1985.
  • R. F. Denno, M. S. McClure, and J. R. Ott, “Interspecific interactions in phytophagous insects: competition reexamined and resurrected,” Annual Review of Entomology, vol. 40, pp. 297–331, 1995.
  • D. Tilman, “Resource competition between planktonic algae: an experimental and theoretical approach,” Ecology, vol. 58, no. 2, pp. 338–348, 1977.
  • D. Tilman, “The importance of the mechanisms of interspecific competition,” The American Naturalist, vol. 129, no. 5, pp. 769–774, 1978.
  • J. P. Grover, Resource Competition, Chapman & Hall, London, UK, 1997.
  • I. P. Martines, H. V. Kojouharov, and J. P. Grover, “A chemostat model of resource competition and allelopathy,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 573–582, 2009.
  • B. Liu and L. Zhang, “Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input,” Applied Mathematics and Computation, vol. 214, no. 1, pp. 155–162, 2009.
  • O. A. Ebraheen, A. D. Fordyce, and D. Niall, “A 3-species competition model for bio-control,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9690–9698, 2012.
  • A. A. S. Zaghrout and F. M. Kandil, “Competition between three microbial populations for a single limiting resource in continuous culture,” Applied Mathematics and Computation, vol. 92, no. 2-3, pp. 271–281, 1998.
  • A. J. Lotka, Elementary of Physical Biology, Williams and Wilkins, Baltimore, Md, USA, 1925.
  • V. Volterra, “Fluctuations in the abundance of a species considered mathematically,” Nature, vol. 118, no. 2972, pp. 558–560, 1926.
  • D. Tilman, “Competition and biodiversity in spatially structured habitats,” Ecology, vol. 75, no. 1, pp. 2–16, 1994.
  • J. A. Leon and D. B. Tumpson, “Competition between two species for two complementary or substitutable resources,” Journal of Theoretical Biology, vol. 50, no. 1, pp. 185–201, 1975.
  • S. B. Hsu, “Limiting behavior for competing species,” SIAM Journal on Applied Mathematics, vol. 34, no. 4, pp. 760–763, 1978.
  • S. B. Hsu, S. Hubbell, and P. Waltman, “A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms,” SIAM Journal on Applied Mathematics, vol. 32, no. 2, pp. 366–383, 1977.
  • S. R. Hansen and S. P. Hubbell, “Single-nutrient microbial competition: qualitative agreement between experimental and theoretically forecast outcomes,” Science, vol. 207, no. 4438, pp. 1491–1493, 1980.
  • J. S. Bale, G. J. Masters, I. D. Hodkinson et al., “Herbivory in global climate change research: direct effects of rising temperature on insect herbivores,” Global Change Biology, vol. 8, no. 1, pp. 1–16, 2002.
  • M. Ladányi and L. Horváth, “A review of the potential climate change impact on insect populations,” Applied Ecology and Environmental Research, vol. 8, no. 1, pp. 143–152, 2010.
  • W. J. O'Brien, “The dynamic of nutrient limitation of phytoplankton algae: a model reconsidered,” Ecology, vol. 55, no. 1, pp. 135–141, 1974.
  • R. Levins, “Coexistence in a variable environment,” The American Naturalist, vol. 114, no. 6, pp. 765–783, 1979.
  • P. Chesson, “Multispecies competition in variable environments,” Theoretical Population Biology, vol. 45, no. 3, pp. 227–276, 1994.
  • S. B. Hsu and P. Waltman, “Competition in the chemostat when one competitor produces a toxin,” Japan Journal of Industrial and Applied Mathematics, vol. 15, no. 3, pp. 471–490, 1998.
  • S. B. Hsu, Y.-S. Li, and P. Waltman, “Competition in the presence of a lethal external inhibitor,” Mathematical Biosciences, vol. 167, no. 2, pp. 177–199, 2000.
  • M. Smale, D. Byerlee, and T. Jayne, “Maize Revolution in Sub-Saharan Africa,” The World Bank, 2001.
  • R. Kfir, W. A. Overholt, Z. R. Khan, and A. Polaszek, “Biology and management of economically important lepidopteran cereal stem borers in Africa,” Annual Review of Entomology, vol. 47, pp. 701–731, 2002.
  • R. James and R. Washington, “Changes in African temperature and precipitation associated with degrees of global warming,” Climatic Change, vol. 117, no. 4, pp. 859–872, 2013. \endinput