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Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat

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Abstract

Recent research indicates that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. We also analyze the subsystem that results when the resistant competitor is absent. The model takes the form of an SIS epidemic model. Criteria for the global stability of the disease free and endemic steady states are obtained for both the subsystem as well as for the full competition model. However, for certain parameter ranges, bi-stability, and/or multiple periodic orbits is possible and both disease induced oscillations and competition induced oscillations are possible. It is proved that persistence of the vulnerable and resistant populations can occur, but only when the disease is endemic in the population. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf, saddle-node, and homoclinic bifurcations, and a hysteresis effect. An explicit expression for the basic reproduction number for the epidemic is given in terms of biologically meaningful parameters. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Katherine Northcott & Gail S. K. Wolkowicz

  2. Department of Mathematics, Lahore University of Management Science (LUMS), LUMS, DHA Phase 2, Lahore, Pakistan

    Mudassar Imran

Authors
  1. Katherine Northcott
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  2. Mudassar Imran
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  3. Gail S. K. Wolkowicz
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Correspondence to Gail S. K. Wolkowicz.

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Northcott, K., Imran, M. & Wolkowicz, G.S.K. Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat. J. Math. Biol. 64, 1043–1086 (2012). https://doi.org/10.1007/s00285-011-0439-z

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  • DOI: https://doi.org/10.1007/s00285-011-0439-z

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