[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Stability Switches in a Host–Pathogen Model as the Length of a Time Delay Increases

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

The destabilising effects of a time delay in mathematical models are well known. However, delays are not necessarily destabilising. In this paper, we explore an example of a biological system where a time delay can be both stabilising and destabilising. This example is a host–pathogen model, incorporating density-dependent prophylaxis (DDP). DDP describes when individual hosts invest more in immunity when population densities are high, due to the increased risk of infection in crowded conditions. In this system, as the delay length increases, there are a finite number of switches between stable and unstable behaviour. These stability switches are demonstrated and characterised using a combination of numerical methods and analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  • Anderson, R.M., May, R.M.: Population biology of infectious diseases: part I. Nature 280, 361–367 (1979)

    Article  Google Scholar 

  • Anderson, R.M., May, R.M.: The population dynamics of microparasites and their invertebrate hosts. Philos. Trans. R. Soc. Lond. B 291, 451–524 (1981)

    Article  Google Scholar 

  • Barnes, A.I., Siva-Jothy, M.T.: Density-dependent prophylaxis in the mealworm beetle Tenebrio molitor L. (Coleoptera: Tenebrionidae): cuticular melanization is an indicator of investment in immunity. Proc. R. Soc. Lond. B 267, 177–182 (2000)

    Article  Google Scholar 

  • Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Bowers, R.G., Begon, M., Hodgkinson, D.E.: Host–pathogen population cycles in forest insects? Lessons from simple models reconsidered. Oikos 67, 529–538 (1993)

    Article  Google Scholar 

  • Colijn, C., Mackey, M.C.: A mathematical model of hematopoiesis, I: periodic chronic myelognous leukemia. J. Theor. Biol. 237, 117–132 (2005)

    Article  MathSciNet  Google Scholar 

  • Cooke, K.L., van den Driessche, P.: On zeroes of some transcendental equations. Funkc. Ekvacioj 29, 77–90 (1986)

    MATH  Google Scholar 

  • Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28, 1–21 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Fowler, A.C., Kalamangalam, G.P.: The role of the central chemoreceptor in causing periodic breathing. IMA J. Math. Appl. Med. Biol. 17, 147–167 (2000)

    Article  MATH  Google Scholar 

  • Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  • Hastings, A.: Age-dependent predation is not a simple process, I: continuous time models. Theor. Popul. Biol. 23, 347–362 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Kraaijeveld, A.R., Godfray, H.C.J.: Trade-off between parasitoid resistance and larval competitive ability in Drosophila melanogaster. Nature 389, 278–280 (1997)

    Article  Google Scholar 

  • Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  • Kunimi, Y., Yamada, E.: Relationship between larval phase and susceptibility of the armyworm, Pseudaletia separata Walker (Lepidoptera: Noctuidae) to a nuclear polyhedrosis virus and a granulosis virus. Appl. Entomol. Zool. 25, 289–297 (1990)

    Google Scholar 

  • Lee, M.S., Hsu, C.S.: On the τ-decomposition method of stability analysis for retarded dynamical systems. SIAM J. Control Optim. 7, 242–259 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  • Mackey, M.C., Milton, J.G.: Dynamical diseases. Ann. N.Y. Acad. Sci. 504, 16–32 (1987)

    Article  Google Scholar 

  • Mahaffy, J.M.: A test for stability of linear differential delay equations. Q. Appl. Math. 40, 193–202 (1982)

    MathSciNet  MATH  Google Scholar 

  • May, R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973)

    Article  Google Scholar 

  • Mitchell, S.E., Read, A.F.: Poor maternal environment enhances offspring disease resistance in an invertebrate. Proc. R. Soc. Lond. B 272, 2601–2607 (2005)

    Article  Google Scholar 

  • Mufti, I.H.: A note on the stability of an equation of third order with time lag. IEEE Trans. Autom. Control 9, 190–191 (1964)

    Article  Google Scholar 

  • Murray, J.D.: Mathematical Biology I: An Introduction, 3rd edn. Springer, New York (2002)

    Google Scholar 

  • Nicholson, A.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)

    Article  Google Scholar 

  • Råberg, L., Grahn, M., Hasselquist, D., Svensson, E.: On the adaptive significance of stress-induced immunosuppression. Proc. R. Soc. Lond. B 265, 1637–1641 (1998)

    Article  Google Scholar 

  • Reeson, A.F., Wilson, K., Gunn, A., Hails, R.S., Goulson, D.: Baculovirus resistance in the noctuid Spodoptera exempta is phenotypically plastic and responds to population density. Proc. R. Soc. Lond. B 265, 1787–1791 (1998)

    Article  Google Scholar 

  • Reynolds, J.J.H., White, A., Sherratt, J.A., Boots, M.: The population dynamical consequences of density-dependent prophylaxis. J. Theor. Biol. 288, 1–8 (2011)

    Article  MathSciNet  Google Scholar 

  • Ruiz-González, M.X., Moret, Y., Brown, M.J.F.: Rapid induction of immune density-dependent prophylaxis in adult social insects. Biol. Lett. 5, 781–783 (2009)

    Article  Google Scholar 

  • Ryder, J.J., Webberley, K.M., Boots, M., Knell, R.J.: Measuring the transmission dynamics of a sexually transmitted disease. Proc. Natl. Acad. Sci. USA 102, 15140–15143 (2005)

    Article  Google Scholar 

  • Ryder, J.J., Miller, M.R., White, A., Knell, R.J., Boots, M.: Host–parasite population dynamics under combined frequency- and density-dependent transmission. Oikos 116, 2017–2026 (2007)

    Article  Google Scholar 

  • White, A., Bowers, R.G., Begon, M.: Population cycles in self-regulated insect pathogen systems: resolving conflicting predictions. Am. Nat. 148, 220–225 (1996)

    Article  Google Scholar 

  • Wilson, K., Reeson, A.F.: Density-dependent prophylaxis: evidence from lepidoptera–baculovirus interactions? Ecol. Entomol. 23, 100–101 (1998)

    Article  Google Scholar 

  • Wilson, K., Thomas, M.B., Blanford, S., Doggett, M., Simpson, S.J., Moore, S.L.: Coping with crowds: density-dependent disease resistance in desert locusts. Proc. Natl. Acad. Sci. USA 99, 5471–5475 (2002)

    Article  Google Scholar 

  • Xiao, Y., Bowers, R.G., Tang, S.: The effect of delayed host self-regulation on host–pathogen population cycles in forest insects. J. Theor. Biol. 258, 240–249 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jennifer J. H. Reynolds.

Additional information

Communicated by P.K. Maini.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Reynolds, J.J.H., Sherratt, J.A. & White, A. Stability Switches in a Host–Pathogen Model as the Length of a Time Delay Increases. J Nonlinear Sci 23, 1073–1087 (2013). https://doi.org/10.1007/s00332-013-9179-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-013-9179-0

Keywords

Mathematics Subject Classification

Navigation