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.
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Anderson, R.M., May, R.M.: Population biology of infectious diseases: part I. Nature 280, 361–367 (1979)
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)
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)
Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)
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)
Colijn, C., Mackey, M.C.: A mathematical model of hematopoiesis, I: periodic chronic myelognous leukemia. J. Theor. Biol. 237, 117–132 (2005)
Cooke, K.L., van den Driessche, P.: On zeroes of some transcendental equations. Funkc. Ekvacioj 29, 77–90 (1986)
Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28, 1–21 (2002)
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)
Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)
Hastings, A.: Age-dependent predation is not a simple process, I: continuous time models. Theor. Popul. Biol. 23, 347–362 (1983)
Kraaijeveld, A.R., Godfray, H.C.J.: Trade-off between parasitoid resistance and larval competitive ability in Drosophila melanogaster. Nature 389, 278–280 (1997)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)
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)
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)
Mackey, M.C., Milton, J.G.: Dynamical diseases. Ann. N.Y. Acad. Sci. 504, 16–32 (1987)
Mahaffy, J.M.: A test for stability of linear differential delay equations. Q. Appl. Math. 40, 193–202 (1982)
May, R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973)
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)
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)
Murray, J.D.: Mathematical Biology I: An Introduction, 3rd edn. Springer, New York (2002)
Nicholson, A.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)
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)
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)
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)
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)
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)
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)
White, A., Bowers, R.G., Begon, M.: Population cycles in self-regulated insect pathogen systems: resolving conflicting predictions. Am. Nat. 148, 220–225 (1996)
Wilson, K., Reeson, A.F.: Density-dependent prophylaxis: evidence from lepidoptera–baculovirus interactions? Ecol. Entomol. 23, 100–101 (1998)
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)
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)
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Communicated by P.K. Maini.
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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
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DOI: https://doi.org/10.1007/s00332-013-9179-0