A Generalist Predator Regulating Spread of a Wildlife Disease: Exploring Two Infection Transmission Scenarios

M. Sen¹ ; M. Banerjee² ; A. Morozov^3, 4

¹ Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore
² Department of Mathematics and Statistics, I. I. T. Kanpur, India
³ Department of Mathematics, University of Leicester, UK
⁴ Shirshov Institute of Oceanology, Moscow, Russia

Mathematical modelling of natural phenomena, Tome 10 (2015) no. 2, pp. 74-95.

Résumé

Ecoepidemiology is a well-developed branch of theoretical ecology, which explores interplay between the trophic interactions and the disease spread. In most ecoepidemiological models, however, the authors assume the predator to be a specialist, which consumes only a single prey species. In few existing papers, in which the predator was suggested to be a generalist, the alternative food supply was always considered to be constant. This is obviously a simplification of reality, since predators can often choose between a number of different prey. Consumption of these alternative prey can dramatically change their densities and strongly influence the model predictions. In this paper, we try to bridge the gap and explore a generic eco-epidemiological system with a generalist predator, where the densities of all prey are dynamical variables. The model consists of two prey species, one of which is subject to an infectious disease, and a predator, which consumes both prey species. We investigate two main scenarios of infection transmission mode: (i) the disease transmission rate is predator independent and (ii) the transmission rate is a function of predator density. For both scenarios we fulfil an extensive bifurcation analysis. We show that including a second dynamical prey in the system can drastically change the dynamics of the single prey case. In particular, the presence of a second prey impedes disease spread by decreasing the basic reproduction number and can result in a substantial drop of the disease prevalence. We demonstrate that with efficient consumption of the second prey species by the predator, the predator-dependent disease transmission can not destabilize interactions, as in the case with a specialist predator. Interestingly, even if the population of the second prey eventually vanishes and only one prey species finally remains, the system with two prey species may exhibit different properties to those of the single prey system.

Détail

DOI : 10.1051/mmnp/201510206

Bibliographie

[1] S. Altizer, C.L. Nunn, P.H. Thrall, J.L. Gittleman, J. Antonovics, A.A. Cunningham, A.P. Dobson, V. Ezenwa, K.E. Jones, A.B. Pedersen, M. Poss, J.R.C. Pulliam Annu. Rev. Ecol. Evol. Syst. 2003 517 547

[2] R.M. Anderson, R. M. May Nature 1979 361 367

[3] R. M. Anderson, R. M. May Philos. Trans. R. Soc. London B 1986 533 570

[4] M. Begon, J.L. Harper, C.R. Townsend. Ecology. Oxford, Blackwell Science, 2002.

[5] I. Cote, R. Poulin Behavioral Ecology 1995 159 163

[6] O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz J. Math. Biol. 1990 365 382

[7] M.A. Duffy, L. Sivars-Becker Ecol. Lett. 2007 44 53

[8] A. El-Gohary, A.S. Al-Ruzaiza Chaos Solitons & Fractals 2007 443 453

[9] M. Friend Hydrobiologia 2002 293 306

[10] S. Gakkhar, R.K. Naji Chaos Solitons & Fractals 2003 639 649

[11] W. Gentleman, A. Leising, B. Frost, S. Strom, J. Murray Deep Sea Res. II 2003 2847 2875

[12] M.E. Gilpin Am. Nat. 1979 306 308

[13] K.P. Hadeler, H.I. Freedman J. Math. Biol. 1989 609 631

[14] M. Haque, E. Venturino Theor. Pop. Biol. 2006 273 288

[15] M. Haque, E. Venturino J. Biol. Sys. 2008 445 461

[16] V. Hutson, G.T. Vickers Math. Biosci. 1983 253 269

[17] P.T.J. Johnson, D.E. Stanton, E.R. Preu, K.J. Forshay, S.R. Carpenter Ecology 2006 1973 1980

[18] Z. Kabata. Parasites and diseases of fish cultured in the tropics. London, Taylor and Francis, 1985.

[19] A. Klebanoff, A. Hastings Math. Biosci. 1994 221 233

[20] B. Krasnov, I.S. Khokhlova, G.I. Shenbrot Ecology 2002 164 175

[21] R. Kortet, M.J. Rantala, A. Hedrick Evol. Ecol. Res. 2007 185 197

[22] H. Malchow, F.M. Hilker, S.V. Petrovskii, K. Brauer Ecol. Compl. 2004 211 223

[23] C. Matz, S. Kjelleberg Trends Microbiol 2005 302 307

[24] H. Mccallum, N. Barlow, J. Hone Trends Ecol. Evol. 2001 295 300

[25] A.Y. Morozov Theor. Ecol. 2012 517 532

[26] A. Morozov, A. Best J. Theor. Biol. 2012 29 36

[27] E. Odum, G.W. BARRETT. Fundamentals of Ecology. Belmont, Thomson Brooks/Cole, 2004.

[28] C. Packer, R.D. Holt, P.J. Hudson, K.D. Lafferty, A.P. Dobson Ecol. Lett. 2003 797 802

[29] M.C. Rigby, J. Jokela Proc. R. Soc. Lond. B Biol. Sci. 2000 171 176

[30] S. Roy, J. Chattopadhyay M2AS 2005 1257 1267

[31] S. Sharma, G. P. Samanta. Analysis of a two prey one predator system with disease in the first prey population. Int. J. Dyna. Cont., (In press) 2014.

[32] S. Van Nouhuys, S.I. Hanski Ecol. Lett. 2000 82

[33] R. R. Vance Am. Nat. 1978 797 813

[34] E. Venturino Rocky Mount. J. Math. 1994 381 402

[35] E. Venturino Ecoepidemiology: a more comprehensive view of population interactions MMNP, (In press) 2015.

[36] E. Venturino, S. Petrovskii Ecol. Compl. 2013 37 47