×

Spatiotemporal dynamics of the epidemic transmission in a predator-prey system. (English) Zbl 1169.92042

Summary: Epidemic transmission is one of the critical density-dependent mechanisms that affect species viability and dynamics. In a predator-prey system, epidemic transmission can strongly affect the success probability of hunting, especially for social animals. Predators, therefore, will suffer from the positive density-dependence, i.e., Allee effect, due to epidemic transmission in the population. The rate of species contacting the epidemic, especially for those endangered or invasive, has largely increased due to the habitat destruction caused by anthropogenic disturbance.
Using ordinary differential equations and cellular automata, we here explored the epidemic transmission in a predator-prey system. The results show that a moderate Allee effect will destabilize the dynamics, but this is not true for the extreme Allee effect (weak or strong). The predator-prey dynamics amazingly stabilize by the extreme Allee effect. Predators suffer the most from the epidemic disease at moderate transmission probabilities. Counter-intuitively, habitat destruction will benefit the control of the epidemic disease. The demographic stochasticity dramatically influences the spatial distribution of the system. The spatial distribution changes from oil-bubble-like (due to local interaction) to aggregated spatially scattered points (due to local interaction and demographic stochasticity). This indicates the possibility of using human disturbance in habitat as a potential epidemic-control method in conservation.

MSC:

92D30 Epidemiology
92D40 Ecology
37N25 Dynamical systems in biology
34A99 General theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Allee, W.C., 1931. Animal Aggregations, a Study in General Sociology. University of Chicago Press, Chicago.
[2] Amarasekare, P., 1998. Allee effect in metapopulation dynamics. Am. Nat. 152, 298–02. · doi:10.1086/286169
[3] Amarasekare, P., 2004. Spatial dynamics of mutualistic interactions. J. Anim. Ecol. 73, 128–40. · doi:10.1046/j.0021-8790.2004.00788.x
[4] Anderson, R.M., May, R.M., 1986. The invasion, persistence, and spread of infectious diseases within animal and plant communities. Philos. Trans. R. Soc. Lond. B 314, 533–70. · doi:10.1098/rstb.1986.0072
[5] Bairagi, N., Roy, P.K., Chattopadhyay, J., 2007. Role of infection on the stability of a predator-prey system with several response functions–a comparative study. J. Theor. Biol. 248, 10–5. · doi:10.1016/j.jtbi.2007.05.005
[6] Bascompte, J., Solé, R.V., 1998. Effects of habitat destruction in a prey-predator metapopulation model. J. Theor. Biol. 195, 383–93. · doi:10.1006/jtbi.1998.0803
[7] Berec, L., Boukal, D.S., Berec, M., 2001. Linking the Allee effect, sexual reproduction, and temperature-dependent sex determination via spatial dynamics. Am. Nat. 157, 217–30. · doi:10.1086/318626
[8] Boccara, N., Cheong, K., 1992. Automata network models for the spread of infectious diseases in a population of moving individuals. J. Phys. A 25, 2447–461. · Zbl 0752.92024 · doi:10.1088/0305-4470/25/9/018
[9] Brännström, Å., Sumpter, D.J.T., 2005. Coupled map lattice approximations for spatially explicit individual-based models of ecology. Bull. Math. Biol. 67, 663–82. · Zbl 1334.92445 · doi:10.1016/j.bulm.2004.09.006
[10] Carlsson-Grańer, U., 2006. Disease dynamics, host specificity and pathogen persistence in isolated host populations. Oikos 112, 174–84. · doi:10.1111/j.0030-1299.2006.13292.x
[11] Chattopadhyay, J., Sarkar, R.R., Ghosal, G., 2002. Removal of infected prey prevent limit cycle oscillations in an infected prey-predator system–a mathematical study. Ecol. Model. 156, 113–21. · doi:10.1016/S0304-3800(02)00133-3
[12] Chen, J.C., Elimelech, M., Kim, A.S., 2005. Monte Carlo simulation of colloidal membrane filtration: Model development with application to characterization of colloid phase transition. J. Membr. Sci. 255, 291–05. · doi:10.1016/j.memsci.2005.02.004
[13] Courchamp, F., Macdonald, D.W., 2001. Crucial importance of pack size in the African wild dog Lycaon Pictus. Anim. Conserv. 4, 169–74. · doi:10.1017/S1367943001001196
[14] Courchamp, F., Clutton-Brock, T., Grenfell, B., 2000. Multipack dynamics and the Allee effect in the African wild dog. Lycaon Pictus. Anim. Conserv. 3, 277–85. · doi:10.1111/j.1469-1795.2000.tb00113.x
[15] Cruickshank, I., Gurney, W.S.C., Veitch, A.R., 1999. The characteristics of epidemics and invasions with thresholds. Theor. Popul. Biol. 56, 279–92. · Zbl 0963.92034 · doi:10.1006/tpbi.1999.1432
[16] Delgado, M., Molina-Becerra, M., Suarez, A., 2005. Relating disease and predation: equilibria of an epidemic model. Math. Methods Appl. Sci. 28, 349–62. · Zbl 1079.34017 · doi:10.1002/mma.573
[17] Deredec, A., Courchamp, F., 2003. Extinction thresholds in host-parasite dynamics. Ann. Zool. Fenn. 40, 115–30.
[18] Fahrig, L., Merrian, G., 1994. Conservation of fragmented populations. Cons. Biol. 8, 50–9. · doi:10.1046/j.1523-1739.1994.08010050.x
[19] Feagin, R.A., Wu, X.B., Feagin, T., 2007. Edge effects in lacunarity analysis. Ecol. Model. 201, 262–68. · doi:10.1016/j.ecolmodel.2006.09.019
[20] Freedman, H.I., Wolkowicz, G.S.K., 1986. Predator-prey system with group defense: the paradox of enrichment revisited. Bull. Math. Biol. 48, 493–08. · Zbl 0612.92017
[21] Gai, F., Hasson, K.C., McDonald, J.C., Anfinrud, P.A., 1998. Chemical dynamics in proteins: The photoisomerization of retinal in bacteriorhodopsin. Science 279, 1886–891. · doi:10.1126/science.279.5358.1886
[22] Gilpin, M.E., Rosenzweig, M.L., 1972. Enriched predator-prey systems: theoretical stability. Science 177, 902–04. · doi:10.1126/science.177.4052.902
[23] Grassly, N.C., Fraser, C., 2006. Seasonal infectious disease epidemiology. Proc. R. Soc. B 273, 2541–550. · doi:10.1098/rspb.2006.3604
[24] Greenhalgh, D., Haque, M., 2007. A predator-prey model with disease in the prey species only. Math. Methods Appl. Sci. 30, 911–29. · Zbl 1115.92049 · doi:10.1002/mma.815
[25] Gulland, F.M.D., 1995. The impact of infectious diseases on wild animal populations-a review. In: Ecology of Infectious Diseases in Natural Populations. Cambridge University Press, Cambridge.
[26] Hadeler, K.P., Freedman, H.I., 1989. Predator-prey populations with parasitic infection. J. Math. Biol. 27, 609–31. · Zbl 0716.92021
[27] Han, L., Ma, Z., 2001. Four-predator prey models with infectious diseases. Math. Comput. Model. 34, 849–58. · Zbl 0999.92032 · doi:10.1016/S0895-7177(01)00104-2
[28] Hanski, I., 2001. Spatially realistic theory of metapopulation ecology. Naturwissenschaften 88, 372–81. · doi:10.1007/s001140100246
[29] Hanski, I., Kuussaari, M., Nieminen, M., 1994. Metapopulation structure and migration in the butterfly Melitaeacinxia. Ecology 75, 747–62. · doi:10.2307/1941732
[30] Hansson, L., 1991. Dispersal and connectivity in metapopulations. In: Gilpin, M., Hanski, I. (Eds.), Metapopulation Dynamics: Brief History and Conceptual Domain, pp. 89–03. Academic, London.
[31] Haque, M., Venturino, E., 2006. The role of transmissible disease in the Holling–Tanner predator-prey model. Theor. Popul. Biol. 70, 273–88. · Zbl 1112.92053 · doi:10.1016/j.tpb.2006.06.007
[32] Haydon, D.T., Laurenson, M.K., Sillero-Zubiri, C., 2002. Integrating epidemiology into population viability analysis: managing the risk posed by rabies and canine distemper to the Ethiopian wolf. Conserv. Biol. 16, 1372–385. · doi:10.1046/j.1523-1739.2002.00559.x
[33] Hethcote, H.W., Wang, W.D., Han, L.T., Ma, Z.E., 2004. A predator-prey model with infected prey. Theor. Popul. Biol. 66, 259–68. · doi:10.1016/j.tpb.2004.06.010
[34] Hui, C., Li, Z.Z., 2003. Dynamical complexity and metapopulation persistence. Ecol. Model. 164, 201–09. · doi:10.1016/S0304-3800(03)00025-5
[35] Hui, C., Li, Z.Z., 2004. Distribution patterns of metapopulation determined by Allee effects. Popul. Ecol. 46, 55–3. · doi:10.1007/s10144-004-0171-2
[36] Hui, C., McGeoch, M.A., 2007a. Spatial patterns of prisoner’s dilemma game in metapopulations. Bull. Math. Biol. 69, 659–76. · Zbl 1139.92322 · doi:10.1007/s11538-006-9145-1
[37] Hui, C., McGeoch, M.A., 2007b. A self-similarity model for the occupancy frequency distribution. Theor. Popul. Biol. 71, 61–0. · Zbl 1118.92059 · doi:10.1016/j.tpb.2006.07.007
[38] Hui, C., Zhang, F., Han, X., Li, Z.Z., 2005. Cooperation evolution and self-regulation dynamics in metapopulation: stage-equilibrium hypothesis. Ecol. Model. 184, 397–12. · doi:10.1016/j.ecolmodel.2004.11.004
[39] Hui, C., McGeoch, M.A., Warren, M., 2006. A spatially explicit approach to estimating species occupancy and spatial correlation. J. Anim. Ecol. 75, 140–47. · doi:10.1111/j.1365-2656.2005.01029.x
[40] Johansen, A., 1994. Spatio-temporal self-organisation in a model of disease spreading. Physica D 78, 186–93. · Zbl 0812.60102 · doi:10.1016/0167-2789(94)90114-7
[41] Jülicher, F., 2006. Statistical physics of active processes in cells. Physica A 369, 185–00. · doi:10.1016/j.physa.2006.04.008
[42] Kermack, W.O., McKendrick, A.G., 1927. Contributions to the mathematical theory of epidemics, part I. Proc. R. Soc. A 115, 700–21. · JFM 53.0517.01 · doi:10.1098/rspa.1927.0118
[43] Kermack, W.O., McKendrick, A.G., 1932. Contributions to the mathematical theory of epidemics. II–The problem of endemicity. Proc. R. Soc. A 138, 55–3. · Zbl 0005.30501 · doi:10.1098/rspa.1932.0171
[44] Keitt, T.H., Lewis, M.A., Holt, R.D., 2001. Allee effects, invasion pinning, and species’ borders. Am. Nat. 157, 203–16. · doi:10.1086/318633
[45] Lamont, B.B., Klinkhamer, P.G.L., Witkowski, E.T.F., 1993. Population fragmentation may reduce fertility to zero in Banksia goodie: a demonstration of the Allee effect. Oecologia 94, 446–50. · doi:10.1007/BF00317122
[46] Li, Z., Gao, M., Hui, C., Han, X., Shi, H., 2005. Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation. Ecol. Model. 185, 245–54. · doi:10.1016/j.ecolmodel.2004.12.008
[47] Liu, Q.X., Jin, Z., 2005. Cellular automata modelling of SEIRS. Chin. Phys. 14(07), 1370–377. · doi:10.1088/1009-1963/14/7/018
[48] Malchow, H., Hiker, F.M., Sarkar, R.R., Brauer, K., 2005. Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection. Math. Comput. Model. 42, 1035–048. · Zbl 1080.92066 · doi:10.1016/j.mcm.2004.10.025
[49] McCarthy, M.A., 1997. The Allee effect, finding mates and theoretical models. Ecol. Model. 103, 99–02. · doi:10.1016/S0304-3800(97)00104-X
[50] Morita, S., Tainaka, K., 2006. Undamped oscillations in prey-predator models on a finite size lattice. Popul. Ecol. 48, 99–05. · doi:10.1007/s10144-006-0257-0
[51] Murray, J.D., 1993. Mathematical Biology. Springer, Berlin. · Zbl 0779.92001
[52] Namba, T., Umemoto, A., Minami, E., 1999. The effects of habitat fragmentation on persistence of source-sink metapopulations in systems with predators and prey or apparent competitors. Theor. Popul. Biol. 56, 123–37. · Zbl 0980.92041 · doi:10.1006/tpbi.1999.1422
[53] Nowak, M.A., Bonhoetter, S., May, R.M., 1994. Spatial games and the maintenance of cooperation. Proc. Natl. Acad. Sci. USA 91, 4877–881. · Zbl 0799.92010 · doi:10.1073/pnas.91.11.4877
[54] Pal, S., Kundu, K., Chattopadhyay, J., 2006. Role of standard incidence in an eco-epidemiological system: a mathematical study. Ecol. Model. 199, 229–39. · doi:10.1016/j.ecolmodel.2006.05.030
[55] Peterson, R.O., Page, R.E., 1987. Wolf density as a predictor of predation rate. Swed. Wildlife Res. 1, 771–73.
[56] Rhodes, C.J., Anderson, R.M., 1996. Persistence and dynamics in lattice models of epidemic spread. J. Theor. Biol. 180, 125–33. · doi:10.1006/jtbi.1996.0088
[57] Rhodes, C.J., Anderson, R.M., 1997. Epidemic thresholds and Vaccination in a lattice model of disease spread. Theor. Popul. Biol. 52, 101–18. · Zbl 0886.92027 · doi:10.1006/tpbi.1997.1323
[58] Sayama, H., 2004. Self-protection and diversity in self-replicating cellular automata. Artif. Life 10, 83–8. · doi:10.1162/106454604322875922
[59] Schreiber, S., Ludwig, K., Herrmann, A., Holzhütter, H.G., 2001. Stochastic simulation of hemagglutinin-mediated fusion pore formation. Biophys. J. 81(3), 1360–372. · doi:10.1016/S0006-3495(01)75792-X
[60] Swihart, R.K., Zhilan, F., Slade, N.A., Mason, D.M., Gehring, T.M., 2001. Effects of habitat destruction and resource supplementation in a predator-prey metapopulation model. J. Theor. Biol. 210, 287–03. · doi:10.1006/jtbi.2001.2304
[61] Szwabiński, J., Pekalski, A., 2006. Effects of random habitat destruction in a predator-prey model. Physica A 360, 59–0. · doi:10.1016/j.physa.2005.05.079
[62] Taylor, C.M., Hastings, A., 2004. Finding optimal control strategies for invasive species: a density-structured model for spartina alterniflora. J. Appl. Ecol. 41, 1049–057. · doi:10.1111/j.0021-8901.2004.00979.x
[63] Tilman, D., Kareiva, P., 1997. Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton.
[64] Tobin, P.C., Bjørnstad, O.N., 2005. Roles of dispersal, stochasticity, and nonlinear dynamics in the spatial structuring of seasonal natural enemy-victim populations. Popul. Ecol. 47, 221–27. · doi:10.1007/s10144-005-0229-9
[65] Venturino, E., 2002. Epidemics in predator-prey models: disease in the predators. J. Math. Appl. Med. Biol. 19, 185–05. · Zbl 1014.92036 · doi:10.1093/imammb/19.3.185
[66] Wang, M.H., Kot, M., 2001. Speeds of invasion in a model with strong or weak Allee effect. Math. Biosci. 171, 83–7. · Zbl 0978.92033 · doi:10.1016/S0025-5564(01)00048-7
[67] Wang, G., Liang, X., Wang, F., 1999. The competitive dynamics of populations subject to an Allee effect. Ecol. Model. 124, 183–92. · doi:10.1016/S0304-3800(99)00160-X
[68] Zhang, F., Hui, C., Han, X., Li, Z., 2005. Evolution of cooperation in patchy habitat under patch decay and isolation. Ecol. Res. 20, 461–69. · doi:10.1007/s11284-005-0072-7
[69] Zhou, S.R., Liu, Y.F., Wang, G., 2005. The stability of predator-prey systems subject to the Allee effects. Theor. Popul. Biol. 67, 23–1. · Zbl 1072.92060 · doi:10.1016/j.tpb.2004.06.007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.