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Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System

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Abstract

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. Results show that a moderate Allee effect will destabilize the dynamics, but it 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 probability. 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). It indicates the possibility of using human disturbance in habitat as a potential epidemic-control method in conservation.

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References

  • Allee, W.C., 1931. Animal Aggregations, a Study in General Sociology. University of Chicago Press, Chicago.

    Google Scholar 

  • Amarasekare, P., 1998. Allee effect in metapopulation dynamics. Am. Nat. 152, 298–02.

    Article  Google Scholar 

  • Amarasekare, P., 2004. Spatial dynamics of mutualistic interactions. J. Anim. Ecol. 73, 128–40.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Bascompte, J., Solé, R.V., 1998. Effects of habitat destruction in a prey-predator metapopulation model. J. Theor. Biol. 195, 383–93.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Carlsson-Grańer, U., 2006. Disease dynamics, host specificity and pathogen persistence in isolated host populations. Oikos 112, 174–84.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Courchamp, F., Macdonald, D.W., 2001. Crucial importance of pack size in the African wild dog Lycaon Pictus. Anim. Conserv. 4, 169–74.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cruickshank, I., Gurney, W.S.C., Veitch, A.R., 1999. The characteristics of epidemics and invasions with thresholds. Theor. Popul. Biol. 56, 279–92.

    Article  MATH  Google Scholar 

  • Delgado, M., Molina-Becerra, M., Suarez, A., 2005. Relating disease and predation: equilibria of an epidemic model. Math. Methods Appl. Sci. 28, 349–62.

    Article  MATH  MathSciNet  Google Scholar 

  • Deredec, A., Courchamp, F., 2003. Extinction thresholds in host-parasite dynamics. Ann. Zool. Fenn. 40, 115–30.

    Google Scholar 

  • Fahrig, L., Merrian, G., 1994. Conservation of fragmented populations. Cons. Biol. 8, 50–9.

    Article  Google Scholar 

  • Feagin, R.A., Wu, X.B., Feagin, T., 2007. Edge effects in lacunarity analysis. Ecol. Model. 201, 262–68.

    Article  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Gilpin, M.E., Rosenzweig, M.L., 1972. Enriched predator-prey systems: theoretical stability. Science 177, 902–04.

    Article  Google Scholar 

  • Grassly, N.C., Fraser, C., 2006. Seasonal infectious disease epidemiology. Proc. R. Soc. B 273, 2541–550.

    Article  Google Scholar 

  • Greenhalgh, D., Haque, M., 2007. A predator-prey model with disease in the prey species only. Math. Methods Appl. Sci. 30, 911–29.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Google Scholar 

  • Hadeler, K.P., Freedman, H.I., 1989. Predator-prey populations with parasitic infection. J. Math. Biol. 27, 609–31.

    MATH  MathSciNet  Google Scholar 

  • Han, L., Ma, Z., 2001. Four-predator prey models with infectious diseases. Math. Comput. Model. 34, 849–58.

    Article  MATH  MathSciNet  Google Scholar 

  • Hanski, I., 2001. Spatially realistic theory of metapopulation ecology. Naturwissenschaften 88, 372–81.

    Article  Google Scholar 

  • Hanski, I., Kuussaari, M., Nieminen, M., 1994. Metapopulation structure and migration in the butterfly Melitaeacinxia. Ecology 75, 747–62.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Haque, M., Venturino, E., 2006. The role of transmissible disease in the Holling–Tanner predator-prey model. Theor. Popul. Biol. 70, 273–88.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hui, C., Li, Z.Z., 2003. Dynamical complexity and metapopulation persistence. Ecol. Model. 164, 201–09.

    Article  Google Scholar 

  • Hui, C., Li, Z.Z., 2004. Distribution patterns of metapopulation determined by Allee effects. Popul. Ecol. 46, 55–3.

    Article  Google Scholar 

  • Hui, C., McGeoch, M.A., 2007a. Spatial patterns of prisoner’s dilemma game in metapopulations. Bull. Math. Biol. 69, 659–76.

    Article  MathSciNet  Google Scholar 

  • Hui, C., McGeoch, M.A., 2007b. A self-similarity model for the occupancy frequency distribution. Theor. Popul. Biol. 71, 61–0.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Johansen, A., 1994. Spatio-temporal self-organisation in a model of disease spreading. Physica D 78, 186–93.

    Article  MATH  Google Scholar 

  • Jülicher, F., 2006. Statistical physics of active processes in cells. Physica A 369, 185–00.

    Article  Google Scholar 

  • Kermack, W.O., McKendrick, A.G., 1927. Contributions to the mathematical theory of epidemics, part I. Proc. R. Soc. A 115, 700–21.

    Article  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Keitt, T.H., Lewis, M.A., Holt, R.D., 2001. Allee effects, invasion pinning, and species’ borders. Am. Nat. 157, 203–16.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Liu, Q.X., Jin, Z., 2005. Cellular automata modelling of SEIRS. Chin. Phys. 14(07), 1370–377.

    Article  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • McCarthy, M.A., 1997. The Allee effect, finding mates and theoretical models. Ecol. Model. 103, 99–02.

    Article  Google Scholar 

  • Morita, S., Tainaka, K., 2006. Undamped oscillations in prey-predator models on a finite size lattice. Popul. Ecol. 48, 99–05.

    Article  Google Scholar 

  • Murray, J.D., 1993. Mathematical Biology. Springer, Berlin.

    Book  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Nowak, M.A., Bonhoetter, S., May, R.M., 1994. Spatial games and the maintenance of cooperation. Proc. Natl. Acad. Sci. USA 91, 4877–881.

    Article  MATH  Google Scholar 

  • Pal, S., Kundu, K., Chattopadhyay, J., 2006. Role of standard incidence in an eco-epidemiological system: a mathematical study. Ecol. Model. 199, 229–39.

    Article  Google Scholar 

  • Peterson, R.O., Page, R.E., 1987. Wolf density as a predictor of predation rate. Swed. Wildlife Res. 1, 771–73.

    Google Scholar 

  • Rhodes, C.J., Anderson, R.M., 1996. Persistence and dynamics in lattice models of epidemic spread. J. Theor. Biol. 180, 125–33.

    Article  Google Scholar 

  • Rhodes, C.J., Anderson, R.M., 1997. Epidemic thresholds and Vaccination in a lattice model of disease spread. Theor. Popul. Biol. 52, 101–18.

    Article  MATH  Google Scholar 

  • Sayama, H., 2004. Self-protection and diversity in self-replicating cellular automata. Artif. Life 10, 83–8.

    Article  Google Scholar 

  • Schreiber, S., Ludwig, K., Herrmann, A., Holzhütter, H.G., 2001. Stochastic simulation of hemagglutinin-mediated fusion pore formation. Biophys. J. 81(3), 1360–372.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Szwabiński, J., Pekalski, A., 2006. Effects of random habitat destruction in a predator-prey model. Physica A 360, 59–0.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Tilman, D., Kareiva, P., 1997. Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Princeton University Press, Princeton.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Venturino, E., 2002. Epidemics in predator-prey models: disease in the predators. J. Math. Appl. Med. Biol. 19, 185–05.

    Article  MATH  Google Scholar 

  • Wang, M.H., Kot, M., 2001. Speeds of invasion in a model with strong or weak Allee effect. Math. Biosci. 171, 83–7.

    Article  MATH  MathSciNet  Google Scholar 

  • Wang, G., Liang, X., Wang, F., 1999. The competitive dynamics of populations subject to an Allee effect. Ecol. Model. 124, 183–92.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Min Su & Zizhen Li

  2. Center of Excellence for Invasion Biology, Department of Botany and Zoology, University of Stellenbosch, Private Bag X1, Matieland, 7602, South Africa

    Cang Hui

  3. Key Laboratory of Arid and Grassland Agroecology of the Ministry of Education, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    YanYu Zhang & Zizhen Li

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  1. Min Su
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  2. Cang Hui
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  3. YanYu Zhang
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  4. Zizhen Li
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Correspondence to Zizhen Li.

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Su, M., Hui, C., Zhang, Y. et al. Spatiotemporal Dynamics of the Epidemic Transmission in a Predator-Prey System. Bull. Math. Biol. 70, 2195–2210 (2008). https://doi.org/10.1007/s11538-008-9340-3

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  • DOI: https://doi.org/10.1007/s11538-008-9340-3

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