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Excitable Population Dynamics, Biological Control Failure, and Spatiotemporal Pattern Formation in a Model Ecosystem

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Abstract

Biological control has been attracting an increasing attention over the last two decades as an environmentally friendly alternative to the more traditional chemical-based control. In this paper, we address robustness of the biological control strategy with respect to fluctuations in the controlling species density. Specifically, we consider a pest being kept under control by its predator. The predator response is assumed to be of Holling type III, which makes the system’s kinetics “excitable.” The system is studied by means of mathematical modeling and extensive numerical simulations. We show that the system response to perturbations in the predator density can be completely different in spatial and non-spatial systems. In the nonspatial system, an overcritical perturbation of the population density results in a pest outbreak that will eventually decay with time, which can be regarded as a success of the biological control strategy. However, in the spatial system, a similar perturbation can drive the system into a self-sustained regime of spatiotemporal pattern formation with a high pest density, which is clearly a biological control failure. We then identify the parameter range where the biological control can still be successful and describe the corresponding regime of the system dynamics. Finally, we identify the main scenarios of the system response to the population density perturbations and reveal the corresponding structure of the parameter space of the system.

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References

  • Alonso, D., Bartumeus, F., Catalan, J., 2002. Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83, 28–4.

    Article  Google Scholar 

  • Belsky, A.J., 1986. Population and community processes in a mosaic grassland in the Serengeti, Tanzania. J. Ecol. 74, 841–56.

    Article  Google Scholar 

  • Briggs, C.J., Hoopes, M.F., 2004. Stabilizing effects in spatial parasitoid-host and predator-prey models: a review. Theor. Popul. Biol. 65, 299–15.

    Article  MATH  Google Scholar 

  • Brodmann, P.A., Wilcox, C.V., Harrison, S., 1997. Mobile parasitoids may restrict the spatial spread of an insect outbreak. J. Anim. Ecol. 66, 65–2.

    Article  Google Scholar 

  • Cosper, E.M., Bricelj, V.M., Carpenter, E.J., 1989. Novel Phytoplankton Blooms. Coastal and Estuarine Studies, vol. 35. Springer, Berlin.

    Google Scholar 

  • Couteron, P., Lejeune, O., 2001. Periodic spotted patterns in semi-arid vegetation explained by a propagation-inhibition model. J. Ecol. 89, 616–28.

    Article  Google Scholar 

  • Crawley, M.J., 1997. Plant-herbivore dynamics. In: Crawley, M.J. (Ed.), Plant Ecology, pp. 463–65. Blackwell, Oxford.

    Google Scholar 

  • Edwards, A.M., Brindley, J., 1999. Zooplankton mortality and the dynamical behavior of plankton population models. Bull. Math. Biol. 61, 202–39.

    Google Scholar 

  • Ermentrout, B., Chen, X., Chen, Z., 1997. Transition fronts and localized structures in bistable reaction-diffusion equations. Physica D 108, 147–67.

    Article  MATH  MathSciNet  Google Scholar 

  • Fagan, W.F., Lewis, M.A., Neubert, M.G., van den Driessche, P., 2002. Invasion theory and biological control. Ecol. Lett. 5, 148–57.

    Article  Google Scholar 

  • Ferran, A., Guige, L., Tourniaire, R., Gambier, J., Fournier, D., 1998. An artificial non-flying mutation to improve the efficiency of the ladybird Harmonia axyridis in the biological control of aphids. Biocontrol 43, 53–4.

    Article  Google Scholar 

  • Fife, P.C., 1976. Pattern formation in reacting and diffusing systems. J. Chem. Phys. 64, 554–64.

    Article  Google Scholar 

  • Franks, P., 2001. Phytoplankton blooms in a fluctuating environment: the roles of plankton response time scales and grazing. J. Plankt. Res. 23, 1433–441.

    Article  Google Scholar 

  • Gurney, W.S.C., Veitch, A.R., Cruickshank, I., McGeachin, G., 1998. Circles and spirals: population persistence in a spatially explicit predator-prey model. Ecology 79, 2516–530.

    Google Scholar 

  • Harrison, S., 1997. Persistent localized outbreaks in the western tussock moth (Orgyia vetusta): the roles of resource quality, predation and poor dispersal. Ecol. Entomol. 22, 158–66.

    Article  Google Scholar 

  • Hajek, A., 2004. Natural Enemies: An Introduction to Biological Control. Cambridge University Press, Cambridge.

    Google Scholar 

  • Hastings, A., Harisson, S., McCann, K., 1997. Unexpected spatial patterns in an insect outbreak match a predator diffusion model. Proc. R. Soc. Lond. B 264, 1837–840.

    Article  Google Scholar 

  • Hosseini, P.R., 2006. Pattern formation and individual-based models: the importance of understanding individual-based movement. Ecol. Model. 194, 357–71.

    Article  Google Scholar 

  • Hunter, A.F., Dwyer, G., 1998. Outbreaks and interacting factors: insect population explosions synthesized and dissected. Integr. Biol. 1, 166–77.

    Article  Google Scholar 

  • Krischer, K., Mikhailov, A., 1994. Bifurcation to travelling spots in reaction-diffusion systems. Phys. Rev. Lett. 73, 3165–168.

    Article  Google Scholar 

  • Lindner, B., García-Ojalvo, J., Neiman, A., Schimansky-Geier, L., 2004. Effects of noise in excitable systems. Phys. Rep. 392, 321–24.

    Article  Google Scholar 

  • Lotka, A.J., 1925. Elements of Physical Biology. Williams and Wilkins, Baltimore.

    MATH  Google Scholar 

  • Ludwig, D., Jones, D.D., Holling, C.S., 1978. Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Ecol. 47, 315–32.

    Article  Google Scholar 

  • Malchow, H., Petrovskii, S.V., Venturino, E., 2008. Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall/CRC Press, London.

    MATH  Google Scholar 

  • May, R.M., 1974. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton.

    Google Scholar 

  • McEvoy, P., Cox, C., Coombs, E., 1991. Successful biological control of Ragwort, Senecio Jacobaea, by introduced insects in Oregon. Ecol. Appl. 1, 430–42.

    Article  Google Scholar 

  • Mendel, Z., Assael, F., Zeidan, S., Zehavi, A., 1998. Classical biological control of Palaeococcus fuscipennis (Burmeister) (Homoptera: Margarodidae) in Israel. Biol. Control 12, 151–57.

    Article  Google Scholar 

  • Mimura, M., Tabata, M., Hosono, Y., 1980. Multiple solutions of two-point boundary value problem of Neumann type with a small parameter. SIAM J. Math. Anal. 11, 613–31.

    Article  MATH  MathSciNet  Google Scholar 

  • Morozov, A.Y., Petrovskii, S.V., Li, B.-L., 2006. Spatiotemporal complexity of the patchy invasion in a predator-prey system with the Allee effect. J. Theor. Biol. 238, 18–5.

    Article  MathSciNet  Google Scholar 

  • Morozov, A.Y., Ruan, S., Li, B.L., 2008. Patterns of patchy spread in multi-species reaction–diffusion models. Ecol. Complex. 5, 313–28.

    Article  Google Scholar 

  • Muratov, C.B., 1996. Self-replication and splitting of domain patterns in reaction–diffusion systems with the fast inhibitor. Phys. Rev. E 54, 3369–376.

    Article  MathSciNet  Google Scholar 

  • Muratov, C.B., Osipov, V.V., 1996a. General theory of instabilities for patterns with sharp interfaces in reaction–diffusion systems. Phys. Rev. E 53, 3101–116.

    Article  MathSciNet  Google Scholar 

  • Muratov, C.B., Osipov, V.V., 1996b. Scenarios of domain pattern formation in a reaction–diffusion system. Phys. Rev. E 54, 4860–879.

    Article  MathSciNet  Google Scholar 

  • Murray, J.D., 1989. Mathematical Biology. Springer, Berlin.

    MATH  Google Scholar 

  • Nisbet, R.M., Gurney, W.S.C., 1982. Modelling Fluctuating Populations. Wiley, Chichester.

    MATH  Google Scholar 

  • Pascual, M., 1993. Diffusion-induced chaos in a spatial predator-prey system. Proc. R. Soc. Lond. B 251, 1–.

    Article  Google Scholar 

  • Pascual, M., Caswell, H., 1997. Environmental heterogeneity and biological pattern in a chaotic predator-prey system. J. Theor. Biol. 185, 1–3.

    Article  Google Scholar 

  • Pearson, J.E., 1993. Complex patterns in a simple system. Science 261, 189–92.

    Article  Google Scholar 

  • Petrovskii, S.V., Li, B.-L., 2001. Increased coupling between sub-populations in a spatially structured environment can lead to population outbreaks. J. Theor. Biol. 212, 549–62.

    Article  Google Scholar 

  • Petrovskii, S.V., Li, B.-L., 2006. Exactly Solvable Models of Biological Invasion. Chapman & Hall/CRC Press, London.

    MATH  Google Scholar 

  • Petrovskii, S.V., Malchow, H., 1999. A minimal model of pattern formation in a prey-predator system. Math. Comput. Model. 29, 49–3.

    Article  MATH  MathSciNet  Google Scholar 

  • Petrovskii, S.V., Malchow, H., 2001. Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol. 59, 157–74.

    Article  MATH  Google Scholar 

  • Petrovskii, S.V., Morozov, A.Y., Venturino, E., 2002. Allee effect makes possible patchy invasion in a predator-prey system. Ecol. Lett. 5, 345–52.

    Article  Google Scholar 

  • Petrovskii, S.V., Malchow, H., Hilker, F., Venturino, E., 2005. Patterns of patchy spread in deterministic and stochastic models of biological invasion and biological control. Biol. Invasions 7, 771–93.

    Article  Google Scholar 

  • Room, P.M., Harley, K.L.S., Forno, I.W., Sands, D.P.A., 1981. Successful biological control of the floating weed salvinia. Nature 294, 78–0.

    Article  Google Scholar 

  • Sapoukhina, N., Tyutyunov, Y., Arditi, R., 2003. The role of prey-taxis in biological control: a spatial theoretical model. Am. Nat. 162, 61–6.

    Article  Google Scholar 

  • Scheffer, M., 1991. Fish and nutrients interplay determines algal biomass: a minimal model. Oikos 62, 271–82.

    Article  Google Scholar 

  • Scheffer, M., 1998. Ecology of Shallow Lakes. Chapman & Hall, London.

    Google Scholar 

  • Segel, L.A., Jackson, J.L., 1972. Dissipative structure: an explanation and an ecological example. J. Theor. Biol. 37, 545–59.

    Article  Google Scholar 

  • Sherratt, J.A., Lewis, M.A., Fowler, A.C., 1995. Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 92, 2524–528.

    Article  MATH  Google Scholar 

  • Sherratt, J.A., Smith, M.J., 2008. Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models. Interface 5, 483–05.

    Article  Google Scholar 

  • Shigesada, N., Kawasaki, K., 1997. Biological Invasions: Theory and Practice. Oxford University Press, Oxford.

    Google Scholar 

  • Smayda, T.J., Shimuza, Y. (Eds.), 1993. Toxic Phytoplankton Blooms in the Sea. Elsevier, Amsterdam.

    Google Scholar 

  • Thomas, J., 1995. Numerical Partial Differential Equations: Finite Difference Methods. Texts in Applied Mathematics, vol. 22. Springer, New York.

    MATH  Google Scholar 

  • Truscott, J.E., Brindley, J., 1994a. Ocean plankton populations as excitable media. Bull. Math. Biol. 56, 981–98.

    MATH  Google Scholar 

  • Truscott, J.E., Brindley, J., 1994b. Equilibria, stability and excitability in a general class of plankton population models. Philos. Trans. R. Soc. Lond. A 347, 703–18.

    Article  MATH  Google Scholar 

  • Volterra, V., 1926. Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–60.

    Article  MATH  Google Scholar 

  • White, L.P., 1970. Brousse tigrée patterns in Southern Niger. J. Ecol. 58, 549–53.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK

    Andrew Morozov & Sergei Petrovskii

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  1. Andrew Morozov
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  2. Sergei Petrovskii
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Correspondence to Sergei Petrovskii.

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A. Morozov is on leave from Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prosp. 36, Moscow 117218, Russia.

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Morozov, A., Petrovskii, S. Excitable Population Dynamics, Biological Control Failure, and Spatiotemporal Pattern Formation in a Model Ecosystem. Bull. Math. Biol. 71, 863–887 (2009). https://doi.org/10.1007/s11538-008-9385-3

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  • DOI: https://doi.org/10.1007/s11538-008-9385-3

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