Abstract
Both community ecology and conservation biology seek further understanding of factors governing the advance of an invasive species. We model biological invasion as an individual-based, stochastic process on a two-dimensional landscape. An ecologically superior invader and a resident species compete for space preemptively. Our general model includes the basic contact process and a variant of the Eden model as special cases. We employ the concept of a “roughened” front to quantify effects of discreteness and stochasticity on invasion; we emphasize the probability distribution of the front-runner’s relative position. That is, we analyze the location of the most advanced invader as the extreme deviation about the front’s mean position. We find that a class of models with different assumptions about neighborhood interactions exhibits universal characteristics. That is, key features of the invasion dynamics span a class of models, independently of locally detailed demographic rules. Our results integrate theories of invasive spatial growth and generate novel hypotheses linking habitat or landscape size (length of the invading front) to invasion velocity, and to the relative position of the most advanced invader.
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Abramowitz, M., Stegun, I.A., 1972. Handbook of Mathematical Functions. National Bureau of Standards, Washington.
Allstadt, A., Caraco, T., Korniss, G., 2007. Ecological invasion: spatial clustering and the critical radius. Evol. Ecol. Res. 9, 1–20.
Andow, D.A., Kareiva, P.M., Levin, S.A., Okubo, A., 1990. Spread of invading organisms. Landsc. Ecol. 4, 177–188.
Antal, T., Droz, M., Györgyi, G., Rácz, Z., 2001. 1/f noise and extreme value statistics. Phys. Rev. Lett. 87, 240601. 4p.
Antal, T., Droz, M., Györgyi, G., Rácz, Z., 2002. Roughness distribution of 1/f α signals. Phys. Rev. E 65, 046140. 12p.
Antonovics, J., McKane, A.J., Newman, T.J., 2006. Spatiotemporal dynamics in marginal populations. Am. Nat. 167, 16–27.
Aronson, D.G., Weinberger, H.F., 1978. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76.
Aylor, D.E., 2003. Spread of plant disease on a continental scale: role of aerial dispersal of pathogens. Ecology 84, 1989–1997.
Barabási, A.-L., Stanley, H.E., 1995. Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge.
ben-Avraham, D., 1998. Fisher waves in the diffusion limited coalescence process. Phys. Lett. A 247, 53–58.
Berman, S.M., 1964. Limit theorems for the maximum term in stationary sequences. Ann. Math. Stat. 35, 502–516.
Bjornstad, O.N., Peltonin, M., Liebhold, A.M., Baltensweiler, W., 2002. Waves of larch budmoth outbreaks in the European Alps. Science 298, 1020–1023.
Blythe, R.A., Evans, M.R., 2001. Slow crossover to Kardar–Parisi–Zhang scaling. Phys. Rev. E 64, 051101, 5 p.
Brú, A., Albertos, S., Subiza, J.L., García-Asenjo, J.L., Brú, I., 2003. The universal dynamics of tumor growth. Biophys. J. 85, 2948–2961.
Cain, M.L., Pacala, S.W., Silander, J.A. Jr., Fortin, M.-J., 1995. Neighborhood models of clonal growth in the white clover Trifolium repens. Am. Nat. 145, 888–917.
Cannas, S.A., Marco, D.E., Montemurro, M.A., 2006. Long range dispersal and spatial pattern formation in biological invasions. Math. Biosci. 203, 155–170.
Cantrell, R.S., Cosner, C., 1991. The effect of spatial heterogeneity in population dynamics. J. Math. Biol. 29, 315–338.
Caraco, T., Glavanakov, S., Chen, G., Flaherty, J.E., Ohsumi, T.K., Szymanski, B.K., 2002. Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease. Am. Nat. 160, 348–359.
Cardy, J., 1996. Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge.
Clark, J.S., Fastie, C., Hurtt, G., Jackson, S.T., Johnson, C., King, G.A., Lewis, M., Lynch, J., Pacala, S., Prentice, C., Schupp, E.W., Webb, T., III, Wyckoff, P., 1998. Reid’s paradox of rapid plant migration. BioScience 48, 13–24.
Clark, J.S., Lewis, M., Horvath, L., 2001. Invasion by extremes: population spread with variation in dispersal and reproduction. Am. Nat. 157, 537–554.
Clark, J.S., Lewis, M., McLachlan, J.S., HilleRisLambers, J., 2003. Estimating population spread: what can we forecast and how well? Ecology 84, 1979–1988.
Comins, H.N., Noble, I.R., 1985. Dispersal, variability, and transient niches: species coexistence in a uniformly variable environment. Am. Nat. 126, 706–723.
Connolly, S.R., Muko, S., 2003. Space preemption, size-dependent competition and the coexistence of clonal growth forms. Ecology 84, 2979–2988.
D’Antonio, C.M., 1993. Mechanisms controlling invasion of coastal plant communities by the alien succulent Carpobrotus edulis. Ecology 74, 83–95.
DeAngelis, D.L., Gross, L.J. (Eds.), 1992. Individual-Based Models and Approaches in Ecology. Routledge, Chapman and Hall, New York.
Doering, C.R., Mueller, C., Smereka, P., 2003. Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A 325, 243–259.
Doi, M., 1976. Stochastic theory of diffusion-controlled reaction. J. Phys. A 9, 1479–1495.
Durrett, R., Levin, S.A., 1994a. Stochastic spatial models: a user’s guide to ecological applications. Philos. Trans. R. Soc. Lond. B 343, 329–350.
Durrett, R., Levin, S.A., 1994b. The importance of being discrete (and spatial). Theor. Popul. Biol. 46, 363–394.
Dwyer, G., 1992. On the spatial spread of insect pathogens: theory and experiment. Ecology 73, 479–494.
Dwyer, G., Elkinton, S., 1995. Host dispersal and the spatial spread of insect pathogens. Ecology 76, 1262–1275.
Dwyer, G., Morris, W.F., 2006. Resource-dependent dispersal and the speed of biological invasions. Am. Nat. 167, 165–176.
Eden, M., 1961. A two-dimensional growth process. In: Neyman, J. (Ed.), 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 223–239. University of California Press, Berkeley.
Ellner, S.P., Sasaki, A., Haraguchi, Y., Matsuda, H., 1998. Speed of invasion in lattice population models: pair-edge approximation. J. Math. Biol. 36, 469–484.
Elton, C.S., 1958. The Ecology of Invasions by Animals and Plants. Methuen, London.
Escudero, C., Buceta, J., de la Rubia, F.J., Lindenberg, K., 2004. Extinction in population dynamics. Phys. Rev. E 69, 021908, 9 p.
Family, F., Vicsek, T., 1985. Scaling of the active zone in the Eden process on percolation networks and the ballistic deposition model. J. Phys. A 18, L75–L81.
Ferrandino, F.J., 1996. Length scale of disease spread: fact or artifact of experimental geometry? Phytopathology 86, 806–811.
Ferreira, S.C. Jr., Alves, S.G., 2006. Pitfalls in the determination of the universality class of radial clusters. J. Stat. Mech. 11, P11007, 11 p.
Fisher, M.C., Koenig, G.L., White, T.J., Sans-Blas, G., Negroni, R., Alvarez, I.G., Wanke, B., Taylor, J.W., 2001. Biogeographic range expansion into South America by Coccidioides immitis mirrors New World patterns of human migration. Proc. Nat. Acad. Sci. USA 98, 4558–4562.
Fisher, R.A., 1937. The wave of advance of advantageous genes. Ann. Eugen. Lond. 7, 355–369.
Fisher, R.A., Tippett, L.H.C., 1928. The frequency distribution of the largest or smallest member of a sample. Proc. Camb. Philos. Soc. 24, 180–191.
Foltin, G., Oerding, K., Rácz, Z., Workman, R.L., Zia, R.K.P., 1994. Width distribution for random-walk interfaces. Phys. Rev. E 50, R639–R642.
Frantzen, J., van den Bosch, F., 2000. Spread of organisms: can travelling and dispersive waves be distinguished? Basic Appl. Ecol. 1, 83–91.
Galambos, J., 1987. The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Krieger Publishing, Malabar.
Galambos, J., Lechner, J., Simin, E. (Eds.), 1994. Extreme Value Theory and Applications. Kluwer, Dordrecht.
Gandhi, A., Levin, S., Orszag, S., 1999. Nucleation and relaxation from meta-stability in spatial ecological models. J. Theor. Biol. 200, 121–146.
Gardiner, C.W., 1985. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd edn. Springer, Berlin.
Guclu, H., Korniss, G., 2004. Extreme fluctuations in small-worlds with relaxational dynamics. Phys. Rev. E 69, 065104(R), 4 p.
Guclu, H., Korniss, G., Toroczkai, Z., 2007. Extreme fluctuations in noisy task-completion landscapes on scale-free networks. Chaos 17, 026104, 13 p.
Gumbel, E.J., 1958. Statistics of Extremes. Columbia University Press, New York.
Halpin-Healy, T., Zhang, Y.-C., 1995. Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phys. Rep. 254, 215–414.
Harris, T.E., 1974. Contact interaction on a lattice. Ann. Probab. 2, 969–988.
Hastings, A., Cuddington, K., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A., Harrison, S., Holland, M., Lambrinos, J., Malvadkar, U., Melbourne, B.A., Moore, K., Taylor, C., Thomson, D., 2005. The spatial spread of invasions: new developments in theory and evidence. Ecol. Lett. 8, 91–101.
Hinrichsen, H., 2000. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815–958.
Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R., 1994. Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29.
Holway, D.A., 1998. Factors governing rate of invasion: a natural experiment using Argentine ants. Oecologia 115, 206–212.
Hoopes, M.F., Hall, L.M., 2002. Edaphic factors and competition affect pattern formation and invasion in a California grassland. Ecol. Appl. 12, 24–39.
Hosono, Y., 1998. The minimal speed of travelling fronts for a diffusive Lotka-Volterra competition model. Bull. Math. Biol. 60, 435–448.
Jullien, R., Botet, R., 1985a. Surface thickness in the Eden model. Phys. Rev. Lett. 54, 2055.
Jullien, R., Botet, R., 1985b. Scaling properties of the surface of the Eden model. J. Phys. A 18, 2279–2287.
Kardar, M., Parisi, G., Zhang, Y.-C., 1986. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892.
Kawasaki, K., Takasu, F., Caswell, H., Shigesada, N., 2006. How does stochasticity in colonization accelerate the speed of invasion in a cellular automaton model? Ecol. Res. 21, 334–345.
Kertész, J., Wolf, D.E., 1988. Noise reduction in Eden models: II. Surface structure and intrinsic width. J. Phys. A, Math. Gen. 21, 747–761.
Kolmogorov, A., Petrovsky, N., Pishkounov, N.S., 1937. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Mosc. Univ. Bull. Math. 1, 1–25.
Korniss, G., Caraco, T., 2005. Spatial dynamics of invasion: the geometry of introduced species. J. Theor. Biol. 233, 137–150.
Korniss, G., Schmittmann, B., 1997. Structure factors and their distributions in driven two-species models. Phys. Rev. E 56, 4072–4084.
Korniss, G., Toroczkai, Z., Novotny, M.A., Rikvold, P.A., 2000. From massively parallel algorithms and fluctuating time horizons to nonequilibrium surface growth. Phys. Rev. Lett. 84, 1351–1354.
Korniss, G., Novotny, M.A., Guclu, H., Toroczkai, Z., Rikvold, P.A., 2003. Suppressing roughness of virtual times in parallel discrete-event simulations. Science 299, 677–679.
Kot, M., Lewis, M.A., van den Driessche, P., 1996. Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042.
Krug, J., Meakin, P., 1990. Universal finite-size effects in the rate of growth processes. J. Phys. A 23, L987–L994.
Lewis, M.A., 1997. Variability, patchiness, and jump dispersal in the spread of an invading population. In: Tilman, D., Kareiva, P. (Eds.), Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, pp. 46–69. Princeton University Press, Princeton.
Lewis, M.A., 2000. Spread rate for a nonlinear stochastic invasion. J. Math. Biol. 41, 430–454.
Lewis, M.A., Li, B., Weinberger, H.F., 2002. Spreading speed and linear determinacy for two-species competition models. J. Math. Biol. 45, 219–233.
Lockwood, J.L., Hoopes, M.F., Marchetti, M., 2007. Invasion Ecology. Blackwell, Malden.
Majumdar, S.N., Comtet, A., 2004. Exact maximal height distribution of fluctuation interfaces. Phys. Rev. Lett. 92, 225501, 4 p.
Majumdar, S.N., Comtet, A., 2005. Airy distribution function: from the area under a Brownian excursion to the maximal height of fluctuating interfaces. J. Stat. Phys. 119, 776–826.
McKane, A.J., Newman, T.J., 2004. Stochastic models in population biology and their deterministic analogues. Phys. Rev. E 70, 041902, 19 p.
Minogue, K.P., Fry, W.E., 1983. Models for the spread of plant disease: some experimental results. Phytopathology 73, 1173–1176.
Mollison, D., Levin, S.A., 1995. Spatial dynamics of parasitism. In: Grenfell, B.T., Dobson, A.P. (Eds.), Ecology of Infectious Diseases in Natural Populations, pp. 384–398. Cambridge University Press, Cambridge.
Moro, E., 2001. Internal fluctuations effects on Fisher waves. Phys. Rev. Lett. 87, 238303, 4 p.
Moro, E., 2003. Emergence of pulled fronts in fermionic microscopic particle models. Phys. Rev. E 68, 025102, 4 p.
Murray, J.D., 2003. Mathematical Biology, vol. 2. Springer, New York.
Nash, D.R., Agassiz, D.J.L., Godfray, H.C.J., Lawton, J.H., 1995. The pattern of spread of invading species: two leaf-mining moths colonizing Great Britain. J. Anim. Ecol. 64, 225–233.
Neubert, M.G., Caswell, H., 2000. Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations. Ecology 81, 1613–1628.
Oborny, B., Meszéna, G., Szabó, G., 2005. Dynamics of populations on the verge of extinction. Oikos 109, 291–296.
O’Malley, L., Allstadt, A., Korniss, G., Caraco, T., 2005. Nucleation and global time scales in ecological invasion under preemptive competition. In: Stocks, N.G., Abbott, D., Morse, R.P. (Eds.), Fluctuations and Noise in Biological, Biophysical, and Biomedical Systems III, pp. 117–124. SPIE, Pullman.
O’Malley, L., Basham, J., Yasi, J.A., Korniss, G., Allstadt, A., Caraco, T., 2006a. Invasive advance of an advantageous mutation: nucleation theory. Theor. Popul. Biol. 70, 464–478.
O’Malley, L., Kozma, B., Korniss, G., Rácz, Z., Caraco, T., 2006b. Fisher waves and front propagation in a two-species invasion model with preemptive competition. Phys. Rev. E 74, 041116, 7 p.
O’Malley, L., Kozma, B., Korniss, G., Rácz, Z., Caraco, T., 2009. Fisher waves and the velocity of front propagation in a two-species invasion model with preemptive competition. In: Landau, D.P., Lewis, S.P., Schüttler, H.-B. (Eds.), Computer Simulation Studies in Condensed Matter Physics XIX, Springer Proceedings in Physics, vol. 123, pp. 73–78. Springer, Heidelberg.
Parker, I.M., Reichard, S.H., 1998. Critical issues in invasion biology for conservation science. In: Fieldler, P.L., Kareiva, P.M. (Eds.), Conservation Biology, 2nd edn., pp. 283–305. Chapman and Hall, New York.
Pechenik, L., Levine, H., 1999. Interfacial velocity corrections due to multiplicative noise. Phys. Rev. E 59, 3893–3900.
Peliti, L., 1985. Path integral approach to birth-death processes on a lattice. J. Phys. (Paris) 46, 1469–1483.
Pimentel, D., Lach, L., Zuniga, R., Morrison, D., 2000. Environmental and economic costs of nonindigenous species in the United States. Bioscience 50, 53–65.
Plischke, M., Rácz, Z., 1985. Dynamic scaling and the surface structure of Eden clusters. Phys. Rev. A 32, 3825–3828.
Plischke, M., Rácz, Z., Liu, D., 1987. Time-reversal invariance and universality of two-dimensional growth models. Phys. Rev. B 35, 3485–3495.
Rácz, Z., Gálfi, L., 1988. Properties of the reaction front in an A+B→C type reaction–diffusion process. Phys. Rev. A 38, 3151–3154.
Raychaudhuri, S., Cranston, M., Przybyla, C., Shapir, Y., 2001. Maximal height scaling of kinetically growing surfaces. Phys. Rev. Lett. 87, 136101, 4 p.
Rosenzweig, M.L., 2001. The four questions: what does the introduction of exotic species do to diversity? Evol. Ecol. Res. 3, 361–371.
Ruesink, J.L., Parker, I.M., Groom, M.J., Kareiva, P.M., 1995. Reducing the risks of nonindigenous introductions: guilty until proven innoent. BioScience 45, 465–477.
Ruiz, G.M., Rawlings, T.K., Dobbs, F.C., Huq, A., Colwell, R., 2000. Global spread of microorganisms by ships. Nature 408, 49.
Schehr, G., Majumdar, S.N., 2006. Universal asymptotic statistics of a maximal relative height in one-dimensional solid-on-solid models. Phys. Rev. E 73, 056103, 10 p.
Schmittmann, B., Zia, R.K.P., 1995. Statistical Mechanics of Driven Diffusive Systems. Phase Transitions and Critical Phenomena, vol. 17. Academic Press, New York.
Schwinning, S., Parsons, A.J., 1996. A spatially explicit population model of stoloniferous N-fixing legumes in mixed pasture with grass. J. Ecol. 84, 815–826.
Shigesada, N., Kawasaki, K., 1997. Biological Invasions: Theory and Practice. Oxford University Press, Oxford.
Shigesada, N., Kawasaki, K., Takeda, Y., 1995. Modeling stratified diffusion in biological invasions. Am. Nat. 146, 229–251.
Silvertown, J., Lines, C.E.M., Dale, M.P., 1994. Spatial competition between grasses—rates of mutual invasion between four species and the interaction with grazing. J. Ecol. 82, 31–38.
Simberloff, D., Relva, M.A., Nuñez, M., 2002. Gringos en el bosque: introduced tree invasion in a native Nothofagus/Austrocedrus forest. Biol. Invasions 4, 35–53.
Snyder, R.E., 2003. How demographic stochasticity can slow biological invasions. Ecology 84, 1333–1339.
Tainaka, K., Kushida, M., Itoh, Y., Yoshimura, J., 2004. Interspecific segregation in a lattice ecosystem with intraspecific competition. J. Phys. Soc. Jpn. 73, 2914–2915.
Thomson, N.A., Ellner, S.P., 2003. Pair-edge approximation for heterogeneous lattice population models. Theor. Popul. Biol. 64, 270–280.
van Baalen, M., Rand, D.A., 1998. The unit of selection in viscous populations and the evolution of altruism. J. Theor. Biol. 193, 631–648.
van den Bosch, F., Hengeveld, R., Metz, J.A.J., 1992. Analysing the velocity of animal range expansion. J. Biogeogr. 19, 135–150.
van Kampen, N.G., 1976. The expansion of the master equation. Adv. Chem. Phys. 34, 245–309.
van Kampen, N.G., 1981. Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam.
van Saarloos, W., 2003. Front propagation into unstable states. Phys. Rep. 386, 29–222.
Weinberger, H.F., Lewis, M.A., Li, B.T., 2002. Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218.
Wilson, W., 1998. Resolving discrepancies between deterministic population models and individual-based simulations. Am. Nat. 151, 116–134.
Wilson, W., de Roos, A.M., McCauley, E., 1993. Spatial instabilities within the diffusive Lotka–Volterra system: individual-based simulation results. Theor. Popul. Biol. 43, 91–127.
Yasi, J., Korniss, G., Caraco, T., 2006. Invasive allele spread under preemptive competition. In: Landau, D.P., Lewis, S.P., Schüttler, H.-B. (Eds.), Computer Simulation Studies in Condensed Matter Physics XVIII, Springer Proceedings in Physics, vol. 105, pp. 165–169. Springer, Heidelberg.
Yurkonis, K.A., Meiners, S.J., 2004. Invasion impacts local species turnover in a successional system. Ecol. Lett. 4, 764–769.
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O’Malley, L., Korniss, G. & Caraco, T. Ecological Invasion, Roughened Fronts, and a Competitor’s Extreme Advance: Integrating Stochastic Spatial-Growth Models. Bull. Math. Biol. 71, 1160–1188 (2009). https://doi.org/10.1007/s11538-009-9398-6
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DOI: https://doi.org/10.1007/s11538-009-9398-6