Bulletin of Mathematical Biology

, Volume 74, Issue 2, pp 453–473 | Cite as

Success rate of a biological invasion in terms of the spatial distribution of the founding population

  • Jimmy Garnier
  • Lionel Roques
  • François Hamel
Original Article


We analyze the role of the spatial distribution of the initial condition in reaction–diffusion models of biological invasion. Our study shows that, in the presence of an Allee effect, the precise shape of the initial (or founding) population is of critical importance for successful invasion. Results are provided for one-dimensional and two-dimensional models. In the one-dimensional case, we consider initial conditions supported by two disjoint intervals of length L/2 and separated by a distance α. Analytical as well as numerical results indicate that the critical size L (α) of the population, where the invasion is successful if and only if L>L (α), is a continuous function of α and tends to increase with α, at least when α is not too small. This result emphasizes the detrimental effect of fragmentation. In the two-dimensional case, we consider more general, stochastically generated initial conditions u 0, and we provide a new and rigorous definition of the rate of fragmentation of u 0. We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of u 0 and the fragmentation rate of u 0. Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a non-linear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.


Reaction–diffusion Allee effect Biological invasions Fragmentation Initial condition 


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  1. Allee, W. C. (1938). The social life of animals. New York: Norton. CrossRefGoogle Scholar
  2. Aronson, D. G., & Weinberger, H. G. (1978). Multidimensional non-linear diffusion arising in population-genetics. Advances in Mathematics, 30(1), 33–76. MathSciNetzbMATHCrossRefGoogle Scholar
  3. Berec, L., Angulo, E., & Courchamp, F. (2007). Multiple Allee effects and population management. Trends in Ecology & Evolution, 22, 185–191. CrossRefGoogle Scholar
  4. Berestycki, H., Hamel, F., & Nadin, G. (2008). Asymptotic spreading in heterogeneous diffusive excitable media. Journal of Functional Analysis, 255(9), 2146–2189. MathSciNetzbMATHCrossRefGoogle Scholar
  5. Berestycki, H., Hamel, F., & Roques, L. (2005). Analysis of the periodically fragmented environment model: I - Species persistence. Journal of Mathematical Biology, 51(1), 75–113. MathSciNetzbMATHCrossRefGoogle Scholar
  6. Berestycki, H., Hamel, F., & Rossi, L. (2007). Liouville-type results for semilinear elliptic equations in unbounded domains. Annali Di Matematica Pura Ed Applicata, 186(3), 469–507. MathSciNetzbMATHCrossRefGoogle Scholar
  7. Cantrell, R. S., & Cosner, C. (2003). Spatial ecology via reaction–diffusion equations. Chichester: Wiley. zbMATHGoogle Scholar
  8. DAISIE (2009). Handbook of alien species in Europe. Dordrecht: Springer. Google Scholar
  9. DAISIE (2010). BioRisk 4: Alien terrestrial arthropods of Europe, vol. 1 and 2. Sofia/Moscow: Pensoft. Google Scholar
  10. Dennis, B. (1989). Allee effects: population growth, critical density, and the chance of extinction. Natural Resource Modeling, 3, 481–538. MathSciNetzbMATHGoogle Scholar
  11. Dobson, A. P., & May, R. M. (1986). Patterns of invasions by pathogens and parasites. In H. A. Mooney & J. A. Drake (Eds.), Ecology of biological invasions of north America and Hawaii (pp. 58–76). New York: Springer. CrossRefGoogle Scholar
  12. Drake, J. M. (2004). Allee effects and the risk of biological invasion. Risk Analysis, 24, 795–802. CrossRefGoogle Scholar
  13. Drury, K. L. S., Drake, J. M., Lodge, D. M., & Dwyer, G. (2007). Immigration events dispersed in space and time: Factors affecting invasion success. Ecological Modelling, 206, 63–78. CrossRefGoogle Scholar
  14. Du, Y., & Matano, H. (2010). Convergence and sharp thresholds for propagation in nonlinear diffusion problems. Journal of the European Mathematical Society, 12, 279–312. MathSciNetzbMATHCrossRefGoogle Scholar
  15. Fahrig, L. (2003). Effects of habitat fragmentation on biodiversity. Annual Review of Ecology, Evolution, and Systematics, 34, 487–515. CrossRefGoogle Scholar
  16. Fife, P. C. (1979). Long-time behavior of solutions of bistable non-linear diffusion equations. Archive for Rational Mechanics and Analysis, 70(1), 31–46. MathSciNetzbMATHCrossRefGoogle Scholar
  17. Fife, P. C., & McLeod, J. (1977). The approach of solutions of nonlinear diffusion equations to traveling front solutions. Archive for Rational Mechanics and Analysis, 65(1), 335–361. MathSciNetzbMATHGoogle Scholar
  18. Fisher, R. A. (1937). The wave of advance of advantageous genes. Annals of Eugenics, 7, 335–369. Google Scholar
  19. Friedman, A. (1964). Partial differential equations of parabolic type. Englewood Cliffs: Prentice-Hall. zbMATHGoogle Scholar
  20. Gardner, R. H., Milne, B. T., Turner, M. G., & O’Neill, R. V. (1987). Neutral models for the analysis of broad-scale landscape pattern. Landscape Ecology, 1, 19–28. CrossRefGoogle Scholar
  21. Hamel, F., Fayard, J., & Roques, L. (2010). Spreading speeds in slowly oscillating environments. Bulletin of Mathematical Biology, 72(5), 1166–1191. MathSciNetzbMATHCrossRefGoogle Scholar
  22. Harary, F., & Harborth, H. (1976). Extremal animals. Journal of Combinatorics, Information & System Sciences, 1, 1–8. MathSciNetzbMATHGoogle Scholar
  23. IUCN (2000). Guidelines for the prevention of biodiversity loss caused by alien invasive species prepared by the Species Survival Commission (SSC) invasive species specialist group. Approved by the 51st Meeting of the IUCN Council, Gland. Google Scholar
  24. IUCN (2002). Policy recommendations papers for sixth meeting of the Conference of the Parties to the Convention on Biological Diversity (COP6). The Hague, Netherlands. Google Scholar
  25. Kanarek, A. R., & Webb, C. T. (2010). Allee effects, adaptive evolution, and invasion success. Evolutionary Applications, 3, 122–135. CrossRefGoogle Scholar
  26. Kanel, J. I. (1964). Stabilization of solutions of the equations of combustion theory with finite initial functions. Matematicheskii Sbornik, 65, 398–413. MathSciNetGoogle Scholar
  27. Keitt, T. H. (2000). Spectral representation of neutral landscapes. Landscape Ecology, 15, 479–494. CrossRefGoogle Scholar
  28. Keitt, T. H., Lewis, M. A., & Holt, R. D. (2001). Allee effects, invasion pinning, and species’ borders. The American Naturalist, 157, 203–216. CrossRefGoogle Scholar
  29. Kenis, M. (2006). Insects-insecta. In R. Wittenberg (Ed.), Invasive alien species in Switzerland. An inventory of alien species and their threat to biodiversity and economy in Switzerland (pp. 131–211). Swiss Confederation—Federal Office for the Environment Environmental Studies. Google Scholar
  30. Kenis, M., Auger-Rozenberg, M.-A., Roques, A., Timms, L., Péré, C., Cock, M. J. W., Settele, J., Augustin, S., & Lopez-Vaamonde, C. (2009). Ecological effects of invasive alien insects. Biological Invasions, 11(1), 21–45. CrossRefGoogle Scholar
  31. Kirkpatrick, M., & Barton, N. H. (1997). Evolution of a species’ range. The American Naturalist, 150, 1–23. CrossRefGoogle Scholar
  32. Kolmogorov, A. N., Petrovsky, I. G., & Piskunov, N. S. (1937). Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin de l’Université d’État de Moscou, Série Internationale A, 1, 1–26. Google Scholar
  33. Kramer, A. M., Dennis, B., Liebhold, A. M., & Drake, J. M. (2009). The evidence for Allee effects. Population Ecology, 51, 341–354. CrossRefGoogle Scholar
  34. Lande, R. (1998). Demographic stochasticity and Allee effect on a scale with isotrophic noise. Oikos, 83, 353–358. CrossRefGoogle Scholar
  35. Leung, B., Drake, J. M., & Lodge, D. M. (2004). Predicting invasions: propagule pressure and the gravity of Allee effects. Ecology, 85, 1651–1660. CrossRefGoogle Scholar
  36. Lewis, M. A., & Kareiva, P. (1993). Allee dynamics and the speed of invading organisms. Theoretical Population Biology, 43, 141–158. zbMATHCrossRefGoogle Scholar
  37. Mccarthy, M. A. (1997). The Allee effect, finding mates and theoretical models. Ecological Modelling, 103(1), 99–102. CrossRefGoogle Scholar
  38. Murray, J. D. (2002). Interdisciplinary applied mathematics: Vol. 17. Mathematical biology (3rd ed.). New York: Springer. zbMATHGoogle Scholar
  39. Pease, C. P., Lande, R., & Bull, J. J. (1989). A model of population growth, dispersal and evolution in a changing environment. Ecology, 70, 1657–1664. CrossRefGoogle Scholar
  40. Protter, M. H., & Weinberger, H. F. (1967). Maximum principles in differential equations. Englewood Cliffs: Prentice-Hall. Google Scholar
  41. Richardson, D. M., Pyšek, P., Rejmánek, M., Barbour, M. G., Panetta, F. Dane, & West, C. J. (2000). Naturalization and invasion of alien plants: concepts and definitions. Diversity and Distributions, 6, 93–107. CrossRefGoogle Scholar
  42. Roques, A., Rabitsch, W., Rasplus, J.-Y., Lopez-Vaamonde, C., Nentwig, W., & Kenis, M. (2009). Alien terrestrial invertebrates of Europe. Dordrecht: Springer. Google Scholar
  43. Roques, L., & Chekroun, M. D. (2007). On population resilience to external perturbations. SIAM Journal on Applied Mathematics, 68(1), 133–153. MathSciNetzbMATHCrossRefGoogle Scholar
  44. Roques, L., & Chekroun, M. D. (2010). Does reaction–diffusion support the duality of fragmentation effect? Ecological Complexity, 7, 100–106. CrossRefGoogle Scholar
  45. Roques, L., & Hamel, F. (2007). Mathematical analysis of the optimal habitat configurations for species persistence. Mathematical Biosciences, 210(1), 34–59. MathSciNetzbMATHCrossRefGoogle Scholar
  46. Roques, L., & Stoica, R. S. (2007). Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments. Journal of Mathematical Biology, 55(2), 189–205. MathSciNetzbMATHCrossRefGoogle Scholar
  47. Roques, L., Roques, A., Berestycki, H., & Kretzschmar, A. (2008). A population facing climate change: joint influences of Allee effects and environmental boundary geometry. Population Ecology, 50(2), 215–225. CrossRefGoogle Scholar
  48. Shigesada, N., & Kawasaki, K. (1997). Biological invasions: theory and practice. Oxford Series in Ecology and Evolution. Oxford: Oxford University Press. Google Scholar
  49. Skellam, J. G. (1951). Random dispersal in theoretical populations. Biometrika, 38, 196–218. MathSciNetzbMATHGoogle Scholar
  50. Turchin, P. (1998). Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer Associates, Sunderland, MA. Google Scholar
  51. Veit, R. R., & Lewis, M. A. (1996). Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America. The American Naturalist, 148, 255–274. CrossRefGoogle Scholar
  52. Vilà, M., Pyšek, B. C. P., Josefsson, M., Genovesi, P., Gollasch, S., Nentwig, W., Olenin, S., Roques, A., Roy, D., Hulme, P. E., & Partners, D. (2009). How well do we understand the impacts of alien species on ecosystem services? A pan-European cross-taxa assessment. Frontiers in Ecology and the Environment. Google Scholar
  53. Walther, G.-R., Roques, A., Hulme, P. E., Sykes, M. T., Pyšek, P., Kűhn, I., Zobel, M., Bacher, S., Botta-Dukát, Z., Bugmann, H., Czúcz, B., Dauber, J., Hickler, T., Jarošík, V., Kenis, M., Klotz, S., Minchin, D., Moora, M., Nentwig, W., Ott, J., Panov, V. E., Reineking, B., Robinet, C., Semenchenko, V., Solarz, W., Thuiller, W., Vilà, M., Vohland, K., & Settele, J. (2009). Alien species in a warmer world: risks and opportunities. Trends in Ecology & Evolution, 24(12), 686–693. CrossRefGoogle Scholar
  54. Yamanaka, T., & Liebhold, A. M. (2009). Mate-location failure, the Allee effect, and the establishment of invading populations. Population Ecology, 51, 337–340. CrossRefGoogle Scholar
  55. Zlatoš, A. (2006). Sharp transition between extinction and propagation of reaction. Journal of the American Mathematical Society, 19, 251–263. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  • Jimmy Garnier
    • 1
    • 2
  • Lionel Roques
    • 1
  • François Hamel
    • 2
    • 3
  1. 1.UR 546 Biostatistique et Processus SpatiauxINRAAvignonFrance
  2. 2.LATP, Faculté des Sciences et TechniquesAix-Marseille UniversitéMarseille Cedex 20France
  3. 3.Institut Universitaire de FranceParisFrance

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