Bulletin of Mathematical Biology

, Volume 67, Issue 2, pp 281–312 | Cite as

The Africanized honey bee dispersal: A mathematical zoom

  • Diomar Cristina Mistro
  • Luiz Alberto Díaz Rodrigues
  • Wilson Castro FerreiraJr.


A general mathematical model for population dispersal featuring long range taxis is presented and exemplified by the dispersal episode of the Africanized honey bees (Apis mellifera adansonii) throughout the American Continent. The mathematical model is a discrete-time and nonlocal model represented by an integrodifference recursion. A newtaxis concept is defined and introduced into the mathematical model by an appropriate modification of the redistribution kernel. The model is capable of predicting the natural barrier for the expansion of the Africanized honey bees in the southern part of the Continent due to low winter temperatures. It also describes a sensitive expansion velocity with respect to the quality of resources, which can explain the AHB’s astounding spread rate, by using two different kinds of population dynamics strategies, one for a resourceful environment and the other for poor regions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allen, E.J., Allen, L.J.S., Xiaoning, G., 1996. Dispersal and competition models for plants. J. Math. Biol. 34, 455–481.MATHMathSciNetGoogle Scholar
  2. Andersen, M., 1991. Properties of some density-dependent integrodifference equation population models. Math. Biosci. 104, 135–157.MATHMathSciNetCrossRefGoogle Scholar
  3. Andow, D.A., Kareiva, P.M., Levin, S.A., Okubo, A., 1993. Spread of invading organisms: patterns of spread. Landscape Ecol. 4, 177–188.CrossRefGoogle Scholar
  4. Bowman, J., Cappuccino, N., Fahrig, L., 2002. Patch size and population density: the effect of immigration behavior. Conservation Ecol. 6. Online: http://www.consecol.org/vol6/issl/art9.
  5. Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E., 2001. Self-Organization in Biological Systems. Princeton University Press.Google Scholar
  6. Edelstein-Keshet, L., 1988. Mathematical Models in Biology. Random House.Google Scholar
  7. Fletcher, D.J.C., 1978. The African bee, Apis mellifera adansonii, in Africa. Annu. Rev. Entomol. 23, 151–171.CrossRefGoogle Scholar
  8. Gonçalves, L.S., 1974. The introduction of the African bees (Apis mellifera adansonii) into Brazil and some comments on their spread in South America. Am. Bee J. 114, 414–416.Google Scholar
  9. Hart, D.R., Gardner, R.H., 1997. A spatial model for the spread of invading organisms subject to competition. J. Math. Biol. 35, 935–948.MATHMathSciNetCrossRefGoogle Scholar
  10. Heinrich, B., Esch, H., 1994. Thermoregulation in bees. Am. Sci. 82, 164–170.Google Scholar
  11. Hinderer, T.E., Collins, A.M., 1991. Foraging behavior and honey production. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 235–257.Google Scholar
  12. Hinderer, T.E., Hellmich, R.L., 1991. The process of africanization. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 95–117.Google Scholar
  13. Keller, E.F., Segel, L.A., 1970. Initial of slime mold aggregation viewed as a instability. J. Theor. Biol. 26, 399–415.CrossRefGoogle Scholar
  14. Keller, E.F., Segel, L.A., 1971. Model for chemotaxis. J. Theor. Biol. 30, 225–234.CrossRefGoogle Scholar
  15. Kerr, W.E., del Rio, S.L., Barrionuevo, M.D., 1982. Distribuição da Abelha Africanizada e seus Limites ao Sul. Ciênc. Cultura 34, 1439–1442.Google Scholar
  16. Kerr W.E., Gonçalves L.S., Blotta L.F., Maciel H.B., 1972. Biologia Comparada entre as Abelhas Italianas (Apis mellifera ligustica), Africana (Apis mellifera adansonii) e suas Híbridas. 1o Congresso Brasileiro de Apicultura, Florianópolis, Brazil.Google Scholar
  17. Kot, M., 1992. Discrete-time travelling waves: ecological examples. J. Math. Biol. 30, 413–430.MATHMathSciNetCrossRefGoogle Scholar
  18. Kot, M., Lewis, M.A., van den Driessche, P., 1996. Dispersal data and the spread of invading organisms. Ecology 77, 2027–2042.CrossRefGoogle Scholar
  19. Kot, M., Schaffer, W.M., 1986. Discrete-time growth-dispersal models. Math. Biosci. 80, 109–136.MATHMathSciNetCrossRefGoogle Scholar
  20. Lindauer, M., 1961. Communication Among Social Bees. Harvard University Press, Cambridge.Google Scholar
  21. Mistro, D.C. 1998. Dispersão de Abelhas Africanizadas: Um Zoom Matemático, Tese de Doutorado, Universidade Estadual de Campinas, Brasil.Google Scholar
  22. Moritz, R.F.A., Southwick, E.E., 1992. Bees as Superorganisms: An Evolution Reality. Springer-Verlag, New York.Google Scholar
  23. Myerscough, M.R., 2003. Dancing for a decision: a matrix model for nest-site choice by honeybees. Proc. R. Soc. Lond. B, Online 3 February 2003.Google Scholar
  24. Murray, J.D., 1989. Mathematical Biology. Springer-Verlag, Berlin.Google Scholar
  25. Neubert, M., Kot, M., Lewis, M.A., 2000. Invasion speeds in fluctuating environments. Proc. R. Soc. Lond. B 267, 1603–1610.CrossRefGoogle Scholar
  26. Okubo, A., Levin, S.A., 2001. Diffusion and Ecological Problems. Modern Perspectives. Springer.Google Scholar
  27. Otis, G.W., 1991. Population biology of the Africanized honey bee. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 213–233.Google Scholar
  28. Otis, G.W., 1982. Population biology of the Africanized honey bee. In: Jaisson, P. (Ed.), Social Insects in The Tropics. University of Paris, pp. 209–219.Google Scholar
  29. Ratnieks, F.L., 1991. Africanized bees: natural selection for colonizing ability. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 119–135.Google Scholar
  30. Ricker, W.E., 1954. Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623.Google Scholar
  31. Rodrigues, L.A.D. 1998. O Acaso e a Decisão: Modelos Matemáticos para Dispersào Populacional, Tese de Doutorado, Universidade Estadual de Campinas, Brasil.Google Scholar
  32. Roubik, D.W., 1989. Ecology and Natural History of Tropical Bees. Cambridge University Press.Google Scholar
  33. Roubik, D.W., 1991. Aspects of Africanized honey bee ecology in tropical America. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 259–281.Google Scholar
  34. Seeley, T.D., 1985. The information-center strategy of honeybee foraging. Fortsch. Zool. 31, 75–90.Google Scholar
  35. Seeley, T.D., 1989. The honey bee colony as a superorganism. Am. Sci. 77, 546–553.Google Scholar
  36. Seeley, T.D., 1995. TheWisdom of the Hive—The Social Physiology of Honey Bee Colonies. Harvard University Press.Google Scholar
  37. Seeley, T.D., Visscher, P.K., 1985. Survival of honeybees in cold climates: the critical timing of colony growth and reproduction. Ecol. Entomol. 10, 81–88.Google Scholar
  38. Segel, L.A., 1984. Mathematical models for cellular behavior. In: Levin, S.A. (Ed.), Studies in Mathematical Biology. Mathematical Association of America, pp. 156–190.Google Scholar
  39. Sheppard, W.S., Hinderer, T.E., Mazzoli, J.A., Stelzer, J.A., Shimanuki, H., 1991. Gene flow between African and European-derived honey bee populations in Argentina. Nature 349, 782–784.CrossRefGoogle Scholar
  40. Shigesada, N., 1984. Spatial distribution of rapidly dispersing animals in hetorogeneous environments. In: Levin, S.A., Hallam, T.T. (Eds.), Mathematical Ecology Lectures Notes in Biomathematics, vol. 54. Springer-Verlag, pp. 478–491.Google Scholar
  41. Shigesada, N., Kawasaki, K., 1997. Biological Invasions: Theory and Practice. Oxford University Press.Google Scholar
  42. Shigesada, N., Kawasaki, K., Teramoto, E., 1986. Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160.MATHMathSciNetCrossRefGoogle Scholar
  43. Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis. CRC Press.Google Scholar
  44. Spivak, M., Fletcher, D.J.C., Breed, M.D., 1991. The ‘African’ Honeybee. Westview Press, Boulder.Google Scholar
  45. Stevens, S.S., 1970. Neural events and the psychophysical law. Science 170, 1043–1050.Google Scholar
  46. Taylor, O.R., 1977. The past and possible future spread of Africanized honey bees in the Americas. Bee World 58, 19–30.Google Scholar
  47. Turchin, P., 1998. Quantitative Analysis of Movement, Measuring and Modelling Population Redistribution in Animals and Plants. Sinauer Associates, Inc. Publishers.Google Scholar
  48. Veit, R.R., Lewis, M.A., 1996. Dispersal, population growth, and the Allee efect: dynamics of the house finch invasion of Eastern North America. Am. Naturalist 148, 255–274.CrossRefGoogle Scholar
  49. Visscher, P.K., Camazine, S., 1999. Collective decisions and cognition in bees. Nature 397, 400.CrossRefGoogle Scholar
  50. Wang, M.H., Kot, M., Neubert, M.G., 2002. Integrodifference equations, Allee effects, and invasions. J. Math. Biol. 44, 150–168.MATHMathSciNetCrossRefGoogle Scholar
  51. Weinberger, H., 1982. Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.MATHMathSciNetCrossRefGoogle Scholar
  52. Weinberger, H., 2002. On spreading speed and travelling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45, 511–548.MATHMathSciNetCrossRefGoogle Scholar
  53. Wilson, E.O., 1971. The Insect Societies, Cambridge. The Belknap of Harvard University Press.Google Scholar
  54. Wilson, E.O., 1975. Sociobiology. Harvard University Press, Cambridge.Google Scholar
  55. Winston, M.L., 1987. The Biology of the Honey Bee. Harvard University Press.Google Scholar
  56. Winston, M.L., 1991. The inside story: internal colony dynamics of Africanized bees. In: Spivak, M., Fletcher, D.J.C., Breed, M. (Eds.), The ‘African’ Honey Bee. Westview Press, pp. 201–212.Google Scholar
  57. Winston, M.L., 1992. The biology and management of Africanized honey bee. Annu. Rev. Entomol. 37, 173–193.CrossRefGoogle Scholar
  58. Wolfe, J.M., Alvarez, G.A., Horowitz, T.S., 2000. Attention is fast and volition is slow: a random scan is a quicker way to find items than a systematic search. Nature 406, 691–692.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  • Diomar Cristina Mistro
    • 1
  • Luiz Alberto Díaz Rodrigues
    • 1
  • Wilson Castro FerreiraJr.
    • 2
  1. 1.Departamento de Matemática-CCNEUniversidade Federal de Santa MariaSanta Maria/RSBrazil
  2. 2.Departamento de Matemática Aplicada-IMECCUniversidade Estadual de CampinasCampinas/SPBrazil

Personalised recommendations