×

zbMATH — the first resource for mathematics

The velocity of spatial population expansion. (English) Zbl 0732.92026
The authors study the velocity with which an invading population spreads over space and give modelling frameworks that allow them to calculate the (asymptotic) velocity of population expansion from experimentally observed quantities like the probability for an individual to survive to a certain age and the settlement pattern of juveniles. The approximation formulas presented are used to study some real-life examples. No proofs are given but the interesting results are based on formal and numerical calculations and heuristic arguments.

MSC:
92D40 Ecology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ammerman, A. J., Cavalli-Sforza, L. L.: The neolithic transition and the genetics of populations in Europe. Princeton University Press 1984.
[2] Anderson, R. M. (ed.): Population dynamics of infectious diseases. Theory and applications. London: Chapman and Hall 1982.
[3] Andow, D. A., Kareiva, P. M., Levin, S. A., Okubo, A.: Spread of invading organisms: patterns of spread. In: Kim, K. C. (ed.) Evolution of insect pests: the pattern of variations. New York: Wiley.
[4] Andow, D. A., Kareiva, P. M., Levin, S. A., Okubo, A.: Spread of invading organisms, submitted.
[5] Andral, L., Artois, M., Aubert, M. F. A., Blancou, J.: Radio-pistage de renards enrages. Comp. Immunol. Microbiol. Infect. Diseases 5, 284-291 (1982).
[6] Andral, L., Toma, B.: La rage en France en 1976. Rec. Med. vet. 153, 503-508 (1977).
[7] Anonymous: Ecology of biological invasions. SCOPE Newsletter 23, 1-5 (1985).
[8] Aronson, D. G., Weinberger, H. F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. A. (ed.) Partial differential equations and related topics. (Lect. Notes Math., vol. 446, pp. 5-49) Berlin Heidelberg New York: Springer 1975. · Zbl 0325.35050
[9] Aronson, D. G., Weinberger, H. F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33-76 (1978). · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[10] Artimo, A.: The dispersal and the acclimatisation of the muskrat Ondatra zibetica (L.), in Finland. Papers on game research 21, 1-101 (1960).
[11] Bacon, P. J. (ed.): Population dynamics of rabies in wildlife. New York: Academic Press 1985.
[12] Ball, F. G.: Some statistical problems in the epidemiology of fox rabies. Thesis 1981, University of Nottingham.
[13] Becker, K.: Populationsstudien an Bismratten (Ondatra zibethica L.) I Zoologische Beitrage 13, 369-396 (1967).
[14] Berger, J.: Model of rabies control. In: Berger, J., Buhler, W., Repges, R., Tautu, P. (eds.) Mathematical models in medicine. (Lest. Notes Biomath., vol. 11, pp. 75-88) Berlin Heidelberg New York: Springer 1976.
[15] Bögel, K., Moegle, H.: Characteristics of the spread of a wildlife rabies epidemic in Europe. Biogeographica 8, 251-258 (1980).
[16] Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44, 190 (1983). · Zbl 0517.60083
[17] Broadbent, S. R., Kendall, D. G.: The random walk of Trichostrongylus retortaeformis. Biometrika 9, 460-465 (1953).
[18] Browning, J. A., Frey, K. J.: Multiline cultivars as a means of disease control. Annu. Rev. Phytopathol. 7, 355-382 (1969). · doi:10.1146/annurev.py.07.090169.002035
[19] Caughley, G.: Liberation, dispersal and distribution of Himalayas Thar (Hemitragus jemlahicus) in New Zealand. New Zealand J. Sci. 13, 220-239 (1970).
[20] Creegan, P., Lui, R.: Some ramarks about the wave speed and travelling wave solutions of a nonlinear integral generator. J. Math. Biol. 20, 59-68 (1984). · Zbl 0556.45002 · doi:10.1007/BF00275861
[21] Diekmann, O.: Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109-130 (1978). · Zbl 0415.92020 · doi:10.1007/BF02450783
[22] Diekmann, O.: Run for your life. A note on the asymptotic speed of propagation of an epidemic. J. Differ. Equations 33, 58-73 (1979). · Zbl 0404.45013 · doi:10.1016/0022-0396(79)90080-9
[23] Diekmann, O.: Dynamics in biomathematical perspective. In: Hazewinkel, M., Lenstra, J. K., Meertens, L. G. L. (eds.) Mathematics and computer sicence II. (CWI Monographs vol. 4, pp. 23-50) 1986.
[24] Diekmann, O., Temme, N. M.: Nonlinear diffusion problems. Amsterdam: Mathematical Centre 1976. · Zbl 0357.35002
[25] Doude van Troostwijk, W. J.: The muskrat (Ondatra zibethicus L.) in the Netherlands, its ecological aspects and their consequences for man. Thesis, State University of Leiden.
[26] Errington, P. L.: Muskrat populations. Iowa: Iowa State University.
[27] Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 355-369 (1937). · JFM 63.1111.04
[28] Hadeler, K. P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251-263 (1975). · Zbl 0343.92009 · doi:10.1007/BF00277154
[29] Hengeveld, R.: Dynamics of biological invasions. London: Chapman and Hall 1989.
[30] Hoffman, M.: Die Bisamratte. Leipzig: Academische Verlagsgesellschaft 1958.
[31] Källen, A., Arcuri, P., Murray, J. D.: A simple model for the spatial spread and control of rabies. J. Theor. Biol. 116, 377-393 (1985). · doi:10.1016/S0022-5193(85)80276-9
[32] Kendall, M. G., Stuart, A.: The advanced theory of statistics, vol. I. London: Griffin 1958. · Zbl 0416.62001
[33] Kendall, D. G.: Mathematical models of the spread of infection. In: Mathematics and computer sicence in biology and medicine (Medical Research Council, London, pp. 213-224) 1965.
[34] Keyfitz, N.: Introduction to the mathematics of population. Reading, Mass.: Addison Wesley 1968.
[35] Kolmogorov, A., Petrovsky, I., Piscounov, N.: Etude de l’équation de la diffusion avec croissance de la quantité de matiere et son application a un probleme biologique. Mosc, Univ. Math. Bull. 1, 1-25 (1937).
[36] Kornberg, H., Williamson, M. H.: Quantitative aspects of the ecology of biological invasions. London: Royal Society, 1987.
[37] Lambinet, D., Boisvieux, J. F., Mallet, A., Artois, M., Andral, L.: Modele mathématique de la propagation d’une épizootie de rage vulpine. Rev. Epidém. et Santé Publ. 26, 9-28 (1978).
[38] Levin, S. A.: Analysis of risk for invasions and control programs. In: Drake, J., Castri, F. di, Groves, R., Kruger, F., Mooney, H., Rejamenk, M., Williamson, M. (eds.) Biological invasions: a global perspective. Chichester: Wiley, in press.
[39] Lloyd, H. G.: Wildlife rabies in Europe and the British situation. Trans. R. Soc. Trop. Med. Hyg. 70, 179-187 (1976). · doi:10.1016/0035-9203(76)90036-5
[40] Lubina, J. A., Levin, S. A.: The spread of a reinvading species: Range expansion in the California Sea Otter. Am. Nat. 131, 526-543 (1988). · doi:10.1086/284804
[41] MacDonald, D. W.: Rabies and wildlife. Oxford: Oxford University Press 1980.
[42] MacDonald, D. W., Bacon, P. J.: Fox society, contact rate and rabies epizootiology. Comp. Immunol. Microbiol. Infect. Dis. 5, 247-256 (1982). · doi:10.1016/0147-9571(82)90045-5
[43] Mallach, N.: Markierungsversuche zur Analyse des Aktionsrau und der Ortsbewegungen des Bisams (Ondatra zibethica L.) Anzeiger für Schädlingskunde and Pflanzenschutz XLIV9, 129-136 (1971). · doi:10.1007/BF02027531
[44] Metz, J. A. J., Diekmann, O.: The dynamics of physiologically structured populations. (Lect. Notes Biomath., vol. 68) Berlin Heidelberg New York: Springer 1986. · Zbl 0614.92014
[45] Minogue, K. P., Frey, W. E.: Models for the spread of disease: model description. Phytopathology 73, 1168-1173 (1983a). · doi:10.1094/Phyto-73-1168
[46] Minogue, K. P., Frey, W. E.: Models for the spread of plant disease: some experimental results. Phytophathology 73, 1173-1176 (1983b). · doi:10.1094/Phyto-73-1173
[47] Moens, R.: Etude bio-écologique du rat musqué en Belgique. Parasitica 34, 57-121 (1978).
[48] Mollison, D.: The rate of spatial propagation of simple epidemics. In: Le Cam, L. M., Neyman, J., Scott, E. L. (eds.) Proc. Sixth Berkeley Symposium, III, Univ. of California Press, pp. 579-614 (1972). · Zbl 0266.92008
[49] Mollison, D.: Spatial contact models for ecological and epidemic spread. J. Roy. Statist. Soc. B39, 283-326 (1977). · Zbl 0374.60110
[50] Mollison, D., Kuulasmaa, K.: Spatial epidemic models: theory and simulations. In: Bacon, P. J. (ed.) Population dynamics of rabies in wildlife, pp. 291-309. New York: Academic Press 1985.
[51] Mollison, D.: Modelling biological invasions: chance, explanation, prediction. Philos. Trans. R. Soc. Lond. B, 314, 675-693 (1986). · doi:10.1098/rstb.1986.0080
[52] Mooney, H. A., Drake, J. A. (eds.): Ecology of biological invasions of North America and Hawaii. (Ecological Studies, vol. 58) Berlin Heidelberg New York: Springer 1986.
[53] Nobel, J. V.: Geographic and temporal development of plagues. Nature 250, 726-729 (1974). · doi:10.1038/250726a0
[54] Okubo, A.: Diffusion-type models for avian range expansion. In: Quellet, H. (ed.) Acta XIX Congress Internationa lis Ornithologici, vol. 1, pp. 1038-1049. National Museum of Natural Sciences, University of Ottawa Press, Ontario, Canada 1988.
[55] Othmer, H. G., Dunbar, S. R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263-298 (1988). · Zbl 0713.92018 · doi:10.1007/BF00277392
[56] Roughgarden, J.: Theory of population genetics and evolutionary ecology: an introduction. New York: MacMillan 1979.
[57] Sikes, R. K.: Pathogenesis of rabies in wildlife. I. Comparative effect of varying doses of rabies virus inoculated into foxes and skunks. Am. J. Vet. Res. 23, 1041-1047 (1962).
[58] Skellam, J. G.: Random dispersal in theoretical populations. Biometrica 38, 196-218 (1951). · Zbl 0043.14401
[59] Smith, A. D. M.: A continuous time dterministic model of temporal rabies. In: Bacon, P. J. (ed.) Population dynamics of rabies in wildlife. New York: Academic Press 1985.
[60] Steck, F., Wandeler, A.: The epidemiology of fox rabies in Europe. Epidemiol. Rev. 2, 71-96 (1980).
[61] Thieme, H. R.: A model for the spatial spread of an epidemic. J. Math. Biol 4, 337-351 (1977a). · Zbl 0373.92031 · doi:10.1007/BF00275082
[62] Thieme, H. R.: The asymptotic behaviour of solutions of nonlinear integral equations. Math. Z. 157, 141-154 (1977b). · Zbl 0359.45009 · doi:10.1007/BF01215148
[63] Thieme, H. R.: Asymptotic estimates of the solutions of non-linear integral equations and asymptotic speeds for the spread of populations. Jal Reine Angew. Math. 306, 94-121 (1979a). · Zbl 0395.45010 · doi:10.1515/crll.1979.306.94
[64] Thieme, H. R.: Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J. Math. Biol. 8, 173-187 (1979b). · Zbl 0417.92022 · doi:10.1007/BF00279720
[65] Van den Bosch, F., Zadoks, J. C., Metz, J. A. J.: Focus expansion in plant disease. I: The constant rate of focus expansion. Phytopathology 78, 54-58 (1988). · doi:10.1094/Phyto-78-54
[66] Van den Bosch, F., Zadoks, J. C., Metz, J. A. J.: Focus expansion in plant disease. II: Realistic parameter-sparse models. Phytopathology 78, 59-64 (1988). · doi:10.1094/Phyto-78-59
[67] Van den Bosch, F., Frinking, H. D., Metz, J. A. J., Zadoks, J. C.: Focus expansion in plant disease. III: Two experimental examples. Phytopathology 78, 919-925 (1988). · doi:10.1094/Phyto-78-919
[68] Van den Bosch, F., Verhaar, M. A., Buiel, A. A. M., Hoogkamer, W., Zadoks, J. C.: Focus expansion in plant disease. IV: Expansion rates in mixtures of resistant and susceptible hosts. Phytopathology, in press.
[69] Van den Bosch, F., Hengeveld, R., Metz, J. A. J.: Analysing animal range expansion. Preprint (1988).
[70] Verkaik, A. J.: The muskrat in the Netherlands. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 90, 67-72 (1987).
[71] Vincent, J. P., Quéré, J. P.: Quelques donées sur la reproduction et sur la dynamique des populations du rat musqué dans le nord de la France. Ann. Zool. Ecol. anim. 4, 395-415 (1972).
[72] Watt, K. E. F.: Ecology and resource management. New York: McGraw-Hill 1968. · Zbl 0162.46601
[73] Weinberger, H. F.: Asymptotic behaviour of a model in population genetics. In: Chadam, J. M. (ed.) Nonlinear partial differential equations and applications. (Lect. Notes in Maths., vol. 648, pp. 47-98) Berlin Heidelber New York: Springer 1978. · Zbl 0383.35034
[74] Weinberger, H. F.: Long-time behaviour of a class of biological models. SIAM J. Math. Anal. 13, 353-396 (1982). · Zbl 0529.92010 · doi:10.1137/0513028
[75] Williamson, E. J.: The distribution of larvae of randomly moving insects. Aust. J. Biol. Sci. 14, 598-604 (1961).
[76] Williamson, M. H., Brown, K. C.: The analysis and modelling of British invasions. Phil. Trans. R. Soc. London B314, 505-522 (1986).
[77] Wolfe, M. S.: The current status and prospects of multiline cultivars and variety mixtures for disease resistance. Annu. Rev. Phytopathol. 23, 251-273 (1985). · doi:10.1146/annurev.py.23.090185.001343
[78] Zadoks, J. C., Kampmeijer, P.: The role of crop populations and their development, illustrated by means of a simulator Epimul 76. Ann. N.Y. Acad. Sci. 287, 164-190 (1977). · doi:10.1111/j.1749-6632.1977.tb34238.x
[79] Zawolek, M. W.: A physical theory of focus development in plant disease. Agric. Univ. Wageningen Papers. Pudoc, in press.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.