Species Spreading due to Environmental Hostility, Dispersal Adaptation and Allee Effects

Authors: Sanjeeva Balasuriya

Abstract:

A phenomenological model for species spreading which incorporates the Allee effect, a species- maximum attainable growth rate, collective dispersal rate and dispersal adaptability is presented. This builds on a well-established reaction-diffusion model for spatial spreading of invading organisms. The model is phrased in terms of the “hostility" (which quantifies the Allee threshold in relation to environmental sustainability) and dispersal adaptability (which measures how a species is able to adapt its migratory response to environmental conditions). The species- invading/retreating speed and the sharpness of the invading boundary are explicitly characterised in terms of the fundamental parameters, and analysed in detail.

Keywords: Allee effect, dispersal, migration speed, diffusion, invasion.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079152

References:

[1] R. Fisher, "The wave of advance of advantageous genes," Annals of Eugenics, vol. 7, pp. 355-369, 1937.
[2] J. Skellam, "Random dispersal in theoretical populations," Biometrika, vol. 38, pp. 196-218, 1951.
[3] W. Gurney and R. Nisbet, Ecological dynamics. Oxford University Press, 1998.
[4] J. Murray, Mathematical Biology. Springer-Verlag, 1993.
[5] M. Lewis and P. Kareiva, "Allee dynamics and the spread of invading organisms," Theoretical Population Biology, vol. 43, pp. 141-158, 1993.
[6] S. Petrovskii and B.-L. Li, "An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect," Mathematical Biosciences, vol. 186, pp. 79-91, 2003.
[7] R. Almeida, S. Delphim, and M. da S. Costa, "A numerical model to solve single-species invasion problems with Allee effects," Ecological Modelling, vol. 192, pp. 601-617, 2006.
[8] W. Morris and G. Dwyer, "Population consequences of constitutive and inducible plant resistance: herbivore spatial spread," American Naturalist, vol. 149, pp. 1071-1090, 1997.
[9] G. Dwyer and W. Morris, "Resource-dependent dispersal and the speed of biological invasions," American Naturalist, vol. 167, pp. 165-176, 2006.
[10] C. Taylor, H. Davis, J. Civille, F. Grevstad, and A. Hastings, "Consequences of an Allee effect in the invasion of a Pacific estuary by spartina alterniflora," Ecology, vol. 85, pp. 3254-3266, 2004.
[11] W. Allee, The social life of animals. New York: Norton, 1938.
[12] P. Stephens, W. Sutherland, and R. Freckelton, "What is the Allee effect?" Oikos, vol. 87, pp. 185-190, 1999.
[13] P. Stephens and W. Sutherland, "Consequences of the Allee effect for behaviour, ecology and conservation," Trends in Ecology and Evolution, vol. 14, 1999.
[14] F. Courchamp, T. Clutton-Brock, and B. Grenfell, "Inverse density dependence and the Allee effect," Trends in Ecology and Evolution, vol. 14, pp. 405-410, 1999.
[15] D. Johnson, A. Liebhold, P. Tobin, and O. Bjornstad, "Allee effects and pulsed invasion by the gyspy moth," Nature, vol. 444, pp. 361-363, 2006.
[16] J. Berger, "Persistence of different-sized populations: an empirical assessment of rapid extinctions in bighorn sheep," Conservation Biology, vol. 4, pp. 91-98, 1990.
[17] F. Courchamp and D. MacDonald, "Crucial importance of pack size in the African wild dog lycaon pictus," Animal Conservation, vol. 4, pp. 169-174, 2001.
[18] M. Groom, "Allee effects limit population viability of an annual plant," American Naturalist, vol. 151, pp. 487-496, 1998.
[19] B. Lamont, P. Klinkhamer, and E. Witkowski, "Population fragmentation may reduce fertility to zero in Banksia-Goodii: a demonstration of the Allee effect," Oecologia, vol. 94, pp. 446-450, 1993.
[20] E. Angulo, E. Roemer, L. Berec, J. Gascoigne, and F. Courchamp, "Double Allee effects and extinction in the island fox," Conservation Biology, vol. 21, pp. 1082-1091, 2007.
[21] H. Davis, C. Taylor, J. Civille, and D. Strong, "An Allee effect at the front of a plant invasion: Spartina in a Pacific estuary," Journal of Ecology, vol. 92, pp. 321-327, 2004.
[22] S. Balasuriya and G. Gottwald, "Wavepeed in reaction-diffusion systems, with applications to chemotaxis and population pressure," Journal of Mathematical Biology, 2009, in press, doi: 10.1007/s00285-009-0305-4.
[23] S. Fretwell, Populations in a Seasonal Environment. Princeton, New Jersey: Princeton University Press, 1972.
[24] W. Gurney and R. Nisbet, "The regulation of inhomogeneous population," Journal of Theoretical Biology, vol. 52, pp. 441-457, 1975.
[25] N. Shigesada, K. Kawasaki, and E. Teramoto, "Spatial segregation of interacting species," Journal of Theoretical Biology, vol. 79, pp. 83-99, 1979.
[26] D. Morris and J. Diffendorfer, "Reciprocating dispersal by habitatselecting white-footed mice," Oikos, vol. 107, pp. 549-558, 2004.
[27] Wolfram Research Inc., Mathematica, 5th ed. Champaign, Illinois: Wolfram Research, Inc., 2005.