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A Two-Patch Predator-Prey Metapopulation Model

Published online by Cambridge University Press:  28 May 2015

G. Quaglia ,
E. Re ,
M. Rinaldi  and
E. Venturino
G. Quaglia
Affiliation:
Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
E. Re
Affiliation:
Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
M. Rinaldi
Affiliation:
Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
E. Venturino*
Affiliation:
Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
*
Corresponding author. Email: ezio.venturino@unito.it
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Abstract

A minimal model for predator-prey interaction in a composite environment is presented and analysed. We first consider free migrations between two patches for both interacting populations, and then the particular cases where only one-directional migration is allowed and where only one of the two populations can migrate. Our findings indicate that in all cases the ecosystem can never disappear entirely, under the model assumptions. The predator-free equilibrium and the coexistence of all populations are found to be the only feasible stable equilibria. When there are only one-directional migrations, the abandoned patch cannot be repopulated. Other equilibria then arise, with only prey in the second patch, coexistence in the second patch, or prey in both patches but predators only in the second one. For the case of sedentary prey, with predator migration, the prey cannot thrive alone in either of the two environments. However, predators can survive in a prey-free patch due to their ability to migrate into the other patch, provided prey is present there. If only the prey can migrate, the predators may be eliminated from one patch or from both. In the first case, the patch where there are no predators acts as a refuge for the survival of the prey.

Keywords

92D2592D40Complex ecosystemsfragmented habitatsmigrationspopulation modelspredator-preyequilibriastability
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 2 , Issue 3 , August 2012 , pp. 238 - 265
Copyright
Copyright © Global-Science Press 2012

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References

[1]Bay, L. K., Caley, M. J., Crozier, R. H., Meta-population structure in a coral reef fish demonstrated by genetic data on patterns ofmigration, extinction and re-colonization, BMC Evolutionary Biology 8:248 (2008), doi:10.1186/1471-2148-8-248.CrossRefGoogle Scholar
[2]Black, H. L., A North Temperate Bat Community: Structure and Prey Populations, Journal of Mammalogy 55 (1974), 138–157.Google Scholar
[3]Brack, V. Jr., Laval, R. K., Food habits of the Indiana bat in Missouri, Journal of Mammalogy 66 (1985), 308–315.Google Scholar
[4]Cowie, R. J., Hinsley, S. A., Feeding ecology of great tits (Parus major) and blue tits (Parus caeruleus), breeding in suburban gardens, Journal of Animal Ecology 57 (1988), 611–626.Google Scholar
[5]Cronin, J. T., Movement and spatial population structure of a prairie planthopper, Ecology 84 (2003), 1179–1188.Google Scholar
[6]Dewey, T., Ballenger, L. (1999) “Ovis canadensis” (On-line), Animal Diversity Web, http://animaldiversity.ummz.umich.edu/site/accounts/information/Ovis_canadensis.html.Google Scholar
[7]Drechsler, M., Johst, K., Rapid viability analysis for metapopulations in dynamic habitat networks, Proceedings of the Royal Society B: Biological Sciences 277 (2010), 1889–1897.Google Scholar
[8]Abdllaoui, A. El, Auger, P., Kooi, B., De La Parra, R. Bravo, Mchich, R., Effects of density-dependent migrations on stability of a two-patch predator — prey model, Mathematical Biosciences 210 (2007), 335–354.Google Scholar
[9]Gustafson, E. J., Gardner, R. H., The effect of landscape heterogeneity on the probability of patch colonization, Ecology 77 (1996), 94–107.Google Scholar
[10]Gutiérrez, R. J., Harrison, S., Applying metapopulation theory to spotted owl management: A history and critique, in McCollough, D. R. (Ed.) Metapopulations and Wildlife Conservation, Washington: Island Press, 167–185, 1996.Google Scholar
[11]Hanski, I., Single-species spatial dynamics may contribute to long-term rarity and commonness, Ecology 66 (1985), 335–343.Google Scholar
[12]Hanski, I., Moilanen, A., Pakkala, T., Kuussaari, M., Metapopulation persistence of an endangered butterfly: A test of the quantitative incidence function model, Conservation Biology 10 (1996), 578–590.Google Scholar
[13]Kareiva, P., Population dynamics in spatially complex environments: Theory and data, Philosophical Transactions of the Royal Society of London B 330 (1990), 175–190.Google Scholar
[14]Li, Z. Z., Gao, M., Hui, C., Han, X. Z., Shi, H., Impact of predator pursuit and prey evasion on synchrony and spatial patterns in metapopulation, Ecological Modelling 185 (2005), 245–254.Google Scholar
[15]Malchow, H., Petrovskii, S., Venturino, E., Spatiotemporal Patterns in Ecology and Epidemiology, CRC, Boca Raton, 2008.Google Scholar
[16]Moilanen, A., Hanski, I., Habitat destruction and competitive coexistence in a spatially realistic metapopulation model, Journal of Animal Ecology 64 (1995), 141–144.Google Scholar
[17]Moilanen, A., Smith, A., Hanski, I., Long-term dynamics in a metapopulation of the American pika, American Naturalist 152 (1998), 530–542.Google Scholar
[18]Ryall, K. L., Fahrig, L., Response of predators to loss and fragmentation of prey habitat: A review of theory, Ecology 87 (2006), 1086–1093.Google Scholar
[19]Sun, G.Q., Jin, Z., Liu, Q.X., Li, L., Dynamical complexity of a spatial predator — prey model with migration, Ecological Modelling 219 (2008), 248–255.Google Scholar
[20]Venturino, E., Simple metaecoepidemic models, Bulletin of Mathematical Biology, 73 (2011), 917–950.Google Scholar
[21]Yakubu, A. A., Fogarty, M. J., Spatially discrete metapopulation models with directional dispersal, Mathematical Biosciences 204 (2006) 68–101.Google Scholar
[22]Wiens, J. A., Wildlife in patchy environments: metapopulations, mosaics, and management, in McCullough, D. R. (Editor) Metapopulations and Wildlife Conservation, Washington: Island Press, 53–84, 1996.Google Scholar
[23]Wiens, J. A., Metapopulation dynamics and landscape ecology, in Hanski, I. A., Gilpin, M. E. (Eds.) Metapopulation Biology: Ecology, Genetics, and Evolution, San Diego: Academic Press, 43–62, 1997.Google Scholar
[24]Wu, J., Modeling dynamics of patchy landscapes: linking metapopulation theory, landscape ecology and conservation biology, in Yearbook in Systems Ecology (English edition) Beijing: Chinese Academy of Sciences, 1994.Google Scholar