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Biological Invasions in Heterogeneous Environments: The CoupledMap Lattice Framework

Published online by Cambridge University Press:  28 November 2013

L.A.D. Rodrigues  and
D.C. Mistro
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Abstract

Spatial heterogeneity greatly affects the population spread. Although the theory forbiological invasion in heterogeneous spatially continuous habitats have receivedconsiderable attention, spatially discrete models have remained outside of the mainstream.In this study, we formulate and analyze a Coupled Map Lattice model for a single speciespopulation invading a two dimensional heterogeneous environment. The population growthrate and dispersal coefficient depend on the site quality. We first find an analyticalcriterium for the spread success in terms of the population growth rate and the dispersalcoefficient in unfavorable regions. We then implemented our model for two distinct spatialconfigurations: periodical stripe-like and randomized environments. The spread rate iscomputed numerically and it shows a decrease with an increase of the fraction of thehostile sites. However, we observed that invasion success does not depend on the fractionof favorable sites but crucially depends on the connectivity of favorable regions.

Keywords

heterogeneous habitatcoupled map latticesbiological invasionsfragmentation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 96 - 106
Copyright
© EDP Sciences, 2013

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References

Berestycki, H., Hamel, F., Roques, L.. Analysis of the periodically fragmented environment model: II-biological invasions and pulsating travelling fronts. J. Math. Pures Appl. 84 (2005), 1101–1146. CrossRefGoogle Scholar
de-Camino-Beck, T., Lewis, M.A.. Invasion with stage-structured coupled map lattices: Application to the spread of scentless chamomile. Ecol. Model. 220 (2009), 3394–3403. CrossRefGoogle Scholar
Dewhirst, C., Lutscher, F.. Dispersal in heterogeneous habitats: thresholds, spatial scales, and approximate rates of spread. Ecology 90 (2009), 1338–1345. CrossRefGoogle Scholar
Fahrig, L.. Effect of habitat fragmentation on the extinction threshold: a synthesis. Ecol. Appl. 12 (2002), No.2, 346–353. Google Scholar
Hassell, M.P., Comins, H.N., May, R.M.. Spatial structure and chaos in insect population dynamics. Nature 353 (1991), 255–258. CrossRefGoogle Scholar
P. Kareiva. Influence of Vegetation Texture on Herbivore Populations: Resource Concentration and Herbivore Moviment in R. F. Denno and M. S. McClure (eds.), Variable Plants and Herbivores in Natural and Managed Systems, 259–289. Academic Press, New York, 1983.
P. Kareiva.Trivial Movement and Foraging by Crop Colonizers in M. Kogan (ed.) Ecological Theory and Integrated Pest Manegement Practice, 59–82. John Wiley & Sons, New York, 1986.
Kareiva, P., Odell, G.. Swarms of predators exhibit "preytaxis" if individuals use area-restricted search. Am. Nat. 130 (1987), No.2, 233–270. CrossRefGoogle Scholar
Kawasaki, K., Shigesada, N.. An integrodifference model for biological invasions in a periodically fragmented environment. Japan. J. Indust. Appl. Math. 24 (2007), 3–15. CrossRefGoogle Scholar
Kawasaki, K., Asano, K., Shigesada, N.. Impact of Directed Movement on Invasive Spread in Periodic Patchy Environments. Bull. Math. Biol. 74 (2012), 1448–1467. CrossRefGoogle Scholar
Keitt, T.H., Lewis, M.A., Holt, R.D.. Allee effects, invasion pinning, and species borders. Am. Nat. 157 (2001), 203–216. Google Scholar
Kinezaki, N., Kawasaki, K., Takasu, F., Shigesada, N.. Modeling biological invasions into periodically fragmented environments. Theor. Popul. Biol. 64 (2003), 291–302. CrossRefGoogle Scholar
Kinezaki, N., Kawasaki, K., Shigesada, N.. Spatial dynamics of invasion in sinusoidally varying environments. Popul. Ecol. 48 (2006) 263-270. CrossRefGoogle Scholar
Kinezaki, N., Kawasaki, K., Shigesada, N.. The effect of the spatial configuration of habitat fragmentation on invasive spread. Theor. Popul. Biol. 78 (2010), 298–308. CrossRefGoogle Scholar
Levin, S. A.. The Problem of Pattern and Scale in Ecology. Ecology 73 (1992), No.6, 1943–1967. CrossRefGoogle Scholar
Lewis, M. A., Schmitz, G.. Biological Invasion of an Organism with Separate Mobile and Stationary States: Modeling and Analysis. Forma 11 (1996), 1–25. Google Scholar
Lutscher, F., Lewis, M. A., McCauley, E.. The effects of heterogeneity on population persistence and invasion in rivers. Bull. Math. Biol. 68 (2006), No.8, 2129–2160. CrossRefGoogle ScholarPubMed
Méndez, V., Llopis, I., Campos, D., Horsthemke, W.. Extinction and chaotic patterns in map lattices under hostile conditions. Bull. Math. Biol. 72 (2010), 432–443. CrossRefGoogle ScholarPubMed
Mistro, D.C., Rodrigues, L. A. D., Varriale, M. C.. The Role of Spatial Refuges in Coupled Map Lattice Model for Host-Parasitoid Systems. Bull. Math. Biol. 71 (2009), 1934–1953. CrossRefGoogle ScholarPubMed
Mistro, D.C., Rodrigues, L.A.D., Petrovskii, S.. Spatiotemporal complexity of biological invasion in a space- and time-discrete predator-prey system with the strong Allee effect. Ecol. Comp. 9 (2012), 16–32. CrossRefGoogle Scholar
Mollison, D.. Dependence of epidemic and population velocities on basic parameters. Math. Biosci. 107 (1991), 255–287. CrossRefGoogle ScholarPubMed
Morozov, A., Petrovskii, S., Li, B.-L.. Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. J. theor. Biol. 238 (2006), 18–35. CrossRefGoogle Scholar
A. Okubo, S.A. Levin. Diffusion and ecological problems. (2nd edn.) Springer, Berlin, 2001.
Rodrigues, L. A. D., Mistro, D. C., Petrovskii, S. V.. Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space-and time-discrete predator-prey system. Bull. Math. Biol. 73 (2011), 1812–1840. CrossRefGoogle Scholar
Rodrigues, L. A. D., Mistro, D. C., Petrovskii, S. V.. Pattern formation in a space- and time-discrete predator-prey system with a strong Allee effect. Theor. Ecol. 5 (2012), 341–362. CrossRefGoogle Scholar
L. A. D. Rodrigues, M. C. Varriale, W. A. C. Godoy, D. C. Mistro. Spatiotemporal dynamics of an insect population in response to chemical substances. Ecol. Comp. (2013) accepted.
N. Shigesada, K. Kawasaki. Invasion and the range expansion of species: effects of long-distance dispersal. In: Bullock, J., Kenward, R., Hails, R. (Eds.), Dispersal. Blackwell Science, Oxford, pp. 350–373, 2002.
Shigesada, N., Kawasaki, K., Teramoto, E.. Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30 (1986), 143–160. CrossRefGoogle Scholar
N. Shigesada, K. Kawasaki, E. Teramoto. The speeds of traveling frontal waves in Heterogeneous environments. In: Teramoto, E., Yamaguti, M. (Eds.), Mathematical Topics in Population Biology, Morphogenesis and Neurosciences. In: Lecture Notes in Biomathematics, vol. 71. Springer, Berlin, pp. 87–97, 1987.
N. Shigesada, K. Kawasaki. Biological Invasions: Theory and Practice. Oxford University Press, 1997
Van Kirk, R. W., Lewis, M. A.. Integrodifference models for persistence in fragmented habitats. Bull. Math. Biol. 59 (1997), No.1, 107–137. CrossRefGoogle Scholar
Weinberger, H.F.. On spreading speeds and traveling waves for growth and migration models in a periodic habitat. J. Math. Biol. 45 (2002), 511–548. CrossRefGoogle Scholar
Weinberger, H.F., Kawasaki, K., Shigesada, N.. Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions. J. Math. Biol. 57 (2008), 387–411. CrossRefGoogle ScholarPubMed
With, K. A.. The landscape ecology of invasive spread. Cons. Biol. 16 (2002), 1192–1203. CrossRefGoogle Scholar
White, S.M., White, K.A.J.. Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices. J. Theor. Biol. 235 (2005), 463–475. CrossRefGoogle ScholarPubMed