Abstract
In previous chapters, we studied the speed of spread and the existence of traveling waves. In this chapter, we focus on the shape of traveling-wave profiles and more general patterns of spatial spread. We first provide an approximation scheme, based on asymptotic expansion, for the shape of a monotone wave. Then we explore the existence of nonmonotone waves as well as more complicated patterns of spread when the growth function has a stable two-cycle. We generalize the notion of the asymptotic spreading speed and discuss dynamic stabilization.
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References
Andersen, M. (1991). Properties of some density-dependent integrodifference equation population models. Mathematical Biosciences, 104, 135–157.
Bourgeois, A. (2016). Spreading Speeds and Travelling Waves in Integrodifference Equations with Overcompensatory Dynamics. Master’s Thesis, University of Ottawa.
Bourgeois, A., LeBlanc, V., & Lutscher, F. (2018). Spreading phenomena in integrodifference equations with non-monotone growth functions. SIAM Journal on Applied Mathematics, 78(6), 2950–2972.
Bourgeois, A., LeBlanc, V., & Lutscher, F. (2019). Dynamical stabilization and traveling waves in integrodifference equations. Discrete and Continuous Dynamical Systems - Series S, doi: 10.3934/dcdss.2020117
Hsu, S.-B., & Zhao, X.-Q. (2008). Spreading speeds and traveling waves for non-monotone integrodifference equations. SIAM Journal on Mathematical Analysis, 40(2), 776–789.
Kot, M. (1992). Discrete-time travelling waves: Ecological examples. Journal of Mathematical Biology, 30, 413–436.
Kot, M. (2003). Do invading organisms do the wave? Canadian Applied Mathematics Quarterly, 10, 139–170.
Li, B., Lewis, M., & Weinberger, H. (2009). Existence of traveling waves for integral recursions with nonmonotone growth functions. Journal of Mathematical Biology, 58, 323–338.
Lui, R. (1983). Existence and stability of travelling wave solutions of a nonlinear integral operator. Journal of Mathematical Biology, 16, 199–220.
Malchow, H., Petrovskii, S., & Venturino, E. (2008). Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations. London: Chapman & Hall/CRC Press.
Otto, G. (2017). Nonspreading solutions in integro-difference models with Allee and overcompensation effects. Ph.D. Thesis, University of Louisville.
Schreiber, S. (2003). Allee effects, extinctions, and chaotic transients in simple population models. Theoretical Population Biology, 64, 201–209.
Sullivan, L., Li, B., Miller, T., Neubert, M., & Shaw, A. (2017). Density dependence in demography and dispersal generates fluctuating invasion speeds. Proceedings of the National Academy of Sciences, 114(19), 5053–5058.
Weinberger, H. (1982). Long-time behavior of a class of biological models. SIAM Journal on Mathematical Analysis, 13, 353–396.
Yu, Z.-X., & Yuan, R. (2012). Properties of traveling waves for integrodifference equations with nonmonotone growth functions. Zeitschrift für Angewandte Mathematik und Physik, 63, 249–259.
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Lutscher, F. (2019). The Shape of Spatial Spread. In: Integrodifference Equations in Spatial Ecology. Interdisciplinary Applied Mathematics, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-29294-2_11
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DOI: https://doi.org/10.1007/978-3-030-29294-2_11
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