Abstract
IDE models naturally allow a certain temporal variation within a generation since they divide each generation into separate growth and dispersal phases. However, so far we have assumed that the growth phases in all generations are identical and that the same holds for the dispersal phases. In realistic environments, external conditions in subsequent generations may vary substantially so that growth and dispersal behavior could differ. In this chapter, we present some theory on and examples of how to formulate and analyze IDEs with a periodically or randomly varying growth function and dispersal kernel. In the periodic case, much of the previous theory for temporally constant environments can be applied to the period map. In the random case, even the formulation of the problem requires substantially different tools from the theory of stochastic processes. We focus again on the two fundamental questions of population persistence and spread.
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References
Allen, L. (2010). An introduction to stochastic processes with applications to biology. London: Chapman and Hall/CRC.
Bouhours, J., & Lewis, M. (2016). Climate change and integrodifference equations in a stochastic environment. Bulletin of Mathematical Biology, 78, 1866–1903.
Caswell, H., Neubert, M., & Hunter, C. (2011). Demography and dispersal: Invasion speeds and sensitivity analysis in periodic and stochastic environments. Theoretical Ecology, 4, 407–421.
Ding, W., Liang, X., & Xu, B. (2013). Spreading speeds of n-season spatially periodic integro-difference equations. Discrete and Continuous Dynamical Systems - Series A, 33(8), 3443–3472.
Ellner, S. (1984). Asymptotic behavior of some stochastic difference equation population models. Journal of Mathematical Biology, 19, 169–200.
Ellner, S., & Schreiber, S. (2012). Temporally variable dispersal and demography can accelerate the spread of invading species. Theoretical Population Biology, 82(4), 283–298.
Gharouni, A., Barbeau, M., Chassé, J., Wang, L., & Watmough, J. (2017). Stochastic dispersal increases the rate of upstream spread: A case study with green crabs on the northwest atlantic coast. PLoS ONE, 12(9), e0185671.
Gilioli, G., Pasquali, S., Tramontini, S., & Riolo, F. (2013). Modelling local and long-distance dispersal of invasive chestnut gall wasp in Europe. Ecological Modelling, 263, 281–290.
Hardin, D., Takáč, P., & Webb, G. (1988a). Asymptotic properties of a continuous-space discrete-time population model in a random environment. Journal of Mathematical Biology, 26, 361–374.
Hardin, D., Takáč, P., & Webb, G. (1988b). A comparison of dispersal strategies for survival of spatially heterogeneous populations. SIAM Journal on Applied Mathematics, 48, 1396–1423.
Jacobs, G., & Sluckin, T. (2015). Long-range dispersal, stochasticity and the broken accelerating wave of advance. Theoretical Population Biology, 100, 39–55.
Jacobsen, J., & McAdam, T. (2014). A boundary value problem for integrodifference equation models with cyclic kernels. Discrete & Continuous Dynamical Systems - Series B, 19(10), 3139–3207.
Jacobsen, J., Jin, Y., & Lewis, M. (2015). Integrodifference models for persistence in temporally varying river environments. Journal of Mathematical Biology, 70, 549–590.
Kot, M., Medlock, J., Reluga, T., & Walton, D. (2004). Stochasticity, invasions, and branching random walks. Theoretical Population Biology, 66, 175–184.
Latore, J., Gould, P., & Mortimer, A. (1999). Effects of habitat heterogeneity and dispersal strategies on population persistence in annual plants. Ecological Modelling, 123, 127–139.
Lewis, M. (2000). Spread rate for a nonlinear stochastic invasion. Journal of Mathematical Biology, 41, 430–454.
Lewis, M., & Pacala, S. (2000). Modeling and analysis of stochastic invasion processes. Journal of Mathematical Biology, 41, 387–429.
Lewis, M., Petrovskii, S., & Potts, J. (2016). The mathematics behind biological invasions. Berlin: Springer.
Mahdjoub, T., & Menu, F. (2008). Prolonged diapause: A trait increasing invasion speed? Journal of Theoretical Biology, 251, 317–330.
Meyn, S., & Tweedie, R. (2009). Markov chains and stochastic stability. Cambridge: Cambridge University Press.
Neubert, M., & Caswell, H. (2000a). Demography and dispersal: Calculation and sensitivity analysis of invasion speeds for structured populations. Ecology, 81(6), 1613–1628.
Neubert, M., Kot, M., & Lewis, M. (2000). Invasion speeds in fluctuating environments. Proceedings of the Royal Society of London - Series B, 267, 1603–1610.
Neubert, M., & Parker, I. (2004). Projecting rates of spread for invasive species. Risk Analysis, 24(4), 817–831.
Reimer, J., Bonsall, M., & Maini, P. (2017). The critical patch-size of stochastic population models. Journal of Mathematical Biology, 74, 755–782.
Schreiber, S., & Ryan, M. (2011). Invasion speeds for structured populations in fluctuating environments. Theoretical Ecology, 4(4), 423–434.
Snyder, R. (2003). How denographic stochasticity can slow biological invasions. Ecology, 84(5), 1333–1339.
Zhou, Y., & Fagan, W. (2017). A discrete-time model for population persistence in habitats with time-varying sizes. Journal of Mathematical Biology, 75(3), 649–704.
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Lutscher, F. (2019). Temporal Variation. In: Integrodifference Equations in Spatial Ecology. Interdisciplinary Applied Mathematics, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-29294-2_16
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DOI: https://doi.org/10.1007/978-3-030-29294-2_16
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