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Stochasticity and Invasion Dynamics

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The Mathematics Behind Biological Invasions

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 44))

Abstract

Uncertainty is a hallmark of early invasion processes. Mathematical descriptions of this uncertainty can help us assign probabilities to possible invasion outcomes. This chapter starts with a hierarchical model of invasion, describing the process of transport, introduction, and survival to reproduction. The model yields a probability of successful establishment for potential invaders, as well as the distribution of times needed before a successful invasion will occur. An extension of the model includes the possibility of an invasion bottleneck produced by the need for sexual reproduction. Environmental variability has a role to play in invasion success. This aspect is investigated using classical discrete- and continuous-time models for population growth under stochasticity. Here, Jensen’s inequality is applied to show that an invasion taking place amid discrete-time random environmental fluctuations may not succeed, even if it would succeed in a constant environment. Finally, a general, but approximate, method for understanding the impacts of various types of uncertainty (environmental, demographic, and immigration) on invasion success is formulated using stochastic differential equations. This method is then used to model invasion success for populations with an Allee effect. We apply the theory in this chapter to understand the invasive outcomes for aquatic invasive species, such as the Chinese mitten crab and the apple snail.

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Notes

  1. 1.

    For animals, invasion success can be much higher; see [151].

  2. 2.

    A similar approach is sometimes used in deterministic models as well, where it is called a practical stability concept; see Sect. 3.1

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Authors and Affiliations

  1. Department of Mathematics & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada

    Mark A. Lewis

  2. Department of Mathematics, University of Leicester, Leicester, UK

    Sergei V. Petrovskii

  3. School of Mathematics & Statistics, University of Sheffield, Sheffield, UK

    Jonathan R. Potts

Authors
  1. Mark A. Lewis
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  2. Sergei V. Petrovskii
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  3. Jonathan R. Potts
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Lewis, M.A., Petrovskii, S.V., Potts, J.R. (2016). Stochasticity and Invasion Dynamics. In: The Mathematics Behind Biological Invasions. Interdisciplinary Applied Mathematics, vol 44. Springer, Cham. https://doi.org/10.1007/978-3-319-32043-4_7

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