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Integrodifference models for evolutionary processes in biological invasions

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Abstract

Individual variability in dispersal and reproduction abilities can lead to evolutionary processes that may have significant effects on the speed and shape of biological invasions. Spatial sorting, an evolutionary process through which individuals with the highest dispersal ability tend to agglomerate at the leading edge of an invasion front, and spatial selection, spatially heterogeneous forces of selection, are among the fundamental evolutionary forces that can change range expansions. Most mathematical models for these processes are based on reaction-diffusion equations, i.e., time is continuous and dispersal is Gaussian. We develop novel theory for how evolution shapes biological invasions with integrodifference equations, i.e., time is discrete and dispersal can follow a variety of kernels. Our model tracks how the distribution of growth rates and dispersal ability in the population changes from one generation to the next in continuous space. We include mutation between types and a potential trade-off between dispersal ability and growth rate. We perform the analysis of such models in continuous and discrete trait spaces, i.e., we determine the existence of travelling wave solutions, asymptotic spreading speeds and their linear determinacy, as well as the population distributions at the leading edge. We also establish the relation between asymptotic spreading speeds and mutation probabilities. We observe conditions for when spatial sorting emerges and when it does not and also explore conditions where anomalous spreading speeds occur, as well as possible effects of deleterious mutations in the population.

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Notes

  1. Note that a more subtle definition of an anomalous spreading speed was first given by Weinberger et al. (2007). An anomalous speed in their sense can only occur when the linearized operator is reducible and system is not linearly determined. Our model does not fall under their definition.

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Acknowledgements

SP is funded by São Paulo Research Foundation, FAPESP (grants 2018/24037-4 and 2020/15320-4). FL is funded by Natural Sciences and Engineering Research Council of Canada (RGPIN-2016-0495). We thank Fields Institute for funding and organizing the “Workshop on New Mathematical Theory to Understand the Effects of Evolution on Range Expansion”, this work benefited from the event and discussions therein. We thank Thomas Hillen for inspiring discussions. We thank two anonymous reviewers and also the handling editor, Sebastian Schreiber, for their suggestions and corrections to the text.

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  1. Silas Poloni and Frithjof Lutscher have contributed equally to this work.

Authors and Affiliations

  1. Institute of Theoretical Physics, São Paulo State University, São Paulo, SP, 01140-070, Brazil

    Silas Poloni

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Silas Poloni & Frithjof Lutscher

  3. Department of Biology, University of Ottawa, Ottawa, ON, K1N 6N5, Canada

    Frithjof Lutscher

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  1. Silas Poloni
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Both authors contributed equally to conceptualizing the study and to writing and editing the manuscript. SP led the analysis and the numerical simulations.

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Correspondence to Silas Poloni.

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Poloni, S., Lutscher, F. Integrodifference models for evolutionary processes in biological invasions. J. Math. Biol. 87, 10 (2023). https://doi.org/10.1007/s00285-023-01947-z

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