Inside Dynamics of Integrodifference Equations with Mutations

  • Nathan G. MarculisEmail author
  • Mark A. Lewis
Original Article


The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.


Integrodifference equations Mutations Neutral genetic diversity Range expansion Spreading speed 



This research was supported by a grant to MAL from the Natural Science and Engineering Research Council of Canada (Grant No. NET GP 434810-12) to the TRIA Network, with contributions from Alberta Agriculture and Forestry, Foothills Research Institute, Manitoba Conservation and Water Stewardship, Natural Resources Canada-Canadian Forest Service, Northwest Territories Environment and Natural Resources, Ontario Ministry of Natural Resources and Forestry, Saskatchewan Ministry of Environment, West Fraser and Weyerhaeuser. MAL is also grateful for support through NSERC and the Canada Research Chair Program. NGM acknowledges support from NSERC TRIA-Net Collaborative Research Grant and would like to express his thanks to the Lewis Research Group for the many discussions and constructive feedback throughout this work. We are also grateful for the helpful and detailed comments from anonymous reviewers.

Supplementary material


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Copyright information

© Society for Mathematical Biology 2020

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematical and Statistical Sciences & Department of Biological SciencesUniversity of AlbertaEdmontonCanada

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