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On discrete evolutionary dynamics driven by quadratic interactions

Abstract

After an introduction to the general topic of models for a given locus of a diploid population whose quadratic dynamics is determined by a fitness landscape, we consider more specifically the models that can be treated using genetic (or train) algebras. In this setup, any quadratic offspring interaction can produce any type of offspring and after the use of specific changes of basis, we study the evolution and possible stability of some examples. We also consider some examples that cannot be treated using the framework of genetic algebras. Among these are bistochastic matrices.

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Notes

  1. Throughout, a boldface variable, say \(\mathbf {x}\), will represent a column vector and its transpose, say \(\mathbf {x}^{\prime }\), will be a row vector. And \(B^{\prime }\) will denote the transpose of some square matrix B. We put \(|\mathbf {x}|:=\sum _{k=1}^{K}\left| x_{k}\right|\) and \(\mathbf {x}\succeq \mathbf {0}\) means that all entries of \(\mathbf {x}\) are nonnegative.

  2. Symmetric bistochastic matrices is the convex hull of extremal matrices of the form \(\left( P+P^{\prime }\right) /2\) where P is any permutation matrix.

  3. It can indeed be checked here that \(I^{2}=\left\langle \mathbf {c}_{3},..., \mathbf {c}_{K}\right\rangle\), \(I^{3}=\left\langle \mathbf {c}_{4},..., \mathbf {c}_{K}\right\rangle\),...and \(\mathcal {A}I^{2}\subseteq I^{2},\) \(\mathcal {A}I^{3}\subseteq I^{3}\),...

  4. Because this model is Gonshor-compatible, the Lie algebra generated by \(\left\{ E_{1},...,E_{4}\right\}\) is solvable, see below. And all \(E_{i}-E_{j}\) are nilpotent.

  5. This means that for all examples designed in Sect. 3.1, the Lie algebras generated by the \(\left\{ E_{i}\right\}\) which can be built from the \(\left\{ \Gamma _{i}\right\}\) we started from, were solvable and that all \(E_{i}-E_{j}\) were nilpotent.

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Acknowledgments

T. Huillet acknowledges support from the Project Basal PFB 03 of the CONICYT of Chile, from the “Chaire Modélisation mathématique et biodiversité” and together with N. Grosjean, from the labex MME-DII Center of Excellence (Modèles mathématiques et économiques de la dynamique, de l’incertitude et des interactions, ANR-11-LABX-0023-01 project).

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Grosjean, N., Huillet, T. & Rollet, G. On discrete evolutionary dynamics driven by quadratic interactions. Theory Biosci. 135, 187–200 (2016). https://doi.org/10.1007/s12064-016-0232-z

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Keywords

  • Evolutionary dynamics
  • Quadratic interactions
  • Genetic algebras
  • Polymorphism
  • Bistochastic interaction