Skip to main content

On discrete evolutionary dynamics driven by quadratic interactions


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.

This is a preview of subscription content, access via your institution.


  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.


  • Abraham VM (1980) Linearizing quadratic transformations in genetic algebras. Proc Lond Math Soc 40:346–363

    Article  Google Scholar 

  • Abraham VM (1980) The genetic algebra of polyploids. Proc Lond Math Soc 3(40):385–429

    Article  Google Scholar 

  • Andrade R, Catalan A, Labra A (1994) The identity \(\left( x^{2}\right) ^{2}=\varpi \left( x\right) x^{3}\) in baric algebras. In: Gonzalez S (Ed.) Non-associative algebra and its applications. Math. and its Applic., Springer Science+Business Media, B.V

  • Al’pin YA, Koreshkov NA (2000) On the simultaneous triangulability of matrices. Math Notes 68(5)

  • Bürger, R (2000) The mathematical theory of selection, recombination, and mutation. Wiley Series in mathematical and computational biology. Wiley, Chichester, pp xii+409

  • Etherington IMH (1939) Genetic algebras. Proc R Soc Edinb 59:242–258

    Article  Google Scholar 

  • Etherington IMH (1941) Quart J Math (Oxford) 12:1–8

    Article  Google Scholar 

  • Etherington IMH (1941) Non-associative algebra and the symbolism of genetics. Proc R Soc Edinb B 61:24–42

    Google Scholar 

  • Ewens WJ (2004) Mathematical population genetics. I. Theoretical introduction. Second edition. Interdisciplinary applied mathematics, vol 27. Springer, New York

  • Fran F, Irawati I (2015) The condition for a genetic algebra to be a special train algebra. J Multidiscip Eng Sci Technol 2(6):1496–1500

    Google Scholar 

  • Ganikhodzhaev R, Mukhamedov F, Rozikov U (2011) Quadratic stochastic operators and processes: results and open problems. Infin Dimens Anal Quantum Probab Relat Top 14(2):279–335

    Article  Google Scholar 

  • Gonshor H (1960) Special train algebras arising in genetics. Proc Edinb Math Soc 2(12):41–53

    Article  Google Scholar 

  • Gonshor H (1965) Special train algebras arising in genetics, II. Proc Edinb Math Soc 2(14):333–338

    Article  Google Scholar 

  • Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Holgate P (1965) Genetic algebras associated with polyploidy. Proc Edinb Math Soc 15:1–9

    Article  Google Scholar 

  • Holgate P (1981) Population algebras. J R Stat Soc B 43(1):1–19

    Google Scholar 

  • Holgate P (1989) Some infinite-dimensional genetics algebras. Algèbres génétiques (Montpellier, 1985), 35–45, Cahiers Math Montpellier, 38, Univ. Sci. Tech. Languedoc, Montpellier

  • Karlin S (1984) Mathematical models, problems, and controversies of evolutionary theory. Bull Am Math Soc (N.S.) 10(2):221–274

    Article  Google Scholar 

  • Kesten H (1970) Quadratic transformations: a model for population growth. I (and II). Adv Appl Probab, Vol. 2, No. 1 (resp. 2), 1–82, (resp. 179–228)

  • Kesten H (1971) Some nonlinear stochastic growth models. Bull Am Math Soc 77(4)

  • Kingman JFC. Mathematics of genetic diversity. CBMS-NSF Regional Conference Series in Applied Mathematics, 34. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980, pp vii+70. ISBN: 0-89871-166-5

  • Kingman JFC (1961) A matrix inequality. Quart J Math Oxford Ser 12:78–80

    Article  Google Scholar 

  • Lyubich YI (1992) Mathematical structures in population genetics. Vol 22 of Biomathematics, Springer, Berlin

  • McCoy NH (1934) On quasi-commutative matrices. Trans Am Math Soc 36(2):327–340

    Article  Google Scholar 

  • McCoy NH (1936) On the characteristic roots of matric polynomials. Bull Am Math Soc 42(8):592–600

    Article  Google Scholar 

  • Reed ML (1997) Algebraic structure of genetic inheritance. (New Ser) Am Math Soc 34(2):107–130

    Article  Google Scholar 

  • Weissing FJ, van Boven M (2001) Selection and segregation distortion in a sex-differentiated population. Theor Popul Biol 60(4):327–341

    CAS  Article  PubMed  Google Scholar 

  • Wörz-Busekros A (1980) Algebras in genetics, vol 36., Lecture Notes in Biomathematics, Springer, Berlin-Heidelberg-New York

  • Wörz-Busekros A (1981) Relationship between genetic algebras and semicommutative matrices. Linear Algebra Appl 39:111–123

    Article  Google Scholar 

Download references


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).

Author information

Authors and Affiliations


Corresponding author

Correspondence to N. Grosjean.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Grosjean, N., Huillet, T. & Rollet, G. On discrete evolutionary dynamics driven by quadratic interactions. Theory Biosci. 135, 187–200 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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