Quadratic Stochastic Operators and Evolution Algebras

Quadratic stochastic operators (QSOs) are nonlinear mappings that describe the time evolution of probability distributions across a finite set of states. Originating in population genetics and ecology, they model pairwise interactions—such as mating, competition or recombination—by assigning offspring probabilities via quadratic forms. Evolution algebras are non-associative algebras tailored to these dynamics: each generator corresponds to a basis state, and the product encodes the QSO’s reproduction rule. Unlike classical algebras, evolution algebras often lack associativity yet admit a natural baric mapping reflecting total population size. These structures bridge dynamical systems, graph theory and algebraic classification, offering insights into equilibrium behaviour, convergence of iterates and invariant subspaces. Classification efforts have revealed a rich landscape of low-dimensional types and decompositions, while recent work has extended the theory to coalgebraic duals, tensor constructions and derivations. Applications span the modelling of non-Mendelian inheritance, spread of traits in networks, and restoration of ancestral gene frequencies. The interplay between algebraic properties—such as simplicity, perfectness and solvability—and the long-term dynamics of QSOs underpins a unifying framework for discrete evolutionary processes.

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Quadratic Stochastic Operators and Evolution Algebras publication trend

The graph below shows the total number of articles in quadratic stochastic operators and evolution algebras across all publications each year (not limited to Nature Index journals).

Technical terms

Quadratic stochastic operator: A mapping on the simplex of probability vectors defined by a quadratic form that governs pairwise interaction dynamics.

Evolution algebra: A non-associative algebra whose multiplication encodes state-transition probabilities of a QSO, with each basis element reproducing independently.

Genetic algebra: An algebraic structure derived from a QSO, reflecting inheritance rules and often endowed with a baric character to track total mass.

Derivation: A linear map on an algebra satisfying the Leibniz rule, used to study infinitesimal deformations and symmetry properties.

Tensor product: An operation combining two algebras into a higher-dimensional algebra, under which evolution-algebra properties may or may not be preserved.

References

1. Coalgebraic Structure of Genetic Inheritance. Mathematical Biosciences and Engineering (2004).
2. Classifying Evolution Algebras of Dimensions Two and Three. Mathematics (2019).
3. On ξ(s)‐Quadratic Stochastic Operators on Two‐Dimensional Simplex and Their Behavior. Abstract and Applied Analysis (2013).
4. Tensor Product of Evolution Algebras. Mediterranean Journal of Mathematics (2022).
5. Genetic Algebras Associated with ξ(a)-Quadratic Stochastic Operators. Entropy (2023).
6. Derivations and loops of some evolution algebras. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (2023).