Genetic Algebras Associated with ξ^(a)-Quadratic Stochastic Operators

Mathematical population genetics investigates the dynamics of frequency distributions of genetic types (alleles, genotypes, gene collections, etc.) in successive generations under the action of evolutionary forces. To explore the behavior of the population, the discrete dynamical system associated with an evolution operator is the main object of the theory. In ref. [1], a short history of applications of mathematics to solving various problems in population dynamics is given.

On the other hand, there is another theoretical framework to investigate essential properties of population genetics, which is based on an algebraic approach [2,3]. In this scheme, most of the algebras are nonassociative. In the literature (see, for example, [4,5]), plenty of nonassociative algebras (baric, evolution, Bernstein, train, stochastic, etc.) have appeared to model inheritance in genetic systems. Such algebras are referred to as “genetic algebras”. In general, problems of population genetics were started in [6] by employing quadratic stochastic operators (see also [2]). It is worth recalling those operators which present the time evolution of species in biology [7,8]. Namely, let us look at a population consisting of m species (or traits) which are denoted by $I=\{1,2,\cdots ,m\}$. Assume that $x^{\left(0\right)}=\left(x_1^{\left(0\right)},\cdots ,x_m^{\left(0\right)}\right)$ is a probability distribution of species at an initial state, and $p_{ij,k}$ is a probability that individuals in the ith and jth species interbreed to produce an individual from a kth species. Then, a probability distribution $x^{\left(1\right)}=\left(x_1^{\left(1\right)},\cdots ,x_m^{\left(1\right)}\right)$ of the species in the first generation can be found as a total probability, i.e., The correspondence $x^{\left(0\right)}\to x^{\left(1\right)}$ is called the evolution operator or quadratic stochastic operator (QSO). In other words, such an operator describes a distribution of the next generation if the distribution of the current generation was given. Applications of QSOs to population genetics were given in [2,9,10,11]. The reader is referred to [12] for a self-contained exposition of the recent achievements and open problems in the theory of QSOs.

QSOs find applications as discrete-time dynamical systems in various fields, including economics, epidemiology, and social sciences [13,14,15,16,17]. They are valuable tools for analyzing and predicting the behavior of complex systems that undergo discrete changes at fixed time intervals. For instance, in economics, QSOs can assist in modeling and understanding market dynamics, optimizing resource allocation, and predicting economic trends. In epidemiology, QSOs can be employed to simulate disease spread, evaluate intervention strategies, and forecast the progression of infectious diseases. Moreover, in social sciences, QSOs can aid in studying social dynamics, opinion formation, and decision-making processes within populations. By employing QSOs as discrete-time dynamical systems, researchers can gain insights into the intricate dynamics of these complex systems, enabling better understanding, planning, and decision-making.

Each QSO defines an algebraic structure on the vector space ${\mathbb{R}}^m$ containing the simplex (see next section for definitions). The associated algebra is called genetic algebra. A more modern use of the genetic algebra theory for self-fertilization can be found in [18,19]. Therefore, it is the interplay between the purely mathematical structure and the corresponding genetic properties that makes this subject so fascinating. We refer to [2,3,20] for comprehensive references.

In [21,22], new classes of QSO were introduced, which are called ${\xi }^{\left(as\right)}$-QSOs. We notice that such classes of operators depend on the partition of the coupled index set (the coupled trait set) ${\mathbf{P}}_m=\{(i,j):i<j\}\subset I\times I$. Furthermore, certain subclasses of these operators have been intensively explored in [23,24,25]. However, in those investigations, the algebraic structures of genetic algebras associated with ${\xi }^{\left(a\right)}$-QSO are not considered. Therefore, to fill that gap, in the present paper, we are aiming to study certain algebraic properties of genetic algebras corresponding to ${\xi }^{\left(a\right)}$-QSO. We stress that the considered ${\xi }^{\left(a\right)}$-QSOs are different from Lotka–Volterra QSOs, which also have important applications in several branches of sciences [26,27,28,29]. The genetic algebras associated with Lotka–Volterra operators have been intensively explored in [30,31,32,33]. There appeared several works on the derivations of genetic algebras [34,35,36,37]. Interpretations of the derivations have been discussed in [18]. Recently, in [38,39,40,41], derivations of Lotka–Volterra algebras have been described. Furthermore, other types of genetic algebras have been investigated in [42,43,44,45,46,47].

The paper is organized as follows. In Section 2, we collect necessary definitions from the theory of genetic algebras. Section 3 is devoted to the construction of a class of ${\xi }^{\left(a\right)}$-QSO on two-dimensional simplex. Furthermore, in Section 4, we study the associativity of these operators along with their dynamics. The characters of these algebras are described in Section 5. In Section 6, the derivations of genetic algebras associated with ${\xi }^{\left(a\right)}$ are described. Moreover, in Section 7, the dynamics of these operators are discussed.

Recall that a quadratic stochastic operator (QSO) is a mapping of the simplex into itself, of the form where $V\left(x\right)=x^{\prime }=(x_1^{\prime },\cdots ,x_m^{\prime })$, and $P_{ij,k}$ is a coefficient of heredity, which satisfies the following conditions Thus, each quadratic stochastic operator $V:S^{m-1}\to S^{m-1}$ can be uniquely defined by a cubic matrix $\mathcal{P}={\left(P_{ijk}\right)}_{i,j,k=1}^m$ with conditions (3).

A point $x\in S^{m-1}$ is called a k-periodic point of V, if $V^j\left(x\right)\ne x$, $0\le j<k$, $V^k\left(x\right)=x$. If $k=1$, then such a point is called a fixed point of V. The set of fixed points and $k-$ periodic points of V are denoted by $Fix\left(V\right)$ and $Pe{r}_k\left(V\right)$, respectively. For a given point $x^{\left(0\right)}\in S^{m-1}$, a trajectory ${\left\{x^{\left(n\right)}\right\}}_{n=0}^{\infty }$ of V starting from $x^{\left(0\right)}$ is defined by $x^{(n+1)}=V\left(x^{\left(n\right)}\right)$. By ${\omega }_V\left(x^{\left(0\right)}\right)$, we denote a set of omega limiting points of the trajectory ${\left\{x^{\left(n\right)}\right\}}_{n=0}^{\infty }$.

Note that each element $x\in S^{m-1}$ is a probability distribution of the set $I=\{1,\dots ,m\}.$ Let $x=(x_1,\cdots ,x_m)$ and $y=(y_1,\cdots ,y_m)$ be vectors taken from $S^{m-1}$. We say that x is equivalent to y if $x_k=0$⇔$y_k=0$; this relation is denoted by $x\sim y$.

Let $supp\left(x\right)=\{i:x_i\ne 0\}$ be a support of $x\in S^{m-1}$. We say that x is singular to y and denote by $x\perp y$, if $supp\left(x\right)\cap supp\left(y\right)=\varnothing$.

We denote the sets of coupled indexes by For a given pair $(i,j)\in {\mathbf{P}}_m\cup {\Delta }_m$, we set a vector ${\mathbb{P}}_{ij}=\left(P_{ij,1},\cdots ,P_{ij,m}\right)$. It is clear due to condition (3) that ${\mathbb{P}}_{ij}\in S^{m-1}$.

Let $\xi ={\left\{A_i\right\}}_{i=1}^N$ and $\eta ={\left\{B_i\right\}}_{i=1}^M$ be some fixed partitions of ${\mathbf{P}}_m$ and ${\Delta }_m$, respectively, i.e., $A_i\bigcap A_j=\varnothing$, $B_i\bigcap B_j=\varnothing$, and ${\displaystyle \bigcup _{i=1}^N}{A}_i={\mathbf{P}}_m$, ${\displaystyle \bigcup _{i=1}^M}{B}_i={\Delta }_m$, where $N,M\le m$.

A BIOLOGICAL INTERPRETATION OF A ${\xi }^{\left(a\right)}-$ QSO: We treat $I=\{1,\cdots ,m\}$ as a set of all possible traits of the population system. A coefficient $P_{ij,k}$ is a probability that parents in the $i^{th}$ and $j^{th}$ traits interbreed to produce a child from the $k^{th}$ trait. The condition $P_{ij,k}=P_{ji,k}$ means that the gender of parents does not influence the probability of having a child from the $k^{th}$ trait. In this sense, ${\mathbf{P}}_m\cup {\Delta }_m$ is a set of all the possible coupled traits of the parents. A vector ${\mathbb{P}}_{ij}=\left(P_{ij,1},\cdots ,P_{ij,m}\right)$ is a possible distribution of children in a family where the parents are carrying traits from the $i^{th}$ and $j^{th}$ types.

In this section, we are going to define ${\xi }^{\left(a\right)}$-QSO in two-dimensional simplex, i.e., $m=3$. The set ${\mathbf{P}}_3$ has the following possible partitions:

We notice that ${\xi }^{\left(as\right)}$-QSOs corresponding to the partitions ${\xi }_1-{\xi }_4$ have been studied in [22,23,24,25]. Therefore, in the present paper, we concentrate on the partition ${\xi }_5$ and $\eta =\left\{\right(1,1\left)\right(2,2\left)\right(3,3\left)\right\}$, which defines a class of ${\xi }^{\left(a\right)}$-QSO. In the sequel, for the sake of simplicity, we are going to consider the following coefficients ${\left(P_{ij,k}\right)}_{i,j,k=1}^m$ given by the table:

$P_{11}$	$P_{22}$	$P_{33}$	$P_{12}$	$P_{13}$	$P_{23}$
$(\alpha ,1-\alpha ,0)$	$(1-\alpha ,\alpha ,0)$	$(1-\alpha ,\alpha ,0)$	$(1,0,0)$	$(1,0,0)$	$(1,0,0)$
$(\alpha ,0,1-\alpha )$	$(1-\alpha ,0,\alpha )$	$(1-\alpha ,0,\alpha )$	$(0,1,0)$	$(0,1,0)$	$(0,1,0)$
$(0,\alpha ,1-\alpha )$	$(0,1-\alpha ,\alpha )$	$(0,1-\alpha ,\alpha )$	$(0,0,1)$	$(0,0,1)$	$(0,0,1)$

where $\alpha \in [0,1]$.

The corresponding QSOs are listed as follows:

Let V be a QSO, and suppose that $x,y\in {\mathbb{R}}^m$ are arbitrary vectors. Then, one can define a binary rule [9] on ${\mathbb{R}}^m$ by Using (3), one can see that $\mathbf{x}{\circ }_V\mathbf{y}=\mathbf{y}{\circ }_V\mathbf{x}$, i.e., the multiplication is commutative. Certain algebraic properties of such kinds of algebras were investigated in [2,3,20]. In general, genetic algebra is not necessarily associative.

The multiplication (13) in the canonical basis can be represented as follows: It turns out that the multiplication can be given terms of QSO One can check that This algebraic interpretation is useful, e.g., a state $\mathbf{x}$ is an equilibrium precisely when $\mathbf{x}$ is an idempotent element of the algebra.

The algebra A is called associative if

In this section, we are going to investigate the associativity of genetic algebras generated by ${\xi }^{\left(a\right)}$-QSO described in the previous section. To describe such algebras, we are going to consider more general operators which cover all listed ones. For this reason, we are going to evaluate the following table:

$P_{11,1}=a_1$	$P_{11,2}=b_1$	$P_{11,3}=c_1$
$P_{22,1}=a_2$	$P_{22,2}=b_2$	$P_{22,3}=c_2$
$P_{33,1}=a_3$	$P_{33,2}=b_3$	$P_{33,3}=c_3$

where $a_i,b_i,c_i\ge 0$ Furthermore, we assume that the coefficients ${\left(P_{ij,k}\right)}_{ij,k=1}^m$ are given by

Cases	$P_{12}$	$P_{13}$	$P_{23}$
1	(1,0,0)	(1,0,0)	(1,0,0)
2	(0,1,0)	(0,1,0)	(0,1,0)
3	(0,0,1)	(0,0,1)	(0,0,1)

Then, the corresponding QSOs are described as follows:

The obtained operators $W_1,W_2$ and $W_3$, according to (13), generate corresponding genetic algebras which are denoted by $A_1,A_2$ and $A_3$. Therefore, we are going to investigate the associativity of these algebras. Let us list their table of multiplication.

Case I: In this case, we consider the QSO $W_1$; then, for the corresponding genetic algebra $A_1$, the table of multiplication is given by

	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_3$
${\mathbf{e}}_1$	$a_1{\mathbf{e}}_1+b_1{\mathbf{e}}_2+c_1{\mathbf{e}}_3$	${\mathbf{e}}_1$	${\mathbf{e}}_1$
${\mathbf{e}}_2$	${\mathbf{e}}_1$	$a_2{\mathbf{e}}_1+b_2{\mathbf{e}}_2+c_2{\mathbf{e}}_3$	${\mathbf{e}}_1$
${\mathbf{e}}_3$	${\mathbf{e}}_1$	${\mathbf{e}}_1$	$a_3{\mathbf{e}}_1+b_3{\mathbf{e}}_2+c_3{\mathbf{e}}_3$

Case II: Now, let us consider $W_2$, then the algebra $A_2$ has the following table of multiplication:

	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_3$
${\mathbf{e}}_1$	$a_1{\mathbf{e}}_1+b_1{\mathbf{e}}_2+c_1{\mathbf{e}}_3$	${\mathbf{e}}_2$	${\mathbf{e}}_2$
${\mathbf{e}}_2$	${\mathbf{e}}_2$	$a_2{\mathbf{e}}_1+b_2{\mathbf{e}}_2+c_2{\mathbf{e}}_3$	${\mathbf{e}}_2$
${\mathbf{e}}_3$	${\mathbf{e}}_2$	${\mathbf{e}}_2$	$a_3{\mathbf{e}}_1+b_3{\mathbf{e}}_2+c_3{\mathbf{e}}_3$

Case III: Using the same argument, the algebra $A_3$ is defined by $W_3$, and its table of multiplication is given by

	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_3$
${\mathbf{e}}_1$	$a_1{\mathbf{e}}_1+b_1{\mathbf{e}}_2+c_1{\mathbf{e}}_3$	${\mathbf{e}}_3$	${\mathbf{e}}_3$
${\mathbf{e}}_2$	${\mathbf{e}}_3$	$a_2{\mathbf{e}}_1+b_2{\mathbf{e}}_2+c_2{\mathbf{e}}_3$	${\mathbf{e}}_3$
$e_3$	${\mathbf{e}}_3$	${\mathbf{e}}_3$	$a_3{\mathbf{e}}_1+b_3{\mathbf{e}}_2+c_3{\mathbf{e}}_3$

Furthermore, due to the proved theorem, we always consider the genetic algebra $A_1$.