On $\xi^{(s)}$-Quadratic Stochastic Operators on two Dimensional simplex and their behavior

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, it was investigated several classes of QSO. In this paper, we study $\xi^{(s)}$--QSO defined on 2D simplex. We first classify $\xi^{(s)}$--QSO into 20 non-conjugate classes. Further, we investigate the dynamics of three classes of such operators.

One of such systems which relates to the population genetics is given by a quadratic stochastic operator [1]. A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of species in biology, which arises as follows. Consider a population consisting of m species (or traits) 1, 2, · · · , m. We denote a set of all species (traits) by I = {1, 2, · · · , m}. Let x (0) = x (0) 1 , · · · , x (0) m be a probability distribution of species at an initial state and P ij,k be a probability that individuals in the i th and j th species (traits) interbreed to produce an individual from k th species (trait). Then a probability distribution x (1) = x (1) 1 , · · · , x (1) m of the spices (traits) in the first generation can be found as a total probability, i.e., This means that the association x (0) → x (1) defines a mapping V called the evolution operator. The population evolves by starting from an arbitrary state x (0) , then passing to the state x (1) = V (x (0) ) (the first generation), then to the state x (2) = V (x (1) ) = V (V (x (0) )) = V (2) x (0) (the second generation), and so on. Therefore, the evolution states of the population system described by the following discrete dynamical system x (0) , x (1) = V x (0) , x (2) = V (2) x (0) , x (3) = V (3) x (0) · · · In other words, a QSO describes a distribution of the next generation if the distribution of the current generation was given. The fascinating applications of QSO to population genetics were given in [20].
In [11], it was given along self-contained exposition of the recent achievements and open problems in the theory of the QSO. The main problem in the nonlinear operator theory is to study the behavior of nonlinear operators. This problem was not fully finished even in the class of QSO (the QSO is the simplest nonlinear operator). The difficulty of the problem depends on the given cubic matrix (P ijk ) m i,j,k=1 . An asymptotic behavior of the QSO even on the small dimensional simplex is complicated [6,29,30,32,33]. In order to solve this problem, many researchers always introduced a certain class of QSO and studied their behavior. For examples, Volterra-QSO [7,8,9,16,32], permutated Volterra-QSO [12,13], Quasi-Volterra-QSO [5], ℓ-Volterra-QSO [25,26], non-Volterra-QSO [6,30], strictly non-Volterra-QSO [28], F-QSO [27], and non Volterra operators generated by product measure [3,4,24]. However, all these classes together would not cover a set of all QSO. Therefore, there are many classes of QSO which were not studied yet. Recently, in the papers [21,22], a new class of QSO was introduced. This class was called a ξ (s) -QSO. In this paper, we are going to continue the study of ξ (s) -QSO. This class of operators depends on a partition of the coupled index set (the coupled trait set) P m = {(i, j) : i < j} ⊂ I×I. In case of two dimensional simplex (m = 3), the coupled index set (the coupled trait set) P 3 has five possible partitions. The dynamics of ξ (s) -QSO corresponding to the point partition (the maximal partition) of P 3 have been investigated in [21,22]. In the present paper, we are going to describe and classify such operators generated by other three partitions. Further, we also investigate the dynamics of three classes of such operators.
The paper is organized as follows: In Sec. 2, we give some preliminary definitions. In Sec. 3, we discuss the classification of ξ (s) -QSO related to |ξ| = 2. It turns out that some obtained operators are ℓ-Volterra-QSO (see [25,26]), permuted ℓ-Volterra-QSO. The dynamics of ℓ-Volterra-QSO are not fully studied yet. In [25,26], some particular cases have been investigated, which do not cover our operators. Therefore, in further sections, we study dynamics of ℓ-Volterra-QSO and permuted ℓ-Volterra-QSO. In Sec. 4, we study the behavior of ℓ-Volterra-QSO V 13 taken from the class K 1 . In Sec.5, we study the behavior of a permuted ℓ-Volterra-QSO V 4 taken from the class K 4 . Note that V 4 is a permutation of V 13 . In Sec. 6, we study the behavior of a permuted Volterra-QSO V 28 taken from the class K 19 . In the last section, we just highlight the dynamics of Volterra-QSO V 25 taken from the class K 17 which was already studied in [7]- [9].

Preliminaries
Recall that a quadratic stochastic operator (QSO) is a mapping of the simplex and P ij,k is a coefficient of heredity, which satisfies the following conditions Thus, each quadratic stochastic operator V : S m−1 → S m−1 can be uniquely defined by a cubic matrix P = P ijk m i,j,k=1 with conditions (2.3). We denote sets of fixed points and k−periodic points of V : S m−1 → S m−1 by F ix(V ) and P er k (V ), respectively. Due to Brouwer's fixed point theorem, one always has that F ix(V ) = ∅ for any QSO V . For a given point consists of a single point, then the trajectory converges and a limiting point is a fixed point of V : Recall that a Volterra-QSO is defined by (2.2), (2.3) and the additional assumption The biological treatment of condition (2.4) is clear: the offspring repeats the genotype (trait) of one of its parents.
One can see that a Volterra-QSO has the following form: where (2.6) a ki = 2P ik,k − 1 for i = k and a ii = 0, i ∈ I.
Moreover, a ki = −a ik and |a ki | ≤ 1. This kind of operators intensively studied in [2,7,8,9,16,32]. Note that this operator is a discretization of the Lotka-Volterra model [19,34] which models an interacting competing species in the population system. Such a model has received considerable attention in the fields of biology, ecology, mathematics (see for example [14,15,23,34]).
In [25], it has been introduced a notion of ℓ-Volterra-QSO, which generalizes a notion of Volterra-QSO. Let us recall it here.
Remark 2.1. Here, we stress the following points: 1. Note that an ℓ-Volterra-QSO is a Volterra-QSO if and only if ℓ = m.
2. It is known [7] that there is not a periodic trajectory for Volterra-QSO. However, there are such trajectories for ℓ-Volterra-QSO [25].
We call that an operator V is permuted ℓ-Volterra-QSO, if there is a permutation τ of the set I and an ℓ-Volterra-QSO V 0 such that (V (x)) τ (k) = (V 0 (x)) k for any k ∈ I. In other words, V can be represented as follows: We remark that if ℓ = m then a permuted ℓ-Volterra-QSO becomes a permuted Volterra-QSO. Some properties of such operators were studied in [9,10]. The Dynamics of certain class of permuted Volterra-QSO has been investigated in [22]. Note that in [25,26], it has been studied very particular class of ℓ-Volterra-QSO. An asymptotic behavior of permuted ℓ-Volterra-QSO has not been investigated yet. Some particular cases has been considered in [21,22].
In this paper, we are going to introduce a new class of QSO which contain ℓ-Volterra-QSO and permuted ℓ-Volterra-QSO as a particular case.
Note that each element x ∈ S m−1 is a probability distribution of the set I = {1, ..., m}. Let x = (x 1 , · · · , x m ) and y = (y 1 , · · · , y m ) be vectors taken from S m−1 . We say that x is equivalent to y if x k = 0 ⇔ y k = 0. We denote this relation by x ∼ y.
Let supp(x) = {i : x i = 0} be a support of x ∈ S m−1 . We say that x is singular to y and denote by x ⊥ y, if supp(x) ∩ supp(y) = ∅. Note that if x, y ∈ S m−1 then x ⊥ y if and only if (x, y) = 0, here (·, ·) stands for a standard inner product in R m .
We denote sets of coupled indexes by For a given pair (i, j) ∈ P m ∪ ∆ m , we set a vector P ij = (P ij,1 , · · · , P ij,m ). It is clear due to the condition (2.3) that P ij ∈ S m−1 .
A biological interpretation of a ξ (s) −QSO: We treat I = {1, · · · , 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 do not influence to have a child from the k th trait. In this sense, P m ∪ ∆ m is a set of all possible coupled traits of parents. A vector P ij = (P ij,1 , · · · , P ij,m ) is a possible distribution of children in a family while parents are carrying traits from the i th and j th types. A biological meaning of a ξ (s) −QSO is as follows: a set P m of all differently coupled traits of parents is splitted into N groups A 1 , · · · , A N (here N is less than the number m of traits) such that the chance (probability) of having a child from any trait in two different family whose parents' coupled traits belong to the same group A k is simultaneously either positive or zero (the condition (i) of Definition 2.2), meanwhile, two family whose parents' coupled traits belong to two different groups A k and A l cannot have a child from the same trait, simultaneously (the condition (ii) of Definition 2.2). Moreover, the parents which are sharing the same type of traits can have a child from only one type of traits (the condition (iii) of Definition 2.2 and Remark 2.3).

Classification of ξ (s) −QSO on 2D simplex
In this section, we are going to study ξ (s) −QSO in two dimensional simplex, i.e. m = 3. In this case, we have the following possible partitions of P 3 We note that in [21,22], it has been investigated ξ (s) -QSO related to the partition ξ 1 which is the maximal partition of P 3 . In this paper, we are aiming to study ξ (s) -QSO related to the partitions ξ 2 , ξ 3 , ξ 4 . We shall show that these three classes of ξ (s) -QSO are conjugate each other. Therefore, it is enough to study ξ (s) -QSO related to the partition ξ 2 . A class of ξ (s) -QSO related to the partition ξ 5 will be studied in elsewhere in the future.
Let us recall that two operators V 1 , V 2 are called (topologically or linearly) conjugate, if there is a permutation matrix P such that P −1 V 1 P = V 2 . Let π be a permutation of the set I = {1, · · · , m}. For any vector x, we define π(x) = (x π(1) , · · · , x π(m) ). It is easy to check that if π is a permutation of the set I corresponding to the given permutation matrix P then one has that P x = π(x). Therefore, two operators V 1 , V 2 are conjugate if and only if π −1 V 1 π = V 2 for some permutation π. Throughout this paper, we shall consider "conjugate operators" in this sense. We say that two classes K 1 and K 2 of operators are conjugate if every operator taken from K 1 is conjugate to some operator taken from K 2 and vise versus. Proof. We show that two classes of all ξ (s) −QSO corresponding to the partitions ξ 2 and ξ 3 are conjugate each other. Analogously, one can show that two classes of all ξ (s) −QSO corresponding to the partitions ξ 2 and ξ 4 are conjugate each other as well.
Therefore, it is enough to study a class of all ξ (s) -QSO corresponding to the partition ξ 2 . Now, we shall consider some sub-class of a class of all ξ (s) -QSO corresponding to the partition ξ 2 by choosing coefficients (P ij,k ) 3 i,j,k=1 in special forms: where a ∈ [0, 1] and The choices of the cases (I i , II j ), where i, j = 1, 6, will give 36 operators from the class of ξ (s) −QSO corresponding to the partition ξ 2 .
Theorem 4.2. Let V 13 : S 2 → S 2 be a ξ (s) -QSO given by (4.2) and 13 ) be an initial point. Then the following statements hold true: Proof. Let V 13 : S 2 → S 2 be a ξ (s) -QSO given by (4.2), F ix(V 13 ) be an initial point, and {x (n) } ∞ n=0 be a trajectory of V 13 starting from the point x (0) .
Let a = 1 2 . The first equation of (4.3) takes the form x 1 = x 1 . From the second equation of (4.3), we get that x 2 (x 1 − x 3 ) = 0. This yields that x 2 = 0 or x 1 = x 3 . In both cases, the third equation of (4.3) holds true. Therefore, we have that F ix(V 13 ) = Γ 2 l 13 .
Due to Claim, there exists n 0 such that x is decreasing, and hence it converges to x * 2 . Consequently, (x 3 ) ∈ S 1≤3 . This yields that {x (n) 2 } ∞ n=0 is decreasing and hence, it converges to x * 2 . Therefore, (x 3 ) converges to (x 1 , x In the similar manner, one may have that if This completes the proof.

5.
Dynamics of ξ (s) -QSO from the class K 4 We are going to study dynamics of a ξ (s) -QSO V 4 : S 2 → S 2 taken from K 4 : One can immediately see that this operator is a permuted ℓ-Volterra-QSO. As we mentioned, the behavior of such kinds of operators is not studied yet. It is worth mentioning that V 4 is a permutation of V 13 . Let It is clear that Theorem 5.1. Let V 4 : S 2 → S 2 be a ξ (s) -QSO given by (5.1), be an initial point. Then the following statements hold true: be a trajectory of V 4 starting from the point x (0) .
(i) In order to find fixed points of V 4 , we have to solve the following system: From the first equation of the system (5.2), one can find that x 1 = 0 or x 1 = 1 (see Proposition 4.1 (i)). If x 1 = 1 then x 2 = x 3 = 0. If x 1 = 0 then the second equation of the system (5.2) becomes as follows } whenever a = 1 2 . Let a = 1 2 . The system (5.2) then takes the following form So, by letting x 1 = b (any b ∈ [0, 1]), the second equation of the system (5.3) can be written as follows The solutions of the last equation are . One can check that the only solution 2 . Now, we are going to show that the operator V 4 given by (5.1) does not have any order periodic points in the set S 2 \ Γ 1 , where Γ 1 = {x ∈ S 2 : x 1 = 0}. In fact, since the function f a (x) = x 2 + 2ax(1 − x) is increasing (due to Proposition 4.1 (ii)), the first coordinate of V 4 increases along the iteration of V 4 in the set S 2 \ Γ 1 . This means that V 4 doesn't have any order periodic points in the set S 2 \ Γ 1 . Therefore, it is enough to find periodic points of V 4 in Γ 1 . In this case, in order to find 2-periodic points, we have to solve the following system of equations: The solutions of the second equation of the last system are 0, 1, 3± √ 5 2 . Hence, we have that P er(V 4 ) = {e 2 , e 3 } whenever a = 1 2 . Let a = 1 2 . In order to find 2-periodic points of V 4 , we should solve the following system of equations By letting x 1 = c, where c ∈ [0, 1], the second equation of the system (5.4) reduces to the following equation One  (2) . We immediately find (see the above discussion (ii)) that h (2) On the other hand, we have that Therefore, we obtain that ω V 4 (x (0) ) = {e 2 , e 3 } if x (0) 1 = 0. Let 1 2 < a ≤ 1 and 0 < x (0) 1 < 1. In this case, it is clear that x 1 ). Therefore, due to Proposition 4.1 (iv), we have that lim n→∞ x (n) (iv) Let a = 1 2 . Then the operator V 4 takes the following form: Therefore, we shall study the dynamics of V 4 over L c . Let 2 ) be a fixed number. Let us consider the function 6. Dynamics of ξ (s) -QSO from the class K 19 We are going to study dynamics of a ξ (s) -QSO V 28 : S 2 → S 2 taken from K 19 : One can see that this operator is a permuted Volterra-QSO. The behavior of this operator was not studied in [10,12,13]. It is worth mentioning that V 28 is a permutation of V 25 .
for any a = 1 2 . Then 0 ≤ A ≤ 1. In fact, one has that Theorem 6.1. Let V 28 : S 2 → S 2 be a ξ (s) -QSO given by (6.1), be an initial point. Then the following statements hold true: 2 , x (0) 3 / ∈ F ix(V 28 ) P er 2 (V 28 ) be an initial point and {x (n) } ∞ n=0 be a trajectory of V 28 starting from the point x (0) . (i) In order to find fixed points of V 28 , we need to solve the following system of equations: From the first equation of (6.2) one can find that x 1 = 0 or x 1 = 1. If x 1 = 1 then x 2 = x 3 = 0. If x 1 = 0 then x 2 + x 3 = 1. So, the second equation of (6.2) becomes as follows: Let a = 1 2 . Then one can find that solutions of (6.3) are . We can verify that the only solution which lies in the interval [0, 1] is 2 . Then (6.3) has a solution x 2 = 1 2 . This yields that x 3 = 1 2 . Therefore, one has that F ix(V 28 ) = {e 1 , (0, 1 2 , 1 2 )} whenever a = 1 2 . (ii) It is clear that V 28 does not have any order periodic points in S 2 \ Γ 1 (see Proposition 4.1 (ii)), where Γ 1 = {x ∈ S 2 : x 1 = 0}. So, any order periodic points of V 28 lie on Γ 1 (if any). In order to find 2-periodic points of V 28 , we have to solve the equation V 2 28 (x) = x with the condition x 1 = 0. Then, by taking into account x 2 + x 3 = 1, we may get the following equation Let a = 1 2 . Then, the last equation has the solutions {0, 1, ±A}. So, 2-periodic points of V 28 are only e 2 = (0, 1, 0) and e 3 = (0, 0, 1).
Let a = 1 2 . Then, the equation given above becomes an identity x 2 = x 2 . This means that all points of the edge Γ 1 except (0, 1 2 , 1 2 ) are 2-periodic points. (iii) Let a = 1 2 . It is clear that the edge Γ 1 is invariant under V 28 . We want to study the behavior of V 28 over this line. In this case, V 28 | Γ 1 takes the following form: Let us consider the function g a (x 2 ) = (1 − x 2 ) 2 + 2ax 2 (1 − x 2 ), where a = 1 2 . One can easily check that g a is decreasing on [0, 1]. This yields that g (2) a is increasing on [0, 1]. As we already discussed that F ix(g a ) ∩ [0, 1] = {A} and F ix g  a . We immediately find (see the above discussion (ii)) that g (2) a (x 2 ) > x 2 whenever x 2 > A and g (2) a (x 2 ) < x 2 whenever x 2 < A.
(iv) Let a = 1 2 . Since x (0) / ∈ F ix(V 28 ) P er 2 (V 28 ), we have that x (0) 1 > 0. Then, due to Proposition 4.1 (iv), we again has that x Since ω V 28 (x (0) ) is not empty, we obtain that ω V 28 (x (0) ) = {e 1 }. This completes the proof. 7. Dynamics of ξ (s) -QSO from the class K 17 We are going to highlight the dynamics of a ξ (s) -QSO V 25 : S 2 → S 2 taken K 17 a)x 2 x 3 where 0 ≤ a ≤ 1. One can immediately see that the operator (7.1) is a Volterra-QSO. The dynamics of such kinds of operators have been studied in [7,8,9]. By means of the results of the mentioned papers, one can formulate the following