A family of non-Volterra quadratic operators corresponding to permutations

The quadratic stochastic operators frequently arise in many models of mathematical genetics, namely, in the theory of heredity (see [ASZT], [Ber], [BlJaSc], [GGJ], [Gsb], [GMR], [GS], [Jlob], [Jjph], [Jdnc], [JL], [JLM], [JSW], [K], [Khukr], [L], [RZhmn], [RZhukr], [U], [Zus], [ZhRsb]). Consider a biological population and suppose that each individual in this population belongs precisely to one of the species (genotype) $1,\dots,m$. The scale of species is such that the species of the parents $i$ and $j$, unambiguously, determine the probability of every species $k$ for the first generation of direct descendants. Denote this probability, called the heredity coefficient, by $p_{ij,k}=P(k|(i,j))$. It is then obvious that $p_{ij,k}\geq 0$ for all $i,j,k$ and that

The state of the population can be described by the tuple $(x_{1},x_{2},\dots,x_{m})$ of species probabilities, that is, $x_{k}=P(k)$ is the fraction of the species $k$ in the total population. In the case of panmixia (random interbreeding) the parent pairs $i$ and $j$ arise for a fixed state $\mathbf{x}=(x_{1},x_{2},\dots,x_{m})$ with probability $x_{i}x_{j}=P(i,j)=P(i)P(j)$. Hence, the total probability of the species $k$ in the first generation of direct descendants is defined by

The association $\mathbf{x}\mapsto\mathbf{x}^{\prime}$ defines an evolutionary quadratic operator. Thus evolution of a population can be studied as a dynamical system of a quadratic stochastic operator [L]. See [GMR] and [MG] for a review of QSOs. Recently in [Khukr], [RHqtds] a quasi-strictly non-Volterra QSO is studied. We refer the reader to [JRjoca20] for a review on convex combinations of quadratic stochastic operators. The main goal of the present paper is to study a family of operators which contains a convex combination of two non-Volterra QSOs. The paper is organised as follows. In Section 2 we recall definitions and well known results from the theory of Volterra and non-Volterra QSOs. In Section 3 we consider a class of non-Volterra QSOs and study trajectory behaviors of such operators. We show that each QSO from this class has the two fixed points. Moreover, we prove that such operator is ergodic.

Let

be the $(m-1)$-dimensional simplex. A map $V$ of $S^{m-1}$ into itself is called a quadratic stochastic operator (QSO) if

for any $\mathbf{x}\in S^{m-1}$ and for all $k=1,\dots,m$, where

Assume $\{\mathbf{x}^{(n)}\in S^{m-1}:n=0,1,2,\dots\}$ is the trajectory (orbit) of the initial point $\mathbf{x}\in S^{m-1}$, where $\mathbf{x}^{(n+1)}=V(\mathbf{x}^{(n)})$ for all $n=0,1,2,\dots$, with $\mathbf{x}^{(0)}=\mathbf{x}$. One of the main problems in mathematical biology is to study the asymptotic behavior of the trajectories. This problem deeply studied for the Volterra QSOs (see [Gsb], [Gmn]).

Denote the set of all fixed points by $\text{Fix}\,(V)$ and the set of all periodic points of (not necessarily the smallest) period $n$ by $\text{Per}_{n}\,(V)$. Evidently that the set of all iterates of a periodic point form a periodic trajectory (orbit). Let $D_{\mathbf{x}}V(\mathbf{x^{*}})=(\partial V_{i}/\partial x_{j})(\mathbf{x^{*}})$ be a Jacobian of $V$ at the point $\mathbf{x^{*}}$.

Note that the limit point is a fixed point of a QSO. Thus, the fixed points of a QSO describe limit or long run behavior of the trajectories for any initial point. The limit behavior of trajectories and fixed points play an important role in many applied problems (see [BlJaSc], [GGJ], [Gsb], [GMR], [Jlob], [Jjph], [Juzmj], [JL], [JLM], [K], [L], [RZhmn], [Zus], [ZhRsb]). The biological treatment of the regularity of a QSO is rather clear: in the long run the distribution of species in the next generation coincides with the distribution of species in the previous one, i.e., it is stable. For nonlinear dynamical systems (1) Ulam [U] suggested an analogue of a measure-theoretic ergodicity, the following ergodic hypothesis:

On the basis of numerical calculations Ulam, in [U], conjectured that the ergodic theorem holds for any QSO. In [Zus] Zakharevich proved that this conjecture is false in general. Later, in [GZ], a sufficient condition of non-ergodicity for QSOs defined on $S^{2}$ was established. In [GGJ] have shown the correlation between non-ergodicity of Volterra QSOs and rock-paper-scissors games. In [JSW] the random dynamics of Volterra QSOs is studied. The biological treatment of non-ergodicity of a QSO is the following: in the long run the behavior of the distributions of species is unpredictable. Note that a regular QSO is ergodic, but in general from ergodicity does not follow regularity. Let $\omega_{V}\big{(}\mathbf{x}^{(0)}\big{)}$ be the set of limit points of the trajectory

Note that a Lyapunov function is very helpful to describe an upper estimate of $\omega_{V}(\mathbf{x}^{0})$.

The set $\operatorname{supp}(\pi)=\{i\in E_{n}:\pi(i)\neq i\}$ denotes the support of $\pi$ and we let $\operatorname{supp}(k)$ denote the support of the $k$-cycle, that is, the set

Any permutation can be represented in the form of a product of cycles without common elements (i.e. disjoint cycles) and this representation is unique to within the order of the factors. Let $\pi=\tau_{1}\tau_{2}\dots\tau_{q}$ be a permutation of the set $E_{m-1}=\{1,\dots,m-1\}$, where $\tau_{1},\dots,\tau_{q}$ are disjoint cycles and we denote by $\operatorname{ord}(\tau_{i})$ the order of a cycle $\tau_{i}$. Evidently that

The following notations will be used in the below. Let

denote the boundary of $S^{m-1}$ and let $\text{int}\,S^{m-1}=\left\{\mathbf{x}\in S^{m-1}\colon x_{1}x_{2}\dotsm x_{m}>0\right\}$ be the interior of $S^{m-1}$.

Consider a non-Volterra QSO defined on a finite-dimensional simplex which has the form

where $\pi$ is a permutation on the set $E_{m-1}$. It is worth mentioning that if $\pi=(21)(3)$ then the QSO (3) coincides up to the rearrangement of the coordinates with the quasi-strictly non-Volterra QSO which is studied in [Khukr]. Let $s=\text{LCM}\,\big{(}\operatorname{ord}(\tau_{1}),\dots,\operatorname{ord}(\tau_{q})\big{)}$.

Let $V_{1}:=V_{Id}$ and $V_{2}:=V_{\pi}$. Consider the convex combination of the QSOs $V_{1},V_{2}$, that is,

It is easy to see that the operator $V_{\alpha}$ has the form

where $\pi$ is a permutation on the set $E_{m-1}$. It is evident that if $\pi=Id$ then for any $\alpha\in[0,1]$ the operator $V_{\alpha}$ coincides with the QSO $V_{1}$. The dynamics of the operator $V_{1}$ is given in the Theorem 3.1. In the below we consider the cases $\pi\neq Id$. The QSO (4) can be written as follow

where $\pi$ is a permutation on the set $E_{m-1}$. Consider the function $f(x)=2x^{2}-2x+1,\ \ x\in[0,1]$. We define $f^{n}$ to be the $n$-fold composition of $f$ with itself. One can easily verify the statements of the next proposition about dynamics of the function $f(x)$.

Denote $\operatorname{supp}(\mathbf{x})=\{i:\,x_{i}>0\}$ and let $|\operatorname{supp}(\mathbf{x})|$ be its cardinality. In the next Proposition we will describe the invariant sets, all fixed points and we give some Lyapunov function.

In the next Theorem we give the description of the set of limit points of the trajectories.