Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

1 Introduction

Without a doubt, many biological processes can be considered as some nonlinear dynamical systems. From this point of view, the main problem associated with the state of the process in certain time frame is the same as when studying the limit behavior of the trajectories of corresponding dynamical systems. One of the important dynamical systems is the one generated by quadratic stochastic operators (q.s.o. henceforth) on finite-dimensional simplex, which appear in many problems of population genetics [1]. The central problem in this theory is to study the limit behavior of the trajectory(dynamics) of a given initial point. There are many papers that study such operators (see [2] for review). One of the important classes of q.s.o. are Volterra operators [3]. It is proven for this operators that the ω-limit set of any non-fixed initial point from the interior of the simplex belong to the boundary of the simplex. Moreover, ω– limits set (we will definite it later) of any initial point is either a countable set or is a singleton. There are other classes of q.s.o. that are well studied [4–6]. Another interesting class of q.s.o. is the class of doubly stochastic (d.s.q.o. henceforth) ones. It was firstly defined in [3] in terms of majorization. In was later developed in papers [7, 8] and infinite-dimensional case in [9]. These papers basically studied the structure of the class of d.s.q.o., that is, necessary and sufficient conditions for doubly stochasticity and extreme points of the set of doubly stochastic operators. In particular, it is shown that, up to permutation, there are 37 extreme d.s.q.o. on 2D simplex. It is also worth mentioning that d.s.q.o. are very different than Volterra operators since from the classification theorem for d.s.q.o. [8] it follows that the class of d.s.q.o. and Volterra intersect at the identity operator. Nonetheless, the study of the limit behavior (dynamics) of d.s.q.o is still wide open, except for ergodic theorem for these operators [10]. In fact, the paper [2] asks to investigate the above problems as the class of d.s.q.o. Namely, it asks if a given d.s.q.o. in general is regular, i.e. has convergent trajectory for any initial point. We will answer positively to this question under small assumption. This question is interesting because d.s.q.o. have rich applications. One application is that these kinds of maps appear in population genetics problems as being a sub-class of the general q.s.o. In addition, doubly stochastic maps are widely used in economics and statistics. One can see it in [11]. The present paper, therefore, aims to provide a couple of convergence theorems for general d.s.q.o. Importantly, it does not give any clue where the trajectory converges. Therefore, in addition, we classify the dynamics of extreme d.s.q.o. on 2D simplex. A motivation to study extreme d.s.q.o. comes from the fact that any d.s.q.o. can be written as a convex combination of the extremes [7, 8]. Thus we deem we should start with extreme d.s.q.o. Another motivation comes from the paper [12] where general quadratic stochastic operators having coefficients 1 or 0 were classified on 2D simplex. Extreme d.s.q.o.’s have coefficients 1; 1 2 or 0 [7, 8]. So our result will extend the results in [12] in the class of d.s.q.o. on 2D simplex. The paper is organized as follows. In the next section we give some preliminaries concerning majorization and d.s.q.o. In Section 3, we provide a class of Lyapunov functions for d.s.q.o. and provide a general convergence theorem for d.s.q.o. on finite-dimensional simplex. Section 4 focuses on dynamics classifications of extreme d.s.q.o. on 2D simplex. We note that a couple of examples of extreme d.s.q.o. on 2D simplex have been considered in [13].

2 Preliminaries

In this section we provide some definition based on majorization theory and define doubly stochastic operator.

We define the (m – 1)– dimensional simplex as follows.

The set intS^{m – 1} = {x ∈ S^{m – 1} : x_i > 0} is called the (relative) interior of the simplex. The points ek=(0,0,…,   1   ︸k,…,0) are the vertices of the simplex and the scalar vector (1m,1m,…,1m) is the center of the simplex.

A quadratic stochastic operator V : S^m–1 → S^{m – 1} is defined as:

Where the coefficients p_ij,k satisfy the following conditions

If we let A_k = (p_ij,k )_{i, j}, k = 1, m, then the operator can be given in terms of m matrices and we write

where matrices A_i are non-negative and symmetric. For any x = (x₁, x₂, …, x_m) ∈ S^m–1, we define x↓ = (x[i], x[2], …, x[m]) where x[1] ≥ x[2] ≥ ··· ≥ x[m] - nonincreasing rearrangement x. Recall [11, 14] that for two elements x, y of the simplex S^m–1 the element x is majorized by y and written x ≺ y ( or y ≻ x) if the following condition holds

for any k = 1, m – 1. In fact, this definition is referred to as weak majorization [11], the definition of majorization requires ∑i=1mx[i]=∑i=1my[i]. However, since we consider vectors only on the simplex, we may drop this condition.

A matrix P = (p_ij)_{i, j} = 1, m called doubly stochastic (sometimes bistochastic), if

For a doubly stochastic matrix P = (p_ij), if its entries consist of only 0’s and 1’s, then the matrix is a permutation matrix.

A linear map T : S^{m – 1} → S^{m – 1} is said to be T- transform, if T = λI +(1 – λ)P where I is an identity matrix, P is a permutation matrix which is obtained by swapping only two rows of I and 0 ⩽ λ ⩽ 1.

For example, the identity operator, permutation operators (that is the linear operators with permutation matrix), T- transforms are all doubly stochastic. The following is an example of the d.s.q.o.

Let V be doubly stochastic operator and x⁰ ∈ S^m–1. The set

is called the trajectory starting at x⁰. Here, V⁰(x⁰) = x⁰ and Vⁿ(x⁰) = V(V^{n – 1}(x⁰)). We denote by ω(x⁰) the set of limit points of the trajectory starting at x⁰ and it is said to be the ω- limit set of the trajectory starting at x⁰. Notice that the ω- limit set of any point is invariant for V by definition.

The point x⁰ is called p-periodic, if there is a positive integer p such that V^p(x⁰) = x⁰ and Vⁱ(x⁰) ≠ x⁰ ∀i = 1, p – 1. If p = 1, we say that the point is fixed. We just say periodic if the period is irrelevant. By periodic point we always mean a periodic point of a period strictly grater than one.

Notice that the center C=(1m,1m,…,1m) is always fixed for a doubly stochastic V. Indeed, by the definition of the double stochasticity, we have V(C) ≺ C. On the other hand, C ≺ x for any vector x from the simplex.

Therefore, C ≺ V(C) ≺ C implies .V(C)↓ = C↓ = C. Hence, V(C) = C and C is a fixed point. We will use this fact in subsequent sections.

3 The limit behavior of d.s.q.o

In this section we first provide a class of functionals on the simplex that are Lyapunov functions for a d.s.q.o. Then we prove the convergence theorem for d.s.q.o.

A continuous functional given φ : S^m–1 → R is said to be a Lyapunov function for the operator V if the limit lim_{n →∞}φ(Vⁿ(x⁰)) exists along the trajectory {x⁰, V(x⁰), V²(x⁰), …, Vⁿ(x⁰), …}.

The Lyapunov function is considered to be very useful in the study of limit behavior of (discrete) dynamical systems.

Based on the above method one can provide larger class of Lyapunov functions.

An operator V : S^m-1 → S^m–1 is a permutation if there exists a permutation matrix such that Vx = Px ∀x ∈ S^m–1. It is evident that there are exactly m! permutation operators on S^m−1.

Consider V : S² → S² given by

This operator is d.s.q.o. by the classification theorem 2.6 of [8] or one can easily check that Vx ≺ x. One can see that V³(1, 0, 0) = (1, 0, 0), that is, (1, 0, 0) is a 3-periodic point.

Combining the previous two theorems we obtain a very interesting corollary.

Theorem 3.5 fails in higher dimension. Consider the following example in 3D simplex.

4 The classification of extreme points of d.s.q.o. on S²

In this section we give classification of dynamics of extreme d.s.q.o.

In [7, 8] it was shown that the set of all d.s.q.o. form a convex polytope and the extreme points in 2D simplex were fully described in the following way considering a d.s.q.o. V : S² → S² given by V = (A₁|A₂|A₃) where (A₁|A₂|A₃) are non-negative symmetric matrices such that (this follows from the definition of q.s.o.) their sum is a matrix whose all entries equal to 1.

The set of extreme points of a set A we denote by ExtrA.

We recall a couple of facts from [7] and [8].

It is known [7, 8] that d.s.q.o. is extreme A = (aij) ∈ Ext r𝓤₃ if and only if a_ii = 1 or 0, a_ij = 1 or12or 0 andA≠(012121201212120) and there are 25 extreme matrices.

In [7, 8] A d.s.q.o. V = (A₁ |A₂|A₃) is extreme if and only if at least two of matrices A₁, A₂, A₃ are extreme in 𝓤₃.

Using these facts one can find triples (A₁|A₂|A₃) such that V is extreme. There are 37 such triples. For a given extreme d.s.q.o. (A₁|A₂|A₃) if we rearrange matrices A₁, A₂, A₃ we still get extreme d.s.q.o.

Thus, counting permutations, there are 37 * 3! = 222 extreme d.s.q.o.

Now our objective is to classify them in terms of their dynamics. As we mentioned all 222 extreme d.s.q.o. are obtained by permuting component of 37 extreme operators. Thus, to study the dynamics we will be dealing with the same collection of matrices even if we change the components. This implies that it is enough to consider only 37 extreme d.s.q.o. It is worth noting that in general a permuted operator does not have the same dynamics, but it can only be studied as the original operator. For instance, we if consider the identity operator, then all points are fixed. However, if we permute the identity operator (see operator below) then all points are going to be periodic except the center of the simplex. Let’s consider the simplest case - permutation operator given by

The matrix correspondence for this operator is

Note that V³(x, y, z) = (x, y, z) and therefore every initial point is 3-periodic. Thus, it is not difficult to study the dynamics of this operator. We split the remaining 36 operators into two classes: those who have exactly one matrix from (13) in their matrix representation are in class 2 and the rest in class 1.

It is important that operators from the class 2 must have exactly one matrix from (13). Referring to Ganikhodzhaev and Shahidi (2010)’s Theorem 3.4 if an operator has more than 1 matrix from (13) then it becomes a permutation operator.

Because extreme d.s.q.o. are classified in [8] (Theorem 3.4), then it is not difficult to see that in fact the class 2 has 3 (or 18 up to permutation) operators while the class 1 has 33.

We are going to choose one representative from each class and study the limit behavior of the trajectories. We will see that their dynamics is different.

Figure 1shows that the trajectory of initial values of x, y and z for V₁converges to the center(13).

Fig. 1

The trajectory of operator V₁.

We note that the convergence part of this theorem automatically follows from Theorems 3.3 and 3.5 of previous section. However, we will show how one apply Theorem 3.1 of previous section.

Proof. We first show that V₁ has no periodic point on intS². Suppose on the contrary, that intS² is a p-periodic (p > 1) point of V₁. Then by definition

If follows that each V^kx⁰,k = 1, p interchange the components of x⁰. Therefore, if we define φ(x⁰) = x² + y² + z²for x⁰ = (x, y, z) we must have φ(Vx⁰) = φ(x⁰) by Theorem 3.1.

One can easily compute that

hence –zy(x – z)²(x + z) = 0.

Since x, y, z > 0 the above holds only if x =z. Assume x = z = c, 0 < c < 1 then x⁰₌ (c,1 – 2c, c). One can see that V₁(x) = 2c – 3c² so in order for V₁ to permute the components of x⁰ we must have 2c – 3c²₌c or 2c – 3c² = 1 – 2c which only happens if c=13 that is when x⁰ = (x, y, z) = (13,13,13). But (13,13,13) is fixed point and we assumed x⁰ to be periodic (of period greater than one). Thus, as one can recall Theorem 3.3 from the previous section, that the trajectory of any x⁰ ∈ intS² is convergent. Notice that a convergent point is always fixed by V₁.

By solving the system of equations

we find that V₁ has a unique fixed point (13,13,13) on intS². Therefore (13,13,13) must be the convergent point for any x⁰ ∈ intS². □

Fig. 2

The trajectory of operator V₂.

5 Conclusions and future work

In this paper, we discussed an important class of quadratic stochastic operators, which was defined in terms of majorization of vectors, that is, doubly stochastic quadratic operators (d.s.q.o) on finite-dimensional simplex. The theorem of the convergence of d.s.q.o. is proved. We also showed that it has no periodic point is on the interior of the simplex but it may have (in fact infinitely many) periodic points in higher dimensions of the simplex. Moreover, we classified the extreme points of d.s.q.o. on two dimensional 2D simplex. From the analysis, we found three different classes of 222 extreme points of d.s.q.o. on 2D simplex. We found out that 198 extreme d.s.q.o.’s converge to the center of the simplex. In the second class, we found 18 extreme d.s.q.o., which had property as in Theorem 4.5. In the third classification, we had 6 permutation operators which are extreme d.s.q.o. It is very interesting to study dynamics on infinite-dimensional simplex. The problem would be very different if the infinite-dimensional simplex was not compact, the operator would not necessarily have a fixed point and one should choose a proper metric to study the dynamics.