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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 3, pp. 297-302. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230701
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 37N25, 92D10
Evolutionary Behavior in a Two-Locus System
In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping a 3-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a two-locus system. We find the set of all (a continuum set of) fixed points and show that each fixed point is nonhyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point.
Keywords: loci, gamete, dynamical system, fixed point, trajectory, limit point
1. Introduction
In this paper, following [3, p. 68], we define an evolution operator of a population assuming viability selection, random mating and discrete non-overlapping generations. Consider two loci A (with alleles A1, A2) and B (with alleles B1, B2). Then we have four gametes: A1B1, A1B2, A2B1, and A2B2. Denote the frequencies of these gametes by x, y, u, and v, respectively. Thus, the vector (x, y, u, v) can be considered as a state of the system, and therefore, one takes it as
Received December 25, 2022 Accepted June 30, 2023
The work was partially supported by a grant from the IMU-CDC. Rozikov thanks Institut des Hautes Études Scientifiques (IHES), Bures-sur-Yvette, France, for supporting his visit to IHES.
Abduqahhor M. Diyorov dabduqahhor@mail.ru
The Samarkand branch of Tashkent University of Information Technologies st. Ibn Sino 2A, Samarkand, 140100 Uzbekistan
Utkir A. Rozikov rozikovu@yandex.ru
V. I. Romanovskiy Institute of Mathematics University st. 9, Tashkent, 100174 Uzbekistan Central Asian University
st. Milliy Bog 264, Tashkent, 111221 Uzbekistan National University of Uzbekistan University st. 4, Tashkent, 100174 Uzbekistan
A.M. Diyorov, U.A. Rozikov
a probability distribution on the set of gametes, i. e., as an element of a 3-dimensional simplex, S3. Recall that an (m — 1)-dimensional simplex is defined as
Sm-1 = j x = (xt, ...,xm) e Rm: x, ^ 0, j Xi = ^.
Following [3, Section 2.10], we define the frequencies (x', y', u', v') in the next generation as
' x' = x + a ■ (yu — xv),
y' = y — a ■ (yu — xv), W :)yywj> (1.1)
u = u — b ■ (yu — xv), k v' = v + b ■ (yu — xv),
where a, b e [0, 1].
It is easy to see that this quadratic operator, W, maps S3 to itself. Indeed, we have x' + + y' + u' + v' = 1 and each coordinate is nonnegative, for example, we check it for y':
y' = y — a ■ (yu — xv) = y(1 — au) + axv ^ y(1 — au) ^ 0,
these inequalities follow from the conditions that x, y, u, v, a e [0, 1] and, therefore, we have 0 ^ au ^ 1.
The operator (1.1), for any initial point (state) t0 = (x0, y0, u0, v0) e S3, defines its trajectory {tn = (xn, yn, un, vn)}'^)=o as
tn = (x^ y^ ^ vn) = W'n(to), n = 0 1 2, ...
Here Wn is the n-fold composition of W with itself:
Wnt) = W(W(W...(W(■)))...).
s-v-'
n times
One of the main problems in the theory of dynamical systems (see [2]) is to study the sequence {tn}'^)=0 for each initial point t0 e S3.
In general, if a dynamical system is generated by a nonlinear operator, then a complete solution of the main problem may be very difficult. But in this short note we will completely solve this main problem for the nonlinear operator (1.1).
Remark 1. Using 1 = x + y + u + v (on S3), one can rewrite the operator (1.1) as
x' = x2 + xy + xu + (1 — a)xv + ayu, y' = xy + y2 + (1 — a)yu + axv + yv, u' = xu + (1 — b)yu + u2 + bxv + uv, v = (1 — b)xv + yv + byu + uv + v2.
W:
Note that the operator (1.2) is in the form of a quadratic stochastic operator (QSO), i.e., V: Sm 1 ^ ^ Sm-1 defined by
m
V : xk =53 xx"
where Pij,k > 0, £ Pij,k = 1.
In general, the operators have received little attention in the literature, but a large class of QSOs has been investigated (see, for example, [1, 4-7] and references therein). But the operator (1.1) has not been studied yet.
2. The set of limit points
Remark 2. The case a = b = 0 is very trivial, so we will not consider this case.
Recall that a point t e S3 is called a fixed point for W: S3 — S3 if W(t) = t. Denote the set of all fixed points by Fix(W).
It is easy to see that for any a, b e [0, 1], a + b = 0 the set of all fixed points of (1.1) is
Fix(W) = {t = (x, y, u, v) e S3 : yu - xv = 0}.
This is a continuum set of fixed points.
The main problem is completely solved in the following result:
Theorem 1. For any initial point (x0, y0, u0, v0) e S3 the following assertions hold:
1. If (x0 + y0)(u0 + v0) = 0, then (x0, y0, u0, v0) is a fixed point.
2. If (x0 + y0)(u0 + v0) = 0, then the trajectory has the following limit:
hm (x^ yn ^ vn) = (A(xo, u0)(x0 + Уo), A(Уo, v0)(x0 + Уo),
n^-IX
A(x0, u0)(u0 + v0), A(Уo, v0)(u0 + v0)) e Fix(W),
where
. , . bx + au
A(x, u) =
(u0 + v0)a + (x0 + yQ)b'
Proof. We note that for each a G [0, 1] the following set is invariant:
Xa = {t = (x, y, u, v) G S3 : x + y = a, u + v = 1 — ^,
i.e., W(Xa) C Xa. Note also that
S3 = U xa.
a£[0,l]
Part 1 of the theorem follows in the case a = 0 and a = 1. Indeed, for a = 0 we have
X0 = {t = (0, 0, u, v) G S3 : u + v = 1},
and in the case of a = 1 we get
X1 = {t = (x, y, 0, 0) G S3 : x + y = 1}.
Note that in both cases the restriction of the operator on the corresponding set is an id-operator, i.e., all points of the set are fixed points.
Now to prove part 2, we consider the case a G (0, 1).
Since Xa is an invariant, it suffices to study limit points of the operator on sets Xa, for each a G (0, 1) separately. To do this, we reduce operator W on the invariant set Xa (i.e., replace y = a — x, v = 1 — a — u):
(x' = (1 — a + aa)x + aau,
(2.1)
u = (1 — a)bx + (1 — ba)u,
where a, b G [0, 1], a G (0, 1), x G [0, a], u G [0, 1 — a].
It is easy to find the set of all fixed points:
Fix(Wa) = {(x, u) e [0, a] x [0, 1 — a]: (1 — a)x — au = 0}. The operator Wa is a linear operator given by the matrix
(1 — a + aa aa \
. (2.2) (1 — a)b 1 — baj K J
The eigenvalues of the linear operator are
A1 = 1, A2 = 1 — (1 — a) a — ab. (2.3)
For any a, b e [0, 1], a + b = 0, a e (0, 1) we have 0 < (1 — a)a + ab < 1, therefore, 0 < A2 < 1.
By (2.1) we define the trajectory of an initial point (x0, u0) as
(xn+1, un+1) = Ma(xn, un) , n ^
Thus,
(xn, un) = Man(xo, uo)T, n ^ 1. (2.4)
Therefore, we need to find M^. To find it we use a little Cayley-Hamilton theorem1 to obtain the following formula:
\ \n \ \n \n \n
Mn _ A2A1 - A1A2 . J , 2 - A1 . M
lua — \ \ J2 + \ \ mai
A2 — A1 A2 — A1
where I2 is a 2 x 2 unit matrix and A1, \2 are eigenvalues (defined in (2.3)).
By the explicit formula (2.3) we get the following limit:
Ao 1 1 i ab aa \
lim M" = , • I2 - --— • M„ = ----r •
n^™ a A2 — A1 2 A2 — A1 a (1 — a)a + ab \J1 — a)b (1 — a)aj
Using this limit, for any initial point (x0, u0) e [0, a] x [0, 1 — a] we get
bx i au
lim (xn, un) = lim M2{xq, u0f = , • (a, 1 - a) G Fix(TUJ. (2.5)
n^™ n^™ (1 — a) a + ab
By (2.5) we obtain
Lemma 1. For any initial point (x0, y0, u0, v0) e S3 \ (X0 U X1) there exists a e (0, 1) such that (x0, y0, u0, v0) e Xa and the trajectory of this initial point (under operator W, defined in (1.1)) has the following limit:
lim (xn, yn, un, vj = (A(x0, v,0)a, A(y0, v0)a, A(x0, ^)(1 — a), A(y0, v0)(1 — a)) e Fix(W),
n^™
where
bx + au
11LT, U) = — ^
(1 — a) a + ab
In this lemma we note that a = x0 + y0 and 1 — a = u0 + v0, therefore, part 2 of the theorem follows, where the limit point of the trajectory of each initial point is given as a function of the initial point only. The theorem is proved. □
Figure 1 illustrates the theorem.
https://www.freemathhelp.com/forum/threads/formula-for-matrix-raised-to-power-n.55028/ _ RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2023, 19(3), 297 302
i
j
Fig. 1. In S3 the set of fixed points Fix(W) (yellow surface) is shown. For a = 0.3, b = 0.2 five initial points (0.1, 0.2, 0.4, 0.3) (red), (0.4, 0.1, 0.3, 0.2) (green), (0.2, 0.2, 0.4, 0.3) (blue), (0.3, 0.3, 0.3, 0.1) (pink), (0.7, 0.1, 0.1, 0.2) (black) and their trajectories are given. The limit point of each trajectory is a fixed point belonging to the yellow surface
3. Biological interpretations
The results of Theorem 1 have the following biological interpretations: Let t = (xo, y0, u0, vo) G S3 be an initial state (the probability distribution on the set [AB^,, AlB2, A2Bx, A2B2} of gametes). Theorem 1 says that, as a rule, the population tends to an equilibrium state with the passage of time.
Part 1 of Theorem 1 means that, if at an initial time we had only two gametes, then the (initial) state remains unchanged.
Part 2 means that, depending on the initial state, the future of the population is stable: gametes survive with probability
A(xo, uo)(xo + Уo), A(Уo, vo)(xo + Уo), A(xo, uo)(uo + vo), A(Уo, vo)(uo + ^
respectively. From the existence of the limit point of any trajectory and from the explicit form of Fix(^) it follows that
Biologically, this property means [3, p. 69] that the population asymptotically goes to a state of linkage equilibrium with respect to two loci.
Acknowledgments
We thank the referee for carefully reading the manuscript and useful comments. _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2023, 19(3), 297-302_
lim (ynun - xnvn) = 0.
Conflict of interest
The authors declare that they have no conflict of interest.
References
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[6] Rozikov, U.A., Population Dynamics: Algebraic and Probabilistic Approach, Hackensack, N.J.: World Sci. Publ., 2020.
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