On a Family of Non-Volterra Quadratic Operators Acting on a Simplex
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On a Family of Non-Volterra Quadratic Operators Acting on a Simplex Uygun Jamilov1,2 · Manuel Ladra3 Received: 14 January 2020 / Accepted: 16 October 2020 © Springer Nature Switzerland AG 2020
Abstract In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter α and study their trajectory behaviours. We showed that for any α ∈ [0, 1) the trajectories of such operator converge to a fixed point. For α = 1 any trajectory of the operator converges to a periodic trajectory. Keywords Quadratic stochastic operator · Volterra and Non-Volterra operator Mathematics Subject Classification Primary 37N25 · Secondary 92D10
1 Introduction The evolution of a population can be studied by a dynamical system of a quadratic stochastic operator [17]. Such evolution operators frequently arise in many models of mathematical genetics, namely theory of heredity (see e.g. [1,3–13,15,17–19,22,26]). Let E = {1, . . . , m} be a finite set and the set of all probability distributions on E S m−1 = x = (x1 , x2 , . . . , xm ) ∈ Rm : xi ≥ 0, for any i and
m
xi = 1 ,
i=1
B
Uygun Jamilov [email protected]; [email protected] Manuel Ladra [email protected]
1
V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b, University Str., Tashkent, Uzbekistan 100174
2
National University of Uzbekistan, 4, University Str., Tashkent, Uzbekistan 100174
3
University of Santiago de Compostela, 15782 Santiago de Compostela, Spain 0123456789().: V,-vol
95
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U. Jamilov, M. Ladra
the (m − 1)-dimensional simplex. A quadratic stochastic operator (QSO) is a mapping V : S m−1 → S m−1 of the simplex into itself, of the form V (x) = x ∈ S m−1 , where V : xk =
m
pi j,k xi x j , k = 1, . . . , m
(1.1)
i, j=1
and the coefficients pi j,k satisfy pi j,k = p ji,k ≥ 0,
m
pi j,k = 1, i, j, k ∈ E.
(1.2)
k=1
The trajectory {x(n) }, n = 0, 1, 2, . . . , of V for an initial point x(0) ∈ S m−1 is defined by x(n+1) = V x(n) = V n+1 x(0) , n = 0, 1, 2, . . . Denote by ωV x(0) the set of limit points of the trajectory {x(n) }∞ n=0 . The main problem in mathematical population genetics consists of the study of the asymptotical behaviour of the trajectories for a given QSO (see e.g. [17]). In other words, the main task is the description of the set ωV x(0) for any initial point x(0) ∈ S m−1 for a given QSO. This problem is an open problem even in two-dimensional case. A QSO V is called regular if there is the limit limn→∞ V n (x) for any initial x ∈ S m−1 . A QSO V is said to be ergodic if the limit n−1 1 k V (x) n→∞ n
lim
k=0
exists for any x ∈ S m−1 . It is evident that a regular QSO V is ergodic; however, regularity does not follow from the ergodicity. A Volterra QSO is defined by (1.1), (1.2) and with the additional assumption / {i, j}, i, j, k ∈ E. pi j,k = 0 if k ∈
(1.3)
The biological treatment of conditions (1.3) is rather precise: the offspring repeats the genotype of one
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