In  the notion of a quadratic stochastic operator was introduced. Such operators arise in problems of mathematical biology and mathematical genetics [1–7]. Let
be the -dimensional simplex.
A map from into itself is called a quadratic stochastic operator (shortly QSO) if
for any , where
Assume that is the trajectory of the initial point ,where for all , with .
A point is called a fixed point of a QSO if .
A QSO is called regular if for any initial point , the limit exists. 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.
In , Zakharevich proved that this conjecture is false in general. The biological treatment of non-ergodicity of a QSO is the following: in a long run the behavior of the distributions of species is chaotic, i.e. it is unpredictable. Note that a regular QSO is ergodic, but in general from ergodicity does not follow regularity.
Let the set be the interior of and be the set of limit points of the trajectory .
A quadratic stochastic operator is called Volterra if for any .
The biological treatment of such operators is rather clear: the offspring repeats one of its parents. Recall the definition of quasi-Volterra operator following .
Definition . A quadratic stochastic operator (1), (2) is said to be a quasi-Volterra operator if only one coefficient is non-zero when and all others are zero.
All quasi-Volterra QSOs defined on two-dimensional simplex can be divided into two types.
Without lost of generality we may assume then an arbitrary quasi-Volterra operator of first type has the following representation:
Similarly without lost of generality we may assume then an arbitrary quasi-Volterra operator of second type has the following representation:
In  the sets of fixed points of the quasi-Volterra operators (3) and (4) are described and for some special classes of first and second type quasi-Volterra operators it was proven that ergodic hypotheses true.
Let us consider the following quasi-Volterra quadratic stochastic operator
where and .
In general, the main problem of the study of the asymptotic behavior of a quasi-Volterra QSO is also a difficult problem, which remains open, even in the two-dimensional simplex case. Below we consider the case .
It is easy to check that i) the quasi-Volterra QSO (5) is first type and ii) the quasi-Volterra QSO (5) doesn't coincide with quasi-Volterra QSOs which were studied in .
Theorem: Let .
i) If then for any initial point the trajectory of quasi-Volterra QSO (5) converges to vertex ;
ii) If then for any initial point the trajectory of quasi-Volterra QSO (5) converges to vertex ;
iii) If then for any initial point .
Corollary: Assume conditions of items i), ii) of Theorem are satisfied, then a quasi-Volterra QSO (5) is regular operator and it is an ergodic transformation.
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