ON REGULARITY, TRANSITIVITY, AND ERGODIC PRINCIPLE FOR QUADRATIC STOCHASTIC VOLTERRA OPERATORS MANSOOR SABUROV

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1 ON REGULARITY TRANSITIVITY AND ERGODIC PRINCIPLE FOR QUADRATIC STOCHASTIC VOLTERRA OPERATORS MANSOOR SABUROV Departent of Coputational & Theoretical Sciences Faculty of Science International Islaic University Malaysia P.O. Bo Kuantan Pahang Malaysia E-ail: ABSTRACT In this paper we showed an equivalence of notions of regularity transitivity and Ergodic principle for quadratic stochastic Volterra operators acting on the finite diensional siple. Keywords: Quadratic stochastic Volterra operator; regularity; transitivity; Ergodic principle. Periodic Points Vs Regularity The theory of linear operators has been well studied since the last century. The siplest nonlinear operator is a quadratic operator. A quadratic operator is a priary source for investigations of dynaical properties of population genetics. The proble of studying the behavior of trajectories of quadratic stochastic operators was stated in [11] and the application of quadratic stochastic operators to population genetics was studied in [9]. In [5] it was given a long self-contained eposition of the recent achieveents and open probles in the theory of quadratic stochastic operators. A quadratic stochastic operator arises in population genetics as follows: let us consider a population consisting of species. Let (where 0 i be the probability distribution P() i is the probability of i I 1... ) of species in the initial generation and Pij is the probability that individuals in the i th and j th species interbred to produce an individual ore precisely ij P is the conditional probability P i j that i th and j th species interbred successfully then they produce an individual. We suppose that the population has the odel of free population i.e. there is no difference of se and in any generation the parents i and j are independent i.e. P i j (1) probability distribution can be found by the total probability P( i) P( j). Then the 1... (the state) of the species in the first generation 0 0 ( ) P i j P i j P ij i j i j1 i j1 (1) This eans that the association defines a ap V called the evolution operator. Let us provide the precise definition of the evolution quadratic stochastic operator.

2 1 1 Let S : i 1 i 0 i 1 i1 A apping V : S be an 1 S defined as follows 1 1 ij i j i j1 is said to be a quadratic stochastic operator where diensional siple. V : P 1 (1) ( 1) ( 1 ) Pij Pji 0 S Pij 1 i j 1. 1 The population evolves by starting fro an arbitrary state state (1) V then passing to the (in the net generation ) then to the state () V (1) V V V and so on. Thus states of the population described by the following discrete-tie dynaical syste n n (1) () (3) 3 ( ) V V V... V... The ain proble of the theory of the dynaical syste is to classify states ( n) on the behavior of the trajectory. n 1 It is particularly interesting when the trajectory repeats. In this case we say that a periodic point i.e. that ( ) based is is a periodic point if there is the sallest positive integer such. The nuber is called a period of the periodic point. A fied point is a (1) periodic point of period-1 that is a point such that. If the trajectory of every point converges to soe fied point (the liiting fied point ight be depended on the initial point) then a apping is called regular. It is clear that if a apping is regular then it does not have any order periodic points ecept fied points. It turns out that in one diensional case the converse stateent holds true as well. More precisely one of the fascinating results in one diensional nonlinear dynaical syste is that a apping which aps a copact connected subset of the real line into itself is regular if and only if it does not have any order periodic points (see [1]). The ost incredible result is that a apping which aps a copact connected subset of the real line into itself is regular if and only if it does not have any period- points (see [1]). It is natural to see an analogy of these incredible results in the high diensional case. However in general these results do not hold true in the high diensional case. As a counter eaple we can consider the following quadratic stochastic operator V : S V S ()

3 e3 (001) It is easy to chec that this operator has fied points e1 (100) e (010) and c and it does not have any order periodic points. However the trajectory of any point of the interior int S of the siple ecept c does not converge (see [3-4] [8] [9] [1]). More interestingly the arithetic ean (or the Cesaro ean) of the trajectory of the operator () does not converge (see [] [13]). Surprisingly any order arithetic ean (or any order Cesaro ean) of the trajectory of the operator () does not converge (see [10]). The regularity of the nonlinear operator acting on the high diensional space could not be described in ter of an absence of periodic points. Therefore the study of the regularity is independent of interest. Regularity Transitivity and Ergodic Principle The regularity proble was concerned for the operator (1) in [7]. Naely it was studied the following proble: find the nuber 0 such that Pij i j 1 1 iplies the regularity of quadratic stochastic operators (1). It was shown [7] that if then the operator (1) is regular. The ain proble is to find the sallest positive nuber aong all for fied (if any) such that any quadratic stochastic operator under the condition P i j 1 is regular. One can easily chec that if then ij 1 inf 3 7. If 3 the proble reains open. However the regularity proble was studied intensively for another class of quadratic stochastic operators [3-4] which could not be covered by previous cases. Definition [3-4] An operator (1) is called a quadratic stochastic Volterra operator if Pij 0 i j for any i j 1. Any quadratic stochastic Volterra operator can be written in the following for ( V) : 1 ai i (3) i1 where A ai i 1 is a sew-syetric atri with i 11 a. A nonlinear stochastic Volterra operator was studied in [6].

4 Definition [3-4] A sew-syetric atri A is called transversal if all even order leading (principal) inors are nonzero. A quadratic stochastic Volterra operator (3) is called transversal if the corresponding sew-syetric atri A is transversal. Theore [3-4] The set of all transversal quadratic stochastic Volterra operators is an open everywhere-dense subset of the set of all quadratic stochastic Volterra operators. In the sequel we will only consider transversal quadratic stochastic Volterra operators without entioning transversality. The ain approach to study the dynaics of quadratic stochastic Volterra operator is to construct its fied points chart by ean of tournaents [3-4]. A tournaent is a coplete directed graph. A tournaent is called transitive if it does not contain a cycle of length 3. It is clear that if a sew-syetric atri A corresponding to the Volterra operator (3) is transversal then a 0 i j. Therefore we can construct a tournaent ij corresponding to a transversal Volterra operator (3) as follows [3-4]: a tournaent T consists of vertices and an edge directs for i to j if aij 0 otherwise it directs for j to i. A Volterra operator (3) is called transitive if the corresponding tournaent is transitive. Theore [3-4] If a quadratic stochastic Volterra operator (3) is transitive then it is regular. In this paper we want to prove the converse stateent of this theore. Moreover we describe the regularity of quadratic stochastic Volterra operators in ter of the Ergodic principle. In order to this end we introduce the notion of the Ergodic principle for quadratic stochastic operators. Let us fi a nor define its support as follows in the Euclidean space i1 i supp( ) i : i 0. Definition We say that a quadratic stochastic operator V : S the Ergodic principle on the siple has li n n V V y 0 n 1 S if for any n where V V V V y. For every S we S given by (1) satisfies 1 S with supp( ) supp( y) one ( )... ( )... is n ties copositions of V. n

5 The following theore is the ain result. 1 1 Theore Let V : S S be a quadratic stochastic Volterra operator given by (3). The following stateents are equivalent: i) V is regular; ii) V is transitive; iii) V satisfies the Ergodic principle; iv) One has li V n n1 V 0 n for any 1 S. ACKNOWLEDGMENT The author wishes to epress his gratitude to Professor Rasul Ganihodjaev for suggesting the proble and for any stiulating conversations. REFERENCES [1] W. A. Coppel: The solution of equations by iteration. Proc. Cabridge Philos. Soc. 51 (1955) [] N. N. Ganihodzhaev and D. V. Zanin On a necessary condition for the ergodicity of quadratic operators defined on two-diensional siple Russ. Math. Surv.59 (004) [3] R. N. Ganihodzhaev Quadratic stochastic operators Lyapunov functions and tournaents Acad. Sci. Sb. Math. 76 (1993) [4] R. N. Ganihodzhaev A chart of fied points and Lyapunov functions for a class of discrete dynaical systes Math. Notes 56 (1994) [5] R. N. Ganihodzhaev F. M. Muhaedov U.A. Roziov Quadratic stochastic operators and processes: Results and Open Probles Infinite Diensional Analysis Quantu Probability and Related Topics 14 () (011) [6] R. N. Ganihodzhaev and M. Kh. Saburov A generalized odel of nonlinear Volterra type operators and Lyapunov functions Zhurn. Sib. Federal Univ. Mat.-Fiz. Ser. 1 (008) [7] R. N. Ganihodzhaev and A. T. Sarysaov A siple criterion for regularity of quadratic stochastic operators Dol. Aad. Nau UzSSR 11 (1988) 5 6 in Russian. [8] H. Kesten Quadratic transforations: A odel for population growth I II Adv. Appl. Probab. (1970) 1 8; [9] Yu. I. Lyubich Matheatical Structures in Population Genetics (Springer-Verlag 199). [10] M. Kh. Saburov On ergodic theore for quadratic stochastic operators Dol. Acad. Nau Rep. Uzb. 6 (007) 8 11 in Russian. [11] S. M. Ula Probles in Modern Math. (Wiley 1964). [1] S. S. Vallander On the liit behaviour of iteration sequences of certain quadratic transforations Sov. Math. Dol. 13 (197) [13] M. I. Zaharevich The behavior of trajectories and the ergodic hypothesis for quadratic appings of a siple Russ. Math. Surv. 33 (1978)