Quadratic Stochastic Operators Research Articles

Dynamics of Quadratic Volterra-Type Stochastic Operators Corresponding to Strange Tournaments

By studying the dynamics of these operators on the simplex, focusing on the presence of an interior fixed point, we investigate the conditions under which the operators exhibit nonergodic behavior. Through rigorous analysis and numerical simulations, we demonstrate that certain parameter regimes lead to nonergodicity, characterized by the convergence of initial distributions to a limited subset of the simplex. Our findings shed light on the intricate dynamics of quadratic stochastic operators with interior fixed points and provide insights into the emergence of nonergodic behavior in complex dynamical systems. Also, the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point defined in a simplex introduces additional complexity to the already intricate dynamics of such systems. In this context, the presence of an interior fixed point within the simplex further complicates the exploration of the state space and convergence properties of the operator. In this paper, we give sufficiency and necessary conditions for the existence of strange tournaments. Also, we prove the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point, defined in a simplex.

A Class of Two-Sex Population Quadratic Stochastic Operators: b-Bistochastic-Volterra

A Class of Two-Sex Population Quadratic Stochastic Operators: b-Bistochastic-Volterra

Non-Regular Quadratic Stochastic Operators with Countable State Space

Non-Regular Quadratic Stochastic Operators with Countable State Space

Applications of Quadratic Stochastic Operators to Nonlinear Consensus Problems

Applications of Quadratic Stochastic Operators to Nonlinear Consensus Problems

Volterra-Type Quadratic Stochastic Operators with a Homogeneous Tournament

Volterra-Type Quadratic Stochastic Operators with a Homogeneous Tournament

On Dynamical Systems of Quadratic Stochastic Operators Constructed for Bisexual Populations

On Dynamical Systems of Quadratic Stochastic Operators Constructed for Bisexual Populations

On the dynamics of non-Volterra quadratic operators corresponding to permutations

The present paper deals to a family of non-Volterra quadratic stochastic operators depending on a parameter and a permutation and their trajectory behaviours. We find all fixed and periodic points for such non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We proved that the ω- limit set of any trajectory of such non-Volterra quadratic stochastic operator consists of either single point or finite points.

Periodic and Regular Trajectories of Non-Volterra Quadratic Stochastic Operators

Periodic and Regular Trajectories of Non-Volterra Quadratic Stochastic Operators

Discrete time model of sexual systems

Small factors are the biggest contributors to the biggest changes in an ecosystem which may lead to its demise. Simple things such as tweaking the humidity or tampering with the temperature can be the downfall of the habitants and the ecosystem itself. These changes can be observed right now with the consistent temperature increase on Earth along with the slew of issues that come with global warming. A small change in a few degrees in temperature can perish an entire food supply and the habitat of many already endangered beings forcing them to either adapt or die. Within such an ecosystem the main reason it withstands the test of time boils down to one simple thing; reproduction. This creates the natural question of asking oneself: How does an ecosystem replenish itself and is able to resist collapsing? To even begin considering this thought, one must realise that the size of the population is the key to gaining a deeper understanding as two constituents that are important from evolutionary theory are survival and reproduction. The case of asexual reproducers, it is not difficult to see any alterations that can be introduced to increase reproductivity. Meanwhile, for sexual reproducers, the evolution to increase reproductivity can be observed by looking into the generational data of the species. A certain group of time-based dynamic systems that are connected to a sexual system are the point of contention. The suggested model is a dynamic representation of a hermaphrodite population which is described through quadratic stochastic operators. The key findings offer fresh insights into the future of hermaphrodite populations, that is perhaps a probable solution to prevent the decline of endangered or at-risk species. This demonstrates a fresh perspective on reproduction, which is explored through a purely mathematical approach.

A Convex Combination of Two Quadratic Stochastic Operators Acting in the 2D Simplex

A Convex Combination of Two Quadratic Stochastic Operators Acting in the 2D Simplex

Genetic Algebras Associated with ξ(a)-Quadratic Stochastic Operators.

The present paper deals with a class of ξ(a)-quadratic stochastic operators, referred to as QSOs, on a two-dimensional simplex. It investigates the algebraic properties of the genetic algebras associated with ξ(a)-QSOs. Namely, the associativity, characters and derivations of genetic algebras are studied. Moreover, the dynamics of these operators are also explored. Specifically, we focus on a particular partition that results in nine classes, which are further reduced to three nonconjugate classes. Each class gives rise to a genetic algebra denoted as Ai, and it is shown that these algebras are isomorphic. The investigation then delves into analyzing various algebraic properties within these genetic algebras, such as associativity, characters, and derivations. The conditions for associativity and character behavior are provided. Furthermore, a comprehensive analysis of the dynamic behavior of these operators is conducted.

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-PartitionSiti Nurlaili Karima, Nur Zatul Akmar Hamzahb, Nasir Ganikhodajevc, Muhammad Azrin Ahmadd, Norarida Abd RhanieaFaculty of Information and Communication Technology, Universiti Tunku Abdul Rahman, Kampar, Perak, Malaysia; bDepartment of Computational and Theoretical Sciences, International Islamic University Malaysia, Kuantan, Pahang, Malaysia; cInstitute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan; dCentre for Mathematical Sciences, Universiti Malaysia Pahang, Gambang, Pahang, Malaysia; eDepartment of Mathematics, Universiti Teknologi MARA, Cawangan Kelantan, Machang, Kelantan, Malaysia.AbstractThe theory of quadratic stochastic operator (QSO) has been significantly developed since it was introduced in 1920s by Bernstein on population genetics. Over the century, many researchers have studied the behavior of such nonlinear operators by considering different classes of QSO on finite and infinite state spaces. However, all these studies do not comprehensively represent the core prob-lem of QSO; i.e., the trajectory behavior. Recently, a class of QSO called Lebesgue QSO has been introduced and studied. Such an operator got its name based on Lebesgue measure which serves as a probability measure of the QSO. The conditions of the Lebesgue QSO have allowed us to con-sider the possibility of introducing a new measure for such QSO. This research presents a new class of Lebesgue QSO with nonnegative integers parameters generated by a measurable 2-partition on the continual state space X=[,]01. This research aims to study the trajectory behavior of the QSO by reducing its infinite variables into a mapping of one-dimensional simplex. The behavior of such operators will be investigated computationally and analytically, where the computational results conform to the analytical results. We will apply measure and probability theory as well as functional Email addresses:nurlaili@utar.edu.my (Siti Nurlaili Karim), zatulakmar@iium.edu.my (Nur Zatul Akmar Hamzah), n.ganikhodjaev@mathinst.uz (Nasir Ganikhodjaev), azrin@ump.edu.my (Muhammad Azrin Ahmad), norarida@uitm.edu.my(Norarida Abd Rhani)

Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-PartitionSiti Nurlaili Karima, Nur Zatul Akmar Hamzahb, Nasir Ganikhodajevc, Muhammad Azrin Ahmadd, Norarida Abd RhanieaFaculty of Information and Communication Technology, Universiti Tunku Abdul Rahman, Kampar, Perak, Malaysia; bDepartment of Computational and Theoretical Sciences, International Islamic University Malaysia, Kuantan, Pahang, Malaysia; cInstitute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan; dCentre for Mathematical Sciences, Universiti Malaysia Pahang, Gambang, Pahang, Malaysia; eDepartment of Mathematics, Universiti Teknologi MARA, Cawangan Kelantan, Machang, Kelantan, Malaysia.AbstractThe theory of quadratic stochastic operator (QSO) has been significantly developed since it was introduced in 1920s by Bernstein on population genetics. Over the century, many researchers have studied the behavior of such nonlinear operators by considering different classes of QSO on finite and infinite state spaces. However, all these studies do not comprehensively represent the core prob-lem of QSO; i.e., the trajectory behavior. Recently, a class of QSO called Lebesgue QSO has been introduced and studied. Such an operator got its name based on Lebesgue measure which serves as a probability measure of the QSO. The conditions of the Lebesgue QSO have allowed us to con-sider the possibility of introducing a new measure for such QSO. This research presents a new class of Lebesgue QSO with nonnegative integers parameters generated by a measurable 2-partition on the continual state space X=[,]01. This research aims to study the trajectory behavior of the QSO by reducing its infinite variables into a mapping of one-dimensional simplex. The behavior of such operators will be investigated computationally and analytically, where the computational results conform to the analytical results. We will apply measure and probability theory as well as functional Email addresses:nurlaili@utar.edu.my (Siti Nurlaili Karim), zatulakmar@iium.edu.my (Nur Zatul Akmar Hamzah), n.ganikhodjaev@mathinst.uz (Nasir Ganikhodjaev), azrin@ump.edu.my (Muhammad Azrin Ahmad), norarida@uitm.edu.my(Norarida Abd Rhani)

Genetic Algebras generated by b−bistochastic Quadratic Stochastic Operators: The Character and Associativity

In the present paper we consider a genetic algebra induced by b−bistochastic Quadratic Stochastic Operators (QSOs) which is called b−bistochastic genetic algebra. First, we characterize their nontrivial character function on Rn. It turns out that, the given character function is not unique, hence full descriptions of such functions on R1 and R2 are established. Moreover, the defined algebra is commutative but not associative in general, hence, the associativity of b−bistochastic genetic algebras defined on R1 and R2 are described. In this work, the existence of non-trivial derivations on such algebras are given.

ON DISCRETE-TIME MODELS OF NETWORK WORM PROPAGATION GENERATED BY QUADRATIC OPERATORS

In this paper we consider the discrete-time dynamical systems generated by network worm propagation models based on the theory of quadratic stochastic operators(QSO). This approach simultaneously solves two important problems: exploring of the QSO trajectory‘s behavior, we described the set of limit points, thereby completely solved the main problem of dynamical systems (i), we showed a new application of the theory QSOs in worm propagation modelling (ii). We demonstrated that proposed discrete-time biologically-inspired model represents also realistic picture of the worm propagation process and such analytical models can be used in decision of some problems of computer networks.

A class of bijective Lotka–Volterra operators and its application

It is well known that any classical Lotka–Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

On the Dynamics of Geometric Quadratic Stochastic Operator Generated by 2-Partition on Countable State Space

A quadratic stochastic operator (QSO) is frequently acknowledged as the analysis source to investigate dynamical properties and modeling in numerous areas. Countless classes of QSO have been investigated since the operator was introduced in 1920s. The study of QSOs is still an open problem in the nonlinear operator theory field, especially QSOs on infinite state space. We are interested in the dynamics of Geometric QSO generated by a 2-partition defined on countable state space. We first show the system of equations formed from defined Geometric QSO with infinite-dimensional space can be simplified into a one-dimensional setting, corresponding to the number of defined partitions. The trajectory behavior of such a system is investigated by using functional analysis approach, where the operator either converges to a unique fixed point or has a second-order cycle. It is shown that such an operator can be either regular or nonregular for arbitrary initial points depending on the value of parameters. In this research, we present two cases, i.e., two different parameters and three different parameters. We display the form of the fixed point and periodic points of period-2. Moreover, an example for the nonregular transformation will be provided, where such QSO has 2-periodic points.

Asymptotical Behavior of Trajectories of Non-Volterra Quadratic Stochastic Operators

Asymptotical Behavior of Trajectories of Non-Volterra Quadratic Stochastic Operators

Discrete time dynamics of a SIRD reinfection model

This paper deals with a discrete-time dynamical system generated by a modified susceptible–infected–recovered–dead model (SIRD model; nonlinear operator) in three-dimensional simplex. We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator (QSO) that allows for real-time forecasting. The basic reproductive number [Formula: see text] is obtained. We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic. Further, we study the asymptotical behavior of the trajectories of this system and show that SIRD operators have a regularity property.

The dynamics of superposition of non-Volterra quadratic stochastic operators

In the present paper, we study dynamical systems generated by stochastic operators which are superpositions of non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. We showed that such stochastic operators have two fixed points and there are no periodic points except fixed points. We showed that any trajectory of such an operator converges to a fixed point. Thus, these operators are ergodic transformations.