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We consider a hard core (HC) model with a countable set ℤ of spin values on the Cayley tree. This model is defined by a countable set of parameters λ i > 0 , i ∈ ℤ ∖ { 0 }. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model. [ABSTRACT FROM AUTHOR]

Motivated by the China's five element philosophy (CFEP), we construct a permuted Volterra quadratic stochastic operator acting on the four-dimensional simplex. This operator (depending on 10 parameters) is considered as an evolution operator for CFEP. We study the discrete-time dynamical system generated by this operator. Mainly our results related to a symmetric operator (depending on one parameter). We show that this operator has a unique fixed point, which is repeller. Moreover, in the case of non-zero parameter, it has two 5-periodic orbits. We divide the simplex to four subsets: the first set consists a single point (the fixed point); the second (resp. third) set is the set of initial point trajectories of which converge to the first (resp. second) 5-periodic orbit; the fourth subset is the set of initial point trajectories of which do not converge and their sets of limit points are infinite and lie on the boundary of the simplex. We give interpretations of our results to CFEP. [ABSTRACT FROM AUTHOR]

In this paper we study -adic dynamical systems generated by the function in the set of complex -adic numbers. We find an explicit formula for the -fold composition of for any . Using this formula we give fixed points, periodic points, basin of attraction and Siegel disk of each fixed (periodic) point depending on parameters and . [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of a discrete-time dynamical system modeling a trophic network connecting three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator V associated with this dynamical system is of type 4-Volterra quadratic stochastic operator with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this operator. For each situations we give some biological interpretations. [ABSTRACT FROM AUTHOR]

We study the discrete-time dynamical systems associated to a stage-structured wild and sterile mosquito population. We describe all fixed points of the evolution operator (which depends on five parameters) of mosquito population and show that depending on the parameters this operator may have unique, two and infinitely many fixed points. Under some general conditions on parameters we determine type of each fixed point and give the limit points of the dynamical system. Moreover, for a special case of parameters we give full analysis of corresponding dynamical system. We give some biological interpretations of our results. [ABSTRACT FROM AUTHOR]

We consider the Leslie's prey–predator model with discrete time. This model is given by a nonlinear evolution operator depending on five parameters. We show that this operator has two fixed points and define type of each fixed point depending on the parameters. Finding two invariant sets of the evolution operator, we study the dynamical systems generated by the operator on each invariant set. Depending on the parameters, we classify the dynamics between a predator and a prey of the Leslie's model. [ABSTRACT FROM AUTHOR]

In this paper we study dynamical systems generated by an evolution operator of a dioecious population. This evolution operator is a sixparametric, non-linear operator mapping [0, 1]2 to itself. We find all fixed points and under some conditions on parameters we give limit points of trajectories constructed by iterations of the evolution operator. [ABSTRACT FROM AUTHOR]

Recently, continuous-time dynamical systems of mosquito populations have been studied. In this paper, we consider a discrete-time dynamical system, generated by an evolution quadratic operator of a mosquito population, and show that this system has two fixed points, which become saddle points under some conditions on the parameters of the system. We construct an evolution algebra, taking its matrix of structural constants equal to the Jacobian of the quadratic operator at a fixed point. Idempotent and absolute nilpotent elements, simplicity properties, and some limit points of the evolution operator corresponding to the evolution algebra are studied. We give some biological interpretations of our results. [ABSTRACT FROM AUTHOR]

We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters a , b , c , d) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a ℓ -Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if c d (c + d) = 0 , then (under some conditions on a , b) this ℓ -Volterra operator may have up to three or a countable set of fixed points; if c d (c + d) ≠ 0 , then the operator has up to three fixed points. Depending on the parameters, the fixed points may be attracting, repelling or saddle points. The limit behaviors of trajectories of the dynamical system are studied. It is shown that independently on values of parameters and on initial (starting) point, all trajectories converge. Thus, the operator (dynamical system) is regular. We give some biological interpretations of our results. [ABSTRACT FROM AUTHOR]