Leslie’s prey–predator model in discrete time

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.

Ocean ecosystem discrete time dynamics generated by ℓ-Volterra operators

We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters [Formula: see text]) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a [Formula: see text]-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if [Formula: see text], then (under some conditions on [Formula: see text]) this [Formula: see text]-Volterra operator may have up to three or a countable set of fixed points; if [Formula: see text], 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.

Tree-hierarchy of DNA and distribution of Holliday junctions

We define a DNA as a sequence of [Formula: see text]'s and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which 'lives' on the same Cayley tree. This hierarchy has property that each vertex of the Cayley tree belongs only to one of DNA. Then we give a model (energy, Hamiltonian) of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) on a set of admissible configurations. To study thermodynamic properties of the model of DNAs we describe corresponding translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two. We show that there is a critical temperature [Formula: see text] such that (i) if temperature [Formula: see text] then there exists unique TIGM; (ii) if [Formula: see text] then there are two TIGMs; (iii) if [Formula: see text] then there are three TIGMs. Each such measure describes a phase of the set of DNAs. We use these results to study distributions of Holliday junctions and branches of DNAs. In case of very high and very low temperatures we give stationary distributions and typical configurations of the Holliday junctions.