arXiv subjects

Yuri Kozitsky

Publications and source records attributed to Yuri Kozitsky.

At least 37 records · Page 2Linked to original sources

arXiv2019

Stochastic branching at the edge: Individual-based modeling of tumor cell proliferation

An individual-based model of stochastic branching is proposed and studied, in which point particles drift in $\bar{\mathds{R}}_{+}:=[0,+\infty)$ towards the origin (edge) with unit speed, where each of them splits into two particles that instantly appear in $\bar{\mathds{R}}_{+}$ at random positions. During their drift the particles are subject to a random disappearance (death). The model is intended to capture the main features of the proliferation of tumor cells, in which trait $x\in \bar{\mathds{R}}_{+}$ of a given cell is time to its division and the death is caused by therapeutic factors. The main result of the paper is proving the existence of an honest evolution of this kind and finding a condition that involves the death rate and cell cycle distribution parameters, under which the mean size of the population remains bounded in time.

math.DS↗

arXiv2018

Spatially inhomogeneous population dynamics: beyond the mean field approximation

We propose a novel method for numerical modeling of spatially inhomogeneous moment dynamics of populations with nonlocal dispersal and competition in continuous space. It is based on analytically solvable decompositions of the time evolution operator for a coupled set of master equations. This has allowed us -- for the first time in the literature -- to perform moment dynamics simulations of spatially inhomogeneous systems beyond the mean-field approach and to calculate the inhomogeneous pair correlation function using the Kirkwood superposition ansatz. As a result, we revealed a number of new subtle effects, possible in real populations. Namely, for systems with short-range dispersal and mid-range competition, strong clustering of entities at small distances followed by their deep disaggregation at larger separations are observed in the wavefront of density propagation. For populations in which the competition range is much shorter than that of dispersal, the pair correlation function exhibits a long-tail behavior. Remarkably, the latter effect takes place only due to the spatial inhomogeneity and thus was completely unknown before. Moreover, both effects get stronger in the direction of propagation. All these types of behavior are interpreted as a trade-off between the dispersal and competition in the coexistence of reproductive pair correlations and the inhomogeneity of the density of the system.

q-bio.PE↗

arXiv2018

Evolution of infinite populations of immigrants: micro- and mesoscopic description

A model is proposed of an infinite population of entities immigrating to a noncompact habitat, in which the newcomers are repelled by the already existing population. The evolution of such a population is described at micro- and mesoscopic levels. The microscopic states are probability measures on the corresponding configuration space. States of populations without interactions are Poisson measures, fully characterized by their densities. The evolution of micro-states is Markovian and obtained from the Kolmogorov equation with the use of correlation functions. The mesoscopic description is made by a kinetic equation for the densities. We show that the micro-states are approximated by the Poissonian states characterized by the densities obtained from the kinetic equation. Both micro- and mesoscopic descriptions are performed and their interrelations are analyzed, that includes also discussing the problem of the appearance of a spatial diversity in such populations.

math.DS↗

arXiv2018

Random jumps and coalescence in the continuum: evolution of states of an infinite system

The dynamics of an infinite continuum system of randomly jumping and coalescing point particles is studied. The states of the system are probability measures on the corresponding configuration space $Γ$ the evolution of which is constructed in the following way. The evolution of observables $F_0\to F_t$ is obtained from a Kolmogorov-type evolution equation. Then the evolution of states $μ_0\to μ_t$ is defined by the relation $μ_0(F_t) =μ_t(F_0)$ for $F_0$ belonging to a measure-defining class of functions. The main result of the paper is the proof of the existence of the evolution of this type for a bounded time horizon.

math.DS↗

arXiv2018

Self-Regulation in Infinite Populations with Fission-Death Dynamics

The evolution of an infinite population of interacting point entities placed in $\mathbb{R}^d$ is studied. The elementary evolutionary acts are death of an entity with rate that includes a competition term and independent fission into two entities. The population states are probability measures on the corresponding configuration space and the result is the construction of the evolution of states in the class of sub-Poissonian measures, that corresponds to the lack of clusters in such states. This is considered as a self-regulation in the population due to competition.

math.DS↗

arXiv2018

Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals

The basic elements of the mathematical theory of states of thermal equilibrium of infinite systems of quantum anharmonic oscillators (quantum crystals) are outlined. The main concept of this theory is to describe the states of finite portions of the whole system (local states) in terms of stochastically positive KMS systems and path measures. The global states are constructed as Gibbs path measures satisfying the corresponding DLR equation. The multiplicity of such measures is then treated as the existence of phase transitions. This effect can be established by analyzing the properties of the Matsubara functions corresponding to the global states. The equilibrium dynamics of finite subsystems can also be described by means of these functions. Then three basic results of this theory are presented and discussed: (a) a sufficient condition for a phase transition to occur at some temperature; (b) a sufficient condition for the suppression of phase transitions at all temperatures (quantum stabilization); (c) a statement showing how the phase transition can affect the local equilibrium dynamics.

math-ph↗

arXiv2018

Evolution of states of an infinite fission-death system

The evolution of an infinite system of interacting point entities with traits $x\in \mathds{R}^d$ is studied. The elementary acts of the evolution are state-dependent death of an entity with rate that includes a competition term and independent fission in the course of which an entity gives birth to two new entities and simultaneously disappears. The states of the system are probability measures on the corresponding configuration space and the main result of the paper is the construction of the evolution $μ_0\to μ_t$, $t>0$, of states in the class of sub-Poissonian measures.

math.DS↗

arXiv2018

A Curie-Weiss Theory of the Continuum Widom-Rowlinson Model

A version of the continuum Widom-Rowlinson model is introduced and studied. It is a two-component gas of point particles placed in $\mathbf{R}^d$ in which like particles do not interact and unlike particles contained in a given vessel of volume $V$ repel each other with intensity $a/V$. This model is thermodynamically equivalent to a one-component gas with multi-particle interaction. For both models, a rigorous theory of a phase transition is presented and the ways of its construction in the framework of the grand canonical formalism are outlined.

math-ph↗

arXiv2018

A Phase Transition in a Widom-Rowlinson Model with Curie-Weiss Interaction

An analog of the continuum Widom-Rowlinson model is introduced and studied. Its two-component version is a gas of point particles of types 0 and 1 placed in $\mathds{R}^d$, in which like particles do not interact and unlike particles contained in a vessel of volume $V$ repel each other with intensity $a/V$. The one-component version is a gas of particles with multi-particle interactions of Curie-Weiss type. Its thermodynamic behavior is obtained by integrating out the coordinates of one of the components of the two-component version. In the grand canonical setting, a rigorous theory of phase transitions in this model is developed and discussed. In particular, for both versions thermodynamic phases and phase diagrams are explicitly constructed and the equations of state are obtained and analyzed.

math-ph↗

arXiv2017

Self-regulation in the Bolker-Pacala model

The Markov dynamics is studied of an infinite system of point entities placed in $\mathds{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are continuous and integrable we show that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate.

math.DS↗

arXiv2017

Self-regulation in continuum population models

We study the Markov dynamics of an infinite birth-and-death system of point entities placed in $\mathbb{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are just continuous and integrable we prove that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate $m\geq 0$.

math.DS↗

arXiv2016

Evolution of States of a Continuum Jump Model with Attraction

We study a model of an infinite system of point particles in $\mathds{R}^d$ performing random jumps with attraction. The system's states are probability measures on the space of particle configurations, and their evolution is described by means of Kolmogorov and Fokker-Planck equations. Instead of solving these equations directly we deal with correlation functions evolving according to a hierarchical chain of differential equations, derived from the Kolmogorov equation. Under quite natural conditions imposed on the jump kernels -- and analyzed in the paper -- we prove that this chain has a unique classical sub-Poissonian solution on a bounded time interval. This gives a partial answer to the question whether the sub-Poissonicity is consistent with any kind of attraction. We also discuss possibilities to get a complete answer to this question.

math-ph↗

arXiv2016

Evolution of states in a continuum migration model

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $\mathds{R}^d$ in which the constituents appear (immigrate) with rate $b(x)$ and disappear, also due to competition. For this model, we prove the existence of the evolution of states $μ_0 \mapsto μ_t$ such that the moments $μ_t(N_Λ^n)$, $n\in \mathds{N}$, of the number of entities in compact $Λ\subset \mathds{R}^d$ remain bounded for all $t>0$. Under an additional condition, we prove that the density of entities and the second correlation function remain bounded globally in time.

math.DS↗

arXiv2016

A Widom-Rowlinson Jump Dynamics in the Continuum

We study the dynamics of an infinite system of point particles of two types. They perform random jumps in $\mathbf{R}^d$ in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space, the global (in time) evolution of which is constructed by means of correlation functions. It is proved that for each initial sub-Poissonian state $μ_0$, the states evolve $μ_0 \mapsto μ_t$ in such a way that $μ_t$ is sub-Poissonian for all $t>0$. The mesoscopic (approximate) description of the evolution of states is also given. The stability of translation invariant stationary states is studied. In particular, we show that some of such states can be unstable with respect to space-dependent perturbations.

math.DS↗

arXiv2016

The Global Evolution of States of a Continuum Kawasaki Model with Repulsion

An infinite system of point particles performing random jumps in $\mathds{R}^d$ with repulsion is studied. The states of the system are probability measures on the space of particle's configurations. The result of the paper is the construction of the global in time evolution of states with the help of the corresponding correlation functions. It is proved that for each initial sub-Poissonian state $μ_0$, the constructed evolution $μ_0 \mapsto μ_t$ preserves this property. That is, $μ_t$ is sub-Poissonian for all $t>0$.

math-ph↗

arXiv2015

The Microscopic Dynamics of a Spatial Ecological Model

The evolution of states of a spatial ecological model is studied. The model describes an infinite population of point entities placed in $\mathbb{R}^d$ which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system's states are probability measures on the space of configurations of entities, and their evolution is described by means of a hierarchical chain of equations for the corresponding correlation functions derived from the Fokker-Planck equation for measures. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that each correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are also described.

math.DS↗

arXiv2015

Phase Transitions in Continuum Ferromagnets with Unbounded Spins

States of thermal equilibrium of an infinite system of interacting particles in a Euclidean space are studied. The particles bear 'unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. It is proved that the set of such measures contains at least two elements if the activity is big enough.

math-ph↗

arXiv2015

Hierarchical Random Graphs Based on Motifs

Network motifs are characteristic patterns which occur in the networks essentially more frequently than the other patterns. For five motifs found in S. Itzkovitz, U. Alon, Phys. Rev.~E, 2005, 71, 026117-1, hierarchical random graphs are proposed in which the motifs appear at each hierarchical level. A rigorous construction of such graphs is given and a number of their structural properties are analyzed. This includes degree distribution, amenability, clustering, and the small world property. For one of the motifs, annealed phase transitions in the Ising model based on the corresponding graph are also studied.

math-ph↗