arXiv subjects

Georgy P. Karev

Publications and source records attributed to Georgy P. Karev.

At least 19 recordsLinked to original sources

arXiv2018

How trait distributions evolve in heterogeneous populations

We consider the problem of determining the time evolution of a trait distribution in a mathematical model of non-uniform populations with parametric heterogeneity. This means that we consider only heterogeneous populations in which heterogeneity is described by an individual specific parameter that differs in general from individual to individual, but does not change with time for the whole lifespan of this individual. Such a restriction allows obtaining a number of simple and yet important analytical results. In particular we show that initial assumptions on time-dependent behavior of various characteristics, such as the mean, variance, of coefficient of variation, restrict severely possible choices for the exact form of the trait distribution. We illustrate our findings by in-depth analysis of the variance evolution. We also reanalyze a well known mathematical model for gypsy moth population showing that the knowledge of how distributions evolve allows producing oscillatory behaviors for highly heterogeneous populations.

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arXiv2016

Mathematical Modeling of Extinction of Inhomogeneous Populations

Mathematical models of population extinction have a variety of applications in such areas as ecology, paleontology and conservation biology. Here we propose and investigate two types of sub-exponential models of population extinction.The results of performed analysis show that the principle of minimum of information loss is the underlying law for the evolution of a broad class of models of population extinction. Finally, we propose a possible application of this modeling framework to mechanisms underlying time perception.

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arXiv2012

The HKV method of solving of replicator equations and models of biological populations and communities

Replicator equations (RE) are among the basic tools in mathematical theory of selection and evolution. The method of Hidden keystone variables (HKV) for reducing a wide class of RE of high or even infinite dimensionality to low-dimensional escort systems of ODEs, which can be explored analytically in many cases was developed. The method has potential for different applications to models of biological populations and communities

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arXiv2009

On mathematical theory of selection: Continuous time population dynamics

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.

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arXiv2008

Pure cross-diffusion models: Implications for traveling wave solutions

An analysis of traveling wave solutions of pure cross-diffusion systems, i.e., systems that lack reaction and self-diffusion terms, is presented. Using the qualitative theory of phase plane analysis the conditions for existence of different types of wave solutions are formulated. In particular, it is shown that family of wave trains is a generic phenomenon in pure cross-diffusion systems. The results can be used for construction and analysis of different mathematical models describing systems with directional movement.

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arXiv2007

Inhomogeneous maps and mathematical theory of selection

In this paper we develop a theory of general selection systems with discrete time and explore the evolution of selection systems, in particular, inhomogeneous populations. We show that the knowledge of the initial distribution of the selection system allows us to determine explicitly the system distribution at the entire time interval. All statistical characteristics of interest, such as mean values of the fitness or any trait can be predicted effectively for indefinite time and these predictions dramatically depend on the initial distribution. The Fisher Fundamental theorem of natural selection (FTNS) and more general the Price equations are the famous results of the mathematical selection theory. We show that the problem of dynamic insufficiency for the Price equations and for the FTNS can be resolved within the framework of selection systems. Effective formulas for solutions of the Price equations and for the FTNS are derived. Applications of the developed theory to some other problems of mathematical biology (dynamics of inhomogeneous logistic and Ricker model, selection in rotifer populations) are also given. Complex behavior of the total population size, the mean fitness (in contrast to the plain FTNS) and other traits is possible for inhomogeneous populations with density-dependent fitness. The temporary dynamics of these quantities can be investigated with the help of suggested methods.

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arXiv2006

Mathematical modeling of tumor therapy with oncolytic viruses: Effects of parametric heterogeneity on cell dynamics

One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity. Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence, tumor dormancy, an others, appear in heterogeneous setting. We also demonstrate that, within the applied modeling framework, to overcome the adverse effect of tumor cell heterogeneity on cancer progression, a heterogeneous population of an oncolytic virus must be used. Heterogeneity in parameters of the model, such as tumor cell susceptibility to virus infection and virus replication rate, can lead to complex, time-dependent behaviors of the tumor. Thus, irregular, quasi-chaotic behavior of the tumor-virus system can be caused not only by random perturbations but also by the heterogeneity of the tumor and the virus. The modeling approach described here reveals the importance of tumor cell and virus heterogeneity for the outcome of cancer therapy. It should be straightforward to apply these techniques to mathematical modeling of other types of anticancer therapy.

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arXiv2006

Analytical Models of Forest Dynamics

The hierarchical system of forest ecosystem models based on the theory of individual-based (structured) models of populations and communities is briefly described. New self-thinning models are integrated with tree stand models within a hierarchical system of models, aimed at assessing (quasi-) stable states of a forest ecosystem under given environment and management. Within the structural model of succession, the ergodic theorems in biology are proved. They assert that the areas of biocenoses in a climax state of association should be proportional to the proper times of their development in a succession line; the other characteristics of biocenoses should be also proportional to their own proper regeneration times in a climax state. A simple theoretical approach to estimating the deviation of the current state of forest ecosystem from its steady state is derived and applied to data from the Prioksko-Terrasny Reserve. The results obtained can serve as a theoretical basis in numerical processing of data on ecological monitoring of undisturbed forest lands.

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arXiv2006

Inhomogeneous maps: the basic theorems and some applications

Non-linear maps can possess various dynamical behaviors varying from stable steady states and cycles to chaotic oscillations. Most models assume that individuals within a given population are identical ignoring the fundamental role of variation. Here we develop a theory of inhomogeneous maps and apply the general approach to modeling heterogeneous populations with discrete evolutionary time step. We show that the behavior of the inhomogeneous maps may possess complex transition regimes, which depends both on the mean and the variance of the initial parameter distribution. The examples of inhomogeneous models are discussed.

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arXiv2006

On Mathematical Theory of Selection: Discrete-Time Models

Mathematical theory of selection systems is developed for a wide class of dynamical models of inhomogeneous populations with discrete time. The Price equation and its particular case, the Fisher Fundamental theorem of natural selection (FTNS), are well known general results of the theory. It is known that the Price equation being a mathematical identity is not dynamically sufficient, i.e., it does not allow one to predict changes in the mean of a trait beyond the immediate response if only the value of covariance of the trait and fitness at this moment is known. We show that the problem of dynamically insufficiency for the Price equations and for the FTNS can be correctly overcome if to study these equations subject to given initial distribution in framework of exact models of the population dynamics. The knowledge of the entire distribution at any given instant allows making the exact prediction for indefinite time and this prediction dramatically depends on the initial distribution. For these models, the current trait distribution and hence all statistical characteristics of interest, such as mean values of the fitness or any trait could be computed effectively at any time moment.

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arXiv2005

Mathematical modeling of anti-tumor virus therapy: Regimes with complete recovery within the framework of deterministic models

A complete parametric analysis of dynamic regimes of a conceptual model of anti-tumor virus therapy is presented. The role and limitations of mass-action kinetics are discussed. A functional response, which is a function of the ratio of uninfected to infected tumor cells, is proposed to describe the spread of the virus infection in the tumor. One of the main mathematical features of ratio-dependent models is that the origin is a complicated equilibrium point whose characteristics crucially determine the main properties of the model. It is shown that, in a certain area of parameter values, the trajectories of the model form a family of homoclinics to the origin (so-called elliptic sector). Biologically, this means that both infected and uninfected tumor cells can be eliminated with time, and complete recovery is possible as a result of the virus therapy within the framework of deterministic models

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arXiv2005

Modeling the dynamics of inhomogeneous natural rotifer populations under toxicant exposure

Most population models assume that individuals within a given population are identical, that is, the fundamental role of variation is ignored. Here we develop a general approach to modeling heterogeneous populations with discrete evolutionary time step. The theory is applied to population dynamics of natural rotifer populations. We show that under particular conditions the behavior of the inhomogeneous model possesses complex transition regimes, which depends both on the mean and the variance of the initial parameter distribution and the final state of the population depends on the least possible value from the domain of the parameter. The question of persistence of the population is discussed.

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arXiv2005

Mathematical modeling of evolution of horizontally transferred genes

We describe a stochastic birth-and-death model of evolution of horizontally transferred genes in microbial populations. The model is a generalization of the stochastic model described by Berg and Kurland and includes five parameters: the rate of mutational inactivation, selection coefficient, immigration rate (i.e., rate of arrival of a novel sequence from outside of the recipient population), within-population horizontal transmission rate, and population size. The model of Berg and Kurland included four parameters, namely, mutational inactivation, selection coefficient, population size, and transmission rate. However, the effect of transmission was disregarded in the interpretation of the results, and the overall conclusion was that horizontally acquired sequences can be fixed in a population only when they confer a substantial selective advantage onto the recipient and therefore are subject to strong positive selection. By contrast, analysis of the present model in different domains of parameter values shows that, as long as the rate of within-population horizontal transmission is comparable to the mutational inactivation rate and there is even a low rate of immigration, horizontally acquired sequences can be fixed in the population or at least persist for a long time even when they are neutral or slightly deleterious. The available biological data strongly suggest that intense within-population and even between-populations gene flows are realistic for at least some prokaryotic species and environments. Therefore our modeling results are compatible with the notion of a pivotal role of horizontal gene transfer in the evolution of prokaryotes.

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arXiv2005

Dynamics of Heterogeneous Populations and Communities and Evolution of Distributions

Most population models assume that individuals within a given population are identical, that is, the fundamental role of variation is ignored. Inhomogeneous models of populations and communities allow for birth and death rates to vary among individuals; recently, theorems of existence and asymptotic of solutions of such models were investigated. Here we develop another approach to modeling heterogeneous populations by reducing the model to the Cauchy problem for a special system of ODEs. As a result, the total population size and current distribution of the vector-parameter can be found in explicit analytical form or computed effectively. The developed approach is extended to the models of inhomogeneous communities.

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arXiv2005

Biological applications of the theory of birth-and-death processes

In this review, we discuss the applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer, and somatic evolution of cancers. We further describe how empirical data, e.g., distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. It is concluded that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological process, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.

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arXiv2005

Modeling genome evolution with a diffusion approximation of a birth-and-death process

In our previous studies, we developed discrete-space Birth, Death and Innovation Models (BDIM) of genome evolution. These models explain the origin of the characteristic Pareto distribution of paralogous gene family sizes in genomes, and model parameters that provide for the evolution of these distributions within a realistic timeframe have been identified. Here we develop the diffusion version of BDIM whose dynamics is described by the Fokker-Plank equation and the stationary solution could be any specified Pareto function. The diffusion models have time-dependent solutions of a special kind, namely, the generalized self-similar solutions, which describe the transition from one stationary distribution of the system to another; this provides for the possibility of examining the temporal dynamics of genome evolution. Analysis of the generalized self-similar solutions of the diffusion BDIM reveals a biphasic curve of genome growth in which the initial, relatively short, self-accelerating phase is followed by a prolonged phase of slow deceleration. In biological terms, this regime of evolution can be tentatively interpreted as a punctuated-equilibrium-like phenomenon such that whereby evolutionary transitions are accompanied by rapid gene amplification and innovation, followed by slow relaxation to a new stationary state.

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arXiv2005

Simple stochastic birth and death models of genome evolution: Was there enough time for us to evolve?

We show that simple stochastic models of genome evolution lead to power law asymptotics of protein domain family size distribution. These models, called Birth, Death and Innovation Models (BDIM), represent a special class of balanced birth-and-death processes, in which domain duplication and deletion rates are asymptotically equal up to the second order. The simplest, linear BDIM shows an excellent fit to the observed distributions of domain family size in diverse prokaryotic and eukaryotic genomes. However, the stochastic version of the linear BDIM explored here predicts that the actual size of large paralogous families is reached on an unrealistically long timescale. We show that introduction of non-linearity, which might be interpreted as interaction of a particular order between individual family members, allows the model to achieve genome evolution rates that are much better compatible with the current estimates of the rates of individual duplication/loss events.

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arXiv2005

Modeling the dynamics of natural rotifer populations: phase-parametric analysis

A model of the dynamics of natural rotifer populations is described as a discrete nonlinear map depending on three parameters, which reflect characteristics of the population and environment. Model dynamics and their change by variation of these parameters were investigated by methods of bifurcation theory. A phase-parametric portrait of the model was constructed and domains of population persistence (stable equilibrium, periodic and a-periodic oscillations of population size) as well as population extinction were identified and investigated. The criteria for population persistence and approaches to determining critical parameter values are described. The results identify parameter values that lead to population extinction under various environmental conditions. They further illustrate that the likelihood of extinction can be substantially increased by small changes in environmental quality, which shifts populations into new dynamical regimes.

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