arXiv subjects

Artem S. Novozhilov

Publications and source records attributed to Artem S. Novozhilov.

At least 19 recordsLinked to original sources

arXiv2025

A mathematical framework of consumer-resource dynamics: How to incorporate interactions between interactions in evolutionary process

A novel mathematical framework is proposed to describe the ecological and evolutionary consequences of consumer--resource interactions. Both the consumer and resource are assumed to consist of several (sub)species, which interact between themselves in addition to incorporating the deleterious effects of the consumer on the resource. Separating the ecological and evolutionary time scales, we allow our mathematical model to evolve, with the evolutionary steps chosen according to the (divergent) objective functions of the consumer and the resource. Numerical simulations show that the model, along with the expected outcomes of either consumer or resource winning the evolutionary battle, is capable of producing also the (quasi)stationary state of consumer--resource coexistence with monotone growth of both the consumer and resource fitnesses. Such stable states highlight the importance of the intra-population interactions, which, despite the opposite evolutionary goals of the consumer and the resource, lead to long term ecological stability.

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arXiv2022

On a hypercycle equation with infinitely many members

A hypercycle equation with infinitely many types of macromolecules is formulated and studied both analytically and numerically. The resulting model is given by an integro-differential equation of the mixed type. Sufficient conditions for the existence, uniqueness, and non-negativity of solutions are formulated and proved. Analytical evidence is provided for the existence of non-uniform (with respect to the second variable) steady states. Finally, numerical simulations strongly indicate the existence of a stable nonlinear wave in the form of the wave train.

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arXiv2021

Food webs and the principle of evolutionary adaptation

A principle of evolutionary adaptation is applied to the Lotka--Volterra models, in particular to the food webs. We present a relatively simple computational algorithm of optimization with respect to a given criterion. This algorithm boils down to a sequence of easy to solve linear programming problems. As a criterion for the optimization we use the total weighted population size of the given community and an ecological fitness, which is an analogue of the potential energy in physics. We show by computational experiments that it is almost always possible to substantially increase the total weighed population size for an especially simple food web -- food chain; we also show that food chains are evolutionary unstable under the given optimization criteria and, if allowed, evolve into more complicated structures of food webs.

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arXiv2019

Open Quasispecies Models: Stability, Optimization, and Distributed Extension

We suggest a natural approach that leads to a modification of classical quasispecies models and incorporates the possibility of population extinction in addition to growth. The resulting modified models are called open. Their essential properties, regarding in particular equilibrium behavior, are investigated both analytically and numerically. The hallmarks of the quasispecies dynamics, viz. the heterogeneous quasispecies distribution itself and the error threshold phenomenon, can be observed in our models, along with extinction. In order to demonstrate the flexibility of the introduced framework, we study the inverse problem of fitness allocation under the biologically motivated criterion of steady-state fitness maximization. Having in mind the complexity of numerical investigation of high-dimensional quasispecies problems and the fact that the actual number of genotypes or alleles involved in a studied process can be extremely large, we also build continuous-time distributed open quasispecies models. The obtained results may serve as an initial step to developing mathematical models that involve directed therapy against various pathogens.

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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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arXiv2017

Rigorous mathematical analysis of the quasispecies model: From Manfred Eigen to the recent developments

We review the major progress in the rigorous analysis of the classical quasispecies model that usually comes in two related but different forms: the Eigen model and the Crow--Kimura model. The model itself was formulated almost 50 years ago, and in its stationary form represents an easy to formulate eigenvalue problem. Notwithstanding the simplicity of the problem statement, we still lack full understanding of the behavior of the mean population fitness and the quasispecies distribution for an arbitrary fitness landscape. Our main goal in this review is two-fold: First, to highlight a number of impressive mathematical results, including some of the recent ones, which pertain to the mathematical development of the quasispecies theory. Second, to emphasize that, despite these 50 years of vigorous research, there are still very natural both biological and mathematical questions that remain to be addressed within the quasispecies framework. Our hope is that at least some of the approaches we review in this text can be of help for anyone embarking on further analysis of the quasispecies model.

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arXiv2017

Generalized Quasispecies Model on Finite Metric Spaces: Isometry Groups and Spectral Properties of Evolutionary Matrices

The quasispecies model introduced by Eigen in 1971 has close connections with the isometry group of the space of binary sequences relative to the Hamming distance metric. Generalizing this observation we introduce an abstract quasispecies model on a finite metric space $X$ together with a group of isometries $Γ$ acting transitively on $X$. We show that if the domain of the fitness function has a natural decomposition into the union of $t$ $G$-orbits, $G$ being a subgroup of $Γ$, then the dominant eigenvalue of the evolutionary matrix satisfies an algebraic equation of degree at most $t\cdot {\rm rk}_{\mathbf Z} R$, where $R$ is what we call the orbital ring. The general theory is illustrated by two examples, in both of which $X$ is taken to be the metric space of vertices of a regular polytope with the "edge" metric; namely, the case of a regular $m$-gon and of a hyperoctahedron are considered.

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arXiv2017

Adaptive Fitness Landscape for Replicator Systems: To Maximize or not to Maximize

Sewall Wright's adaptive landscape metaphor penetrates a significant part of evolutionary thinking. Supplemented with Fisher's fundamental theorem of natural selection and Kimura's maximum principle, it provides a unifying and intuitive representation of the evolutionary process under the influence of natural selection as the hill climbing on the surface of mean population fitness. On the other hand, it is also well known that for many more or less realistic mathematical models this picture is a sever misrepresentation of what actually occurs. Therefore, we are faced with two questions. First, it is important to identify the cases in which adaptive landscape metaphor actually holds exactly in the models, that is, to identify the conditions under which system's dynamics coincides with the process of searching for a (local) fitness maximum. Second, even if the mean fitness is not maximized in the process of evolution, it is still important to understand the structure of the mean fitness manifold and see the implications of this structure on the system's dynamics. Using as a basic model the classical replicator equation, in this note we attempt to answer these two questions and illustrate our results with simple well studied systems.

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arXiv2016

On Eigen's quasispecies model, two-valued fitness landscapes, and isometry groups acting on finite metric spaces

A two-valued fitness landscape is introduced for the classical Eigen's quasispecies model. This fitness landscape can be considered as a direct generalization of the so-called single or sharply peaked landscape. A general, non permutation invariant quasispecies model is studied, therefore the dimension of the problem is $2^N\times 2^N$, where $N$ is the sequence length. It is shown that if the fitness function is equal to $w+s$ on a $G$-orbit $A$ and is equal to $w$ elsewhere, then the mean population fitness can be found as the largest root of an algebraic equation of degree at most $N+1$. Here $G$ is an arbitrary isometry group acting on the metric space of sequences of zeroes and ones of the length $N$ with the Hamming distance. An explicit form of this exact algebraic equation is given in terms of the spherical growth function of the $G$-orbit $A$. Sufficient conditions for the so-called error threshold for sequences of orbits are given. Motivated by the analysis of the two-valued fitness landscapes an abstract generalization of Eigen's model is introduced such that the sequences are identified with the points of a finite metric space $X$ together with a group of isometries acting transitively on $X$. In particular, a simplicial analogue of the original quasispecies model is discussed, which can be considered as a mathematical model of the switching of the antigenic variants for some bacteria.

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arXiv2016

Solutions with a bounded support promote permanence of a distributed replicator equation

The now classical replicator equation describes a wide variety of biological phenomena, including those in theoretical genetics, evolutionary game theory, or in the theories of the origin of life. Among other questions, the permanence of the replicator equation is well studied in the local, well-mixed case. Inasmuch as the spatial heterogeneities are key to understanding the species coexistence at least in some cases, it is important to supplement the classical theory of the non-distributed replicator equation with a spatially explicit framework. One possible approach, motivated by the porous medium equation, is introduced. It is shown that the solutions to the spatially heterogeneous replicator equation may evolve to equilibrium states that have a bounded support, and, moreover, that these solutions are of paramount importance for the overall system permanence, which is shown to be a more commonplace phenomenon for the spatially explicit equation if compared with the local model.

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arXiv2015

A commentary on the mathematical model of the heterogeneous gypsy moth larva population by G. Dwyer and his co-authors

In this commentary I utilize the general methods of the mathematical theory of heterogeneous populations in order to point out an omission in the analysis of the mathematical model in [Dwyer et al (2000), Am Nat, 156(2):105--120], which led to the conclusion in [Elderd et al (2008) Am Nat, 172(6):829--842] that the original model must be replaced with an alternative one because of the new data. I show that more thorough understanding of the underlying model allows twitching the model parameters to account for the observed data.

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arXiv2015

Exact solutions for the selection-mutation equilibrium in the Crow-Kimura evolutionary model

We reformulate the eigenvalue problem for the selection--mutation equilibrium distribution in the case of a haploid asexually reproduced population in the form of an equation for an unknown probability generating function of this distribution. The special form of this equation in the infinite sequence limit allows us to obtain analytically the steady state distributions for a number of particular cases of the fitness landscape. The general approach is illustrated by examples; theoretical findings are compared with numerical calculations.

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arXiv2014

Linear algebra of the permutation invariant Crow-Kimura model of prebiotic evolution

A particular case of the famous quasispecies model - the Crow-Kimura model with a permutation invariant fitness landscape - is investigated. Using the fact that the mutation matrix in the case of a permutation invariant fitness landscape has a special tridiagonal form, a change of the basis is suggested such that in the new coordinates a number of analytical results can be obtained. In particular, using the eigenvectors of the mutation matrix as the new basis, we show that the quasispecies distribution approaches a binomial one and give simple estimates for the speed of convergence. Another consequence of the suggested approach is a parametric solution to the system of equations determining the quasispecies. Using this parametric solution we show that our approach leads to exact asymptotic results in some cases, which are not covered by the existing methods. In particular, we are able to present not only the limit behavior of the leading eigenvalue (mean population fitness), but also the exact formulas for the limit quasispecies eigenvector for special cases. For instance, this eigenvector has a geometric distribution in the case of the classical single peaked fitness landscape. On the biological side, we propose a mathematical definition, based on the closeness of the quasispecies to the binomial distribution, which can be used as an operational definition of the notorious error threshold. Using this definition, we suggest two approximate formulas to estimate the critical mutation rate after which the quasispecies delocalization occurs.

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arXiv2014

Asymptotic behavior of spatially distributed replicator systems

The question of biological stability (permanence) of a replicator reaction-diffusion system is considered. Sufficient conditions of biological stability are found. It is proved that there are situations when biologically unstable non-distributed replicator system becomes biologically stable in the distributed case. Numerical examples illustrate analytical findings. This manuscript is a continuation of arXiv:1308.5631.

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arXiv2014

On the behavior of the leading eigenvalue of Eigen's evolutionary matrices

We study general properties of the leading eigenvalue $\overline{w}(q)$ of Eigen's evolutionary matrices depending on the probability $q$ of faithful reproduction. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for $\overline{w}(q),\overline{w}'(q),\overline{w}''(q)$ for $q=0,0.5,1$ and prove that the absolute minimum of $\overline{w}(q)$, which always exists, belongs to the interval $[0,0.5]$. For the specific case of a single peaked landscape we also find lower and upper bounds on $\overline{w}(q)$, which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. Therefore, the discussions of the error threshold applied to biological systems must take this fact into account.

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arXiv2013

On diffusive stability of Eigen's quasispecies model

Eigen's quasispecies system with explicit space and global regulation is considered. Limit behavior and stability of the system in a functional space under perturbations of a diffusion matrix with nonnegative spectrum are investigated. It is proven that if the diffusion matrix has only positive eigenvalues then the solutions of the distributed system converge to the equilibrium solution of the corresponding local dynamical system. These results imply that the error threshold does not change if the spatial interactions under the principle of global regulation are taken into account.

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arXiv2013

Replicator equations and space

A reaction--diffusion replicator equation is studied. A novel method to apply the principle of global regulation is used to write down the model with explicit spatial structure. Properties of stationary solutions together with their stability are analyzed analytically, and relationships between stability of the rest points of the non-distributed replicator equation and distributed system are shown. A numerical example is given to show that the spatial variable in this particular model promotes the system's permanence.

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arXiv2012

Epidemiological models with parametric heterogeneity: Deterministic theory for closed populations

We present a unified mathematical approach to epidemiological models with parametric heterogeneity, i.e., to the models that describe individuals in the population as having specific parameter (trait) values that vary from one individuals to another. This is a natural framework to model, e.g., heterogeneity in susceptibility or infectivity of individuals. We review, along with the necessary theory, the results obtained using the discussed approach. In particular, we formulate and analyze an SIR model with distributed susceptibility and infectivity, showing that the epidemiological models for closed populations are well suited to the suggested framework. A number of known results from the literature is derived, including the final epidemic size equation for an SIR model with distributed susceptibility. It is proved that the bottom up approach of the theory of heterogeneous populations with parametric heterogeneity allows to infer the population level description, which was previously used without a firm mechanistic basis; in particular, the power law transmission function is shown to be a consequence of the initial gamma distributed susceptibility and infectivity. We discuss how the general theory can be applied to the modeling goals to include the heterogeneous contact population structure and provide analysis of an SI model with heterogeneous contacts. We conclude with a number of open questions and promising directions, where the theory of heterogeneous populations can lead to important simplifications and generalizations.

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