Artem S Novozhilov

North Dakota State University | NDSU · Department of Mathematics

Food webs and the principle of evolutionary adaptation Alexander S. Bratus a,b, Sergei Drozhzhin a,b, Anastasiia V. Korushkina c, Artem S. Novozhilov d,∗ a Russian University of Transport, Moscow 127994, Russia b Lomonosov Moscow State University, Moscow 119992, Russia c Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State U...

These is a slightly extended version of my lecture notes for an introductory ODE class. The main difference is that the current file contains a sufficient number of exercises for a one semester course and can be used without any supplements.

An updated version of my PDE lecture notes.

Eigen-Shuster replicated system with infinity many members described integra - differential equation with delay and generated nonlinear undamped waves.

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 pro...

Food webs and the principle of evolutionary adaptation Alexander S. Bratus1,2,∗ , Anastasiia V. Korushkina3 , Artem S. Novozhilov4 , † 1Russian University of Transport, Moscow 127994, Russia 2Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow 119992, Russia 3Faculty of Computational Mathematics and Cyber...

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 th...

This chapter focuses on the extremal properties of replicator systems’ fitness landscapes. We discuss a mathematical interpretation of Fisher’s theorem of natural selection and analyze cases that lie beyond its predictions. For general replicator equation, we examine how the location of the maximum depends on the fitness matrix composition. Using a...

These are my lecture notes for an introductory graduate course on ODE. On July 9 2020 I uploaded a somewhat updated version.

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...

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...

This file contains three first chapters of a potential second edition of the book "Dynamical systems and models in biology" by A.S. Bratus, A.S. Novozhilov, and A.P. Platonov, published by Russian publisher FizMatLit in 2010. Please note that the language of the text is Russian.

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,...

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...

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 $\Gamma$ acting transitively on $X$. We...

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,...

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 simplicit...

The standard genetic code (SGC) is virtually universal among extant life forms. Although many deviations from the universal code exist, particularly in organelles and prokaryotes with small genomes, they are limited in scope and obviously secondary. The universality of the code likely results from the combination of a frozen accident, i.e., the del...

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 s...

These are my lecture notes for MATH 329: Intermediate Linear Algebra, which I taught twice during Fall 2016 and Spring 2017. There are two main intertwined goals of this course. First, of course, the student, upon completion of this course, is expected to be proficient with the language and main results of basic linear algebra (such as vector space...

These are my lecture notes for the undergraduate PDE course I taught in Spring 2016.

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 a...

These are lecture notes that I have written for the course in Mathematical Biology, which I taught twice at NDSU.

This is a (unfinished) set of lecture notes I have written for an applied math graduate course that I taught in Summer 2015. There are still quite a few omissions, including the final chapter on epidemic modeling.

These are lecture notes written for introductory ODE course.

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 w...

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 $N$ i...

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 stea...

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 ana...

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 re...

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 evo...

The Crow--Kimura quasispecies model with permutation invariant fitness landscape is investigated. Using the fact that the mutation matrix in the case of permutation invariant fitness landscape has a special form, a change of the basis is found such that in new coordinates a number of analytical results can be obtained. In particular, we show that i...

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 syst...

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 indiv...

A replicator equation with explicit space and global regulation is considered. This model provides a natural framework to follow frequencies of species that are distributed in the space. For this model, analogues to classical notions of the Nash equilibrium and evolutionary stable state are provided. A sufficient condition for a uniform stationary...

Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of the replicator equation should be transformed; in...

The replicator equation is ubiquitous for many areas of mathematical biology. One of major shortcomings of this equation is that it does not allow for an explicit spatial structure. Here we review analytical approaches to include spatial variables to the system. We also provide a concise exposition of the results concerning the appearance of spatia...

Analytical analysis of spatially extended autocatalytic and hypercyclic systems is presented. It is shown that spatially explicit systems in the form of reaction-diffusion equations with global regulation possess the same major qualitative features as the corresponding local models. In particular, using the introduced notion of the stability in the...

Background: The standard genetic code is redundant and has a highly non-random structure. Codons for the same amino acids typically differ only by the nucleotide in the third position, whereas similar amino acids are encoded, mostly, by codon series that differ by a single base substitution in the third or the first position. As a result, the code...

In many epidemiological models a nonlinear transmission function is used in the form of power law relationship. It is constantly argued that such form reflects population heterogeneities including differences in the mixing pattern, susceptibility, and spatial patchiness, although the function itself is considered phenomenological. Comparison with l...

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 ph...

Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm the interaction matrix of the replicator equation should be transformed; in pa...

The expansion of the standard code according to the coevolution theory. Phase 1 amino acids are orange, and phase 2 amino acids are green. The numbers show the order of amino acid appearance in the code according to (99). The arrows define 13 precursor-product pairs of amino acids, their color defines the biosynthetic families of Glu (blue), Asp (d...

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular di...

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 models of natural rotifer population dynamics. We show that under parti...

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller–Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a prio...

An analysis of traveling wave solutions of certain types of chemotactic models (PDE systems with cross-diffusion) is presented. The conditions for existence of front-front, impulse-front, and front-impulse traveling waves are given for a system of a “separable” type. The simplest mathematical models are presented that have an impulse-impulse soluti...

A class of models of biological population and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-...

The standard genetic code table has a distinctly non-random structure, with similar amino acids often encoded by codons series that differ by a single nucleotide substitution, typically, in the third or the first position of the codon. It has been repeatedly argued that this structure of the code results from selective optimization for robustness t...

In this review, we discuss 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 and formal...

Mathematical Appendix

Figure for the Mathematical Appendix

In this review, we discuss 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 and formal...

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 approa...

Oncolytic viruses that specifically target tumor cells are promising anti-cancer therapeutic agents. The interaction between an oncolytic virus and tumor cells is amenable to mathematical modeling using adaptations of techniques employed previously for modeling other types of virus-cell interaction. A complete parametric analysis of dynamic regimes...

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 o...

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, invasion rate (i.e., rate of arrival of a novel sequence...

New methods for analyzing generalized population models are considered. As in [1-3], it is assumed that the individuals of the interacting populations are not identical. The approach suggested in [4, 5] for analyzing isolated heterogeneous populations is extended to communities of populations. A priori constraints on the form of the dependence of t...

In this paper, a mathematical model of the interaction between pollutants and the environment is considered. The interaction process is assumed to be deterministic, and the mathematical model is governed by a system of semilinear parabolic differential equations. The problem of parameter identification for the proposed mathematical model is solved...

Nonlinear model obtained from notorious Ziegler model by adding nonlinear terms is considered. Ziegler's paradox is explained from nonlinear point of view. It is known that our world is nonlinear. In order to describe it, it is necessary to use nonlinear models. But linear models continue playing an independent and auxiliary (linearization) role as...

The problem of simulation of pollution propagation with regard for the interaction of pollutants with organic environment is considered. It is known that organic environment possesses a purifying regenerating property with respect to pollutants. The conceptual model for describing the pollutant-environment interaction is proposed. This model is a n...

Various mathematical models of the interaction between the animate nature and a pollutant are considered. It is known that the animate nature can absorb pollutant up to certain limits (threshold value). Experiments show that the dependence between the emitted quantity of a pollutant and the remaining quantity can be described by a certain function....