All issues

This paper presents a universal method for constructing an agent-based model of complex systems for their further clear computer representation by means of object-oriented programming languages. The method specifies both steps of model developing from the mathematical description of the system to the determined architecture of the program simulating the system. The efficiency of the method is illustrated by the construction of the two simulation models for the complex systems of various origins: the interactive simulation of the stock exchange and space-time simulation of biological species competition.

In the science development an important role the scientific schools are played. This schools are the associations of researchers connected by the common problem, the ideas and the methods used for problems solution. Usually Scientific schools are formed around the leader and the uniting idea.

The several sciences schools were created around academician A. S. Kholodov during his scientific and pedagogical activity.

This review tries to present the main scientific directions in which the bright science collectives with the common frames of reference and approaches to researches were created. In the review this common base is marked out. First, this is development of the group of numerical methods for hyperbolic type systems of partial derivatives differential equations solution — grid and characteristic methods. Secondly, the description of different numerical methods in the undetermined coefficients spaces. This approach developed for all types of partial equations and for ordinary differential equations.

On the basis of A. S. Kholodov’s numerical approaches the research teams working in different subject domains are formed. The fields of interests are including mathematical modeling of the plasma dynamics, deformable solid body dynamics, some problems of biology, biophysics, medical physics and biomechanics. The new field of interest includes solving problem on graphs (such as processes of the electric power transportation, modeling of the traffic flows on a road network etc).

There is the attempt in the present review analyzed the activity of scientific schools from the moment of their origin so far, to trace the connection of A. S. Kholodov’s works with his colleagues and followers works. The complete overview of all the scientific schools created around A. S. Kholodov is impossible due to the huge amount and a variety of the scientific results.

The attempt to connect scientific schools activity with the advent of scientific and educational school in Moscow Institute of Physics and Technology also becomes.

The paper contains the analysis of the domestic publications issued in 2013–2017 years and devoted to cellular automata. The most of them concern on mathematical modeling. Scientometric schedules for 1990–2017 years have proved relevance of subject. The review allows to allocate the main personalities and the scientific directions/schools in modern Russian science, to reveal their originality or secondness in comparison with world science. Due to the authors choice of national publications basis instead of world, the paper claims the completeness and the fact is that about 200 items from the checked 526 references have an importance for science.

In the Annex to the review provides preliminary information about CA — the Game of Life, a theorem about gardens of Eden, elementary CAs (together with the diagram of de Brujin), block Margolus’s CAs, alternating CAs. Attention is paid to three important for modeling semantic traditions of von Neumann, Zuse and Zetlin, as well as to the relationship with the concepts of neural networks and Petri nets. It is allocated conditional 10 works, which should be familiar to any specialist in CA. Some important works of the 1990s and later are listed in the Introduction.

Then the crowd of publications is divided into categories: the modification of the CA and other network models (29 %), Mathematical properties of the CA and the connection with mathematics (5 %), Hardware implementation (3 %), Software implementation (5 %), Data Processing, recognition and Cryptography (8 %), Mechanics, physics and chemistry (20 %), Biology, ecology and medicine (15 %), Economics, urban studies and sociology (15 %). In parentheses the share of subjects in the array are indicated. There is an increase in publications on CA in the humanitarian sphere, as well as the emergence of hybrid approaches, leading away from the classic CA definition.

This article presents the results of an analytical and computer study of the collective dynamic properties of a chain of self-oscillating systems (conditionally — oscillators). It is assumed that the couplings of individual elements of the chain are non-reciprocal, unidirectional. More precisely, it is assumed that each element of the chain is under the influence of the previous one, while the reverse reaction is absent (physically insignificant). This is the main feature of the chain. This system can be interpreted as an active discrete medium with unidirectional transfer, in particular, the transfer of a matter. Such chains can represent mathematical models of real systems having a lattice structure that occur in various fields of natural science and technology: physics, chemistry, biology, radio engineering, economics, etc. They can also represent models of technological and computational processes. Nonlinear self-oscillating systems (conditionally, oscillators) with a wide “spectrum” of potentially possible individual self-oscillations, from periodic to chaotic, were chosen as the “elements” of the lattice. This allows one to explore various dynamic modes of the chain from regular to chaotic, changing the parameters of the elements and not changing the nature of the elements themselves. The joint application of qualitative methods of the theory of dynamical systems and qualitative-numerical methods allows one to obtain a clear picture of all possible dynamic regimes of the chain. The conditions for the existence and stability of spatially-homogeneous dynamic regimes (deterministic and chaotic) of the chain are studied. The analytical results are illustrated by a numerical experiment. The dynamical regimes of the chain are studied under perturbations of parameters at its boundary. The possibility of controlling the dynamic regimes of the chain by turning on the necessary perturbation at the boundary is shown. Various cases of the dynamics of chains comprised of inhomogeneous (different in their parameters) elements are considered. The global chaotic synchronization (of all oscillators in the chain) is studied analytically and numerically.

Bistability is a fundamental property of nonlinear systems and is found in many applied and theoretical studies of biological systems (populations and communities). In the simplest case it is expressed in the coexistence of diametrically opposed alternative stable equilibrium states of the system, and which of them will be achieved depends on the initial conditions. Bistability in simple models can lead to quad-stability as models become more complex, for example, when adding genetic, age and spatial structure. This occurs in different models from completely different subject area and leads to very interesting, often counterintuitive conclusions. In this article, we review such situations. The paper deals with bifurcations leading to bi- and quad-stability in mathematical models of the following biological objects. The first one is the system of two populations coupled by migration and under the action of natural selection, in which all genetic diversity is associated with a single diallelic locus with a significant difference in fitness for homo- and heterozygotes. The second is the system of two limited populations described by the Bazykin model or the Ricker model and coupled by migration. The third is a population with two age stages and density-dependent regulation of birth rate which is determined either only by population density, or additionally depends on the genetic structure of adjacent generations. We found that all these models have similar scenarios for the birth of equilibrium states that correspond to the formation of spatiotemporal inhomogeneity or to the differentiation by phenotypes of individuals from different age stages. Such inhomogeneity is a consequence of local bistability and appears as a result of a combination of pitchfork bifurcation (period doubling) and saddle-node bifurcation.

The work is devoted to the study of the theoretical value of destructive influence on normal tissues of an organism by infrared radiation that goes beyond the treated pathological focus. This situation is possible if the direct laser radiation on the tissues is extremely long-acting. The solution to this problem can be the uniform distribution of heat inside the volume through indirect heating of the liquid, which contributes to minimal damage to the perifocal structures. A non-stationary thermophysical model of the process of heat propagation in biological tissues is presented, allowing to carry out studies of energy transfer from internal liquid contents of Baker's cyst heated by infrared laser radiation of a given specific power through a certain thickness of its wall to surrounding biological tissues. Calculation of the spacetime temperature distribution in the cyst wall and surrounding fat tissue is carried out by the finite-difference method. The time of effective exposure to temperature on the entire thickness of the cyst wall was estimated to be 55 ° C on its outer surface. The safety procedure ensures the exposure duration of this value is not more than 10 seconds.

As a result of the calculations carried out, it is established that there are several operating modes of a surgical laser that meet all the safety requirements with a simultaneous effective procedure. Local one-sided hyperthermia of the synovial membrane and subsequent coagulation of the entire wall thickness due to heat transfer contributes to the elimination of the cavity neoplasm of the popliteal region. With a thickness of 3 mm, the heating mode is satisfactory, under which the exposure time lasts about 200 seconds, and the specific power of the laser radiation in the internal medium of the liquid contents of the Baker cyst is approximately 1.

The review and systematization of current papers on the mathematical modeling of population dynamics allow us to conclude the key interests of authors are two or three main research lines related to the description and analysis of the dynamics of both local structured populations and systems of interacting homogeneous populations as ecological community in physical space. The paper reviews and systematizes scientific studies and results obtained within the framework of dynamics of structured and interacting populations to date. The paper describes the scientific idea progress in the direction of complicating models from the classical Malthus model to the modern models with various factors affecting population dynamics in the issues dealing with modeling the local population size dynamics. In particular, they consider the dynamic effects that arise as a result of taking into account the environmental capacity, density-dependent regulation, the Allee effect, complexity of an age and a stage structures. Particular attention is paid to the multistability of population dynamics. In addition, studies analyzing harvest effect on structured population dynamics and an appearance of the hydra effect are presented. The studies dealing with an appearance and development of spatial dissipative structures in both spatially separated populations and communities with migrations are discussed. Here, special attention is also paid to the frequency and phase multistability of population dynamics, as well as to an appearance of spatial clusters. During the systematization and review of articles on modeling the interacting population dynamics, the focus is on the “prey–predator” community. The key idea and approaches used in current mathematical biology to model a “prey–predator” system with community structure and harvesting are presented. The problems of an appearance and stability of the mosaic structure in communities distributed spatially and coupled by migration are also briefly discussed.

Seasonal migrations and herd instinct are traditionally recognized as wild reindeer (Rangifer tarandus L.) species-specific behavioral signs. These animals are forced to overcome water obstacles during the migrations. Behaviour peculiarities are considered as the result of the selection process, which has chosen among the sets of strategies, as the only evolutionarily stable one, determining the reproduction and biological survival of wild reindeer as a species. Natural processes in the Taimyr population wild reindeer are currently occurring against the background of an increase in the influence of negative factors due to the escalation of the industrial development of the Arctic. That is why the need to identify the ethological features of these animals completely arose. This paper presents the results of applying the classical methods of the theory of optimal control and differential games to the wild reindeer study of the migration patterns in overcoming water barriers, including major rivers. Based on these animals’ ethological features and behavior forms, the herd is presented as a controlled dynamic system, which presents also two classes of individuals: the leader and the rest of the herd, for which their models, describing the trajectories of their movement, are constructed. The models are based on hypotheses, which are the mathematical formalization of some animal behavior patterns. This approach made it possible to find the trajectory of the important one using the methods of the optimal control theory, and in constructing the trajectories of other individuals, apply the principle of control with a guide. Approbation of the obtained results, which can be used in the formation of a common “platform” for the adaptive behavior models systematic construction and as a reserve for the cognitive evolution models fundamental development, is numerically carried out using a model example with observational data on the Werchnyaya Taimyra River.

The repressilator is the first genetic regulatory network in synthetic biology, which was artificially constructed in 2000. It is a closed network of three genetic elements $lacI$, $\lambda cI$ and $tetR$, which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In our previous paper [Bratsun et al., 2018], we proposed a mathematical model of a delayed repressillator and studied its properties within the framework of a deterministic description. We assume that delay can be both natural, i.e. arises during the transcription / translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using gene engineering technologies. In this work, we apply the stochastic description of dynamic processes in a delayed repressilator, which is an important addition to deterministic analysis due to the small number of molecules involved in gene regulation. The stochastic study is carried out numerically using the Gillespie algorithm, which is modified for time delay systems. We present the description of the algorithm, its software implementation, and the results of benchmark simulations for a onegene delayed autorepressor. When studying the behavior of a repressilator, we show that a stochastic description in a number of cases gives new information about the behavior of a system, which does not reduce to deterministic dynamics even when averaged over a large number of realizations. We show that in the subcritical range of parameters, where deterministic analysis predicts the absolute stability of the system, quasi-regular oscillations may be excited due to the nonlinear interaction of noise and delay. Earlier, we have discovered within the framework of the deterministic description, that there exists a long-lived transient regime, which is represented in the phase space by a slow manifold. This mode reflects the process of long-term synchronization of protein pulsations in the work of the repressilator genes. In this work, we show that the transition to the cooperative mode of gene operation occurs a two order of magnitude faster, when the effect of the intrinsic noise is taken into account. We have obtained the probability distribution of moment when the phase trajectory leaves the slow manifold and have determined the most probable time for such a transition. The influence of the intrinsic noise of chemical reactions on the dynamic properties of the repressilator is discussed.

Rugose spiraling whitefly (RSW) is one of the major pests which affects the coconut trees. It feeds on the tree by sucking up the water content as well as the essential nutrients from leaves. It also forms sooty mold in leaves due to which the process of photosynthesis is inhibited. Biocontrol of pest is harmless for trees and crops. The experimental results in literature reveal that Pseudomallada astur is a potential predator for this pest. We investigate the dynamics of predator, Pseudomallada astur’s interaction with rugose spiralling whitefly, Aleurodicus rugioperculatus in coconut trees using a mathematical model. In this system of ordinary differential equation, the pest-predator interaction is modeled using Holling type III functional response. The parametric values are calculated from the experimental results and are tabulated. An approximate analytical solution for the system has been derived. The homotopy analysis method proves to be a suitable method for creating solutions that are valid even for moderate to large parameter values, hence we employ the same to solve this nonlinear model. The $\hbar$-curves, which give the admissible region of $\hbar$, are provided to validate the region of convergence. We have derived the approximate solution at fifth order and stopped at this order since we obtain a more approximate solution in this iteration. Numerical simulation is obtained through MATLAB. The analytical results are compared with numerical simulation and are found to be in good agreement. The biological interpretation of figures implies that the use of a predator reduces the whitefly’s growth to a greater extent.