Specificity of mathematical modeling of living systems. Mathematical biology

Specificity of mathematical modeling of living systems. Mathematical biology
24.09.2019

The method of describing biological systems with an adequate mathematical apparatus. Definition Mat. The device adequately reflecting the work of biological systems is a complex task associated with their classification. The classification of biosystems by complexity (the logarithm of state numbers) can be carried out using the scale, by using the scale, according to the simple systems, there are systems that have up to a thousand states, to complex - from a thousand to millions and to very complex - over a million states. The second most important characteristic of the biosystem is the pattern expressed by the law of the distribution of probability of states. Under this law, it is possible to determine the uncertainty of its work on K. Shannon and the assessment of a relative organization. T about., Biol. Systems can be classified by complexity (max. diversity or maximum uncertainty) and relative organization, i.e., the degree of organization (see biological systems organization).

Classification diagram Biosystems:

Simple systems;

Complex systems;

Very complex systems;

Probabilistic systems;

Probabilistic and deterministic systems;

Deterministic systems.

In fig. A classification chart of biosystems is given in the axes of the highest possible uncertainty characterizing the number of state states and the logarithm number of states, and the level of relative orgation, which characterizes the degree of system organization. The diagram is given the names of the corresponding bands so that, for example, the area under the number 8 means "very complex probabilistic and deterministic biosystems". The study of the biosystems shows that if calculated by the histogram of the distribution of deviations of the studied indicator from its mathematical expectation lies in the range from 1.0 to 0.3, then we can assume that this is a deterministic biosystem. Such systems include control systems. Authorities, mostly hormonal (humoral) management systems. Neuron, organs internal. The spheres, metabolic systems according to certain parameters can also be attributed to deterministic biosystems. Mat. Models of such systems are based on physico-chemical. relations between elements or system bodies. Modeling in this case is the dynamics of changes in the input, intermediate and output indicators. Such, for example, biophysical models of nervous cell, cardiovascular system, blood sugar content control systems and others. Mat. The device adequately describing the behavior of such deterministic biosystems is the theory of Diff. and integral ur. Based on Mat. Models of biosystems can be using the methods of automatic theory control, successfully solve the DFF tasks. diagnosis and optimization of treatment. The field of modeling deterministic biosystems is most fully developed.

If the organized biosystems with respect to the studied indicator (or the indicator system) lies within 0.3 - 0.1, then the systems can be considered probabilistic-deterministic. These include system management systems. Authorities with a pronounced component of the nervous regulation (eg, the pulse rate control system), as well as hormonal regulation systems in the case of pathology. As an adequate Mat. The device can be the presentation of the dynamics of the change of diff indicators. Urms with coeffs, obeying certain distribution laws. Simulation of such biosystems has been developed relatively weakly, although it is considerable interest for the purposes of cybernetics medical.

Probabilistic biosystems are characterized by the value of the organization R in the range from 0.1 to 0. These include systems that determine the interaction of analyzers and behavioral reactions, including training processes in simple conditionally reflex acts and complex relationships between environmental signals and organism reactions. Adequate Mat. Apparatus

to simulate such biosystems, the theory of deterministic and random automata interacting with deterministic and random media, random theory processes.

Mat. Bosystem modeling includes preliminary statistical processing of experimental results (see biological research Mathematical methods), the study of complexity and organized biosystems, the choice of adequate mat. Models and definition of numerical values \u200b\u200bof parameters Mat. Models according to experimental data (see cybernetics biological). The last task is generally very difficult. For deterministic biosystems, whose models can be represented by linear diff. The urms, the definition of the best parameters of the model (coeff. Diff. URNIA) can be carried out by the method of descent (see the gradient method) in the model parameter space, evaluating the integral from the square of the error. In this case, you need to apply the descent procedure to minimize the functional

where t - the period, the characteristic time for the indicator, y is the experimental curve of changes in the biosystem indicator, y - solution Mat. Models. If you need to get the best (in the sense of an integral of the square error) approximation Mat. Models for the operation of the biosystem in several indicators on various internal states of the biosystem or for various characteristic external influences, it is possible, using the descent method in the model parameter space, minimize the amount of private functionals. When using such a procedure for selecting Mat. Models can increase the likelihood of obtaining a single set of coeff. Models corresponding to the adopted structure. With B. s. m. m. It is desirable to obtain not only the quantitative characteristics of the work of the biosystems, its elements and the characteristics of the interconnection of the elements, but also to identify the criteria for the work of the Baosystems, to establish certain general principles of their functioning. Lit.: Glushkov V. M. Introduction to Cybernetics. K., 1964 [Bibliogr. from. 319-322]; Modeling in biology and medicine, in. 1-3. K., 1965-68; Bush R., Sapelller F. Stochastic training models. Per. from English M., 1962. Yu. G. Antommon.


Gomel, 2003



UDC 57.082.14.002.2.

Developed: Starodubtseva M. N., Kuznetsov B. K.

Tutorial on the topic "Mathematical modeling of biological processes"

The manual contains two laboratory work, acquainting physician students with the basics of mathematical modeling of biological processes, one of them (two classes) is implemented in the computer algebra of Mathcad. In the first work, the "modeling of the functioning of the cardiovascular system" considers mathematical modeling of biological processes, including the model of functioning of the cardiovascular system. A systematic approach is considered in modeling the functioning of complex objects, the principles of compiling systems of differential equations describing the behavior of a biological object, as well as concepts such as stable and unstable states, bifurcation, oscillators, synchronization of processes. The practical part of the work contains an algorithm for calculating the parameters of blood circulation at rest and after loading according to experimental data and methods for their statistical analysis. In the second job associated with computer modeling, a description of the user interface, the input language of the MathCAD system, the main calculation methods (calculating arithmetic expressions, the finding of derivatives, integrals, the solution of differential equations and systems of differential equations), the foundations of building graphs, some statistics functions ( Calculation of the average value, standard deviation, finding the linear regression equation and correlation coefficient).

For students of the 1st year of medical higher educational institutions of all faculties.

Reviewers:

Chenkevich S. N.,

professor, D. B.n, Head of the Department of Biophysics of the Physics Faculty of Belgosuniversitta,

Asenchik O. D.,

k.F.-M.N., Head of the Department of Information Technologies of the Gomel State Technical University. P. O. Dryah.

Approved by the Scientific and Methodological Council of the Institute as a tutorial _____________ 2003, Protocol No. ____ on the topic: "Mathematical modeling of biological processes"

Ó Gomel State Medical Institute, 2003

Subject: Mathematical modeling of biological

processes

Laboratory work 1.

Mathematical modeling of biological processes.

Cardiovascular functioning

systems

Classes time - 135 minutes.

Purpose: To explore the modern models of the cardiovascular system and show the effectiveness of the use of the modeling method to assess the condition and identification of the characteristic features of the behavior of complex biological objects.

1.1. Questions theory

1.1.1. Mathematical modeling of biological processes. Biophysics of complex systems.

The functioning of a complex biological system, including the cardiovascular system, is the result of the interaction of the components of its elements and processes occurring in it. It should be borne in mind that according to the general principle of the upward hierarchy of movement types (mechanical - physical - chemical - biological - social), the biological form of motion cannot be fully reduced to the mechanical, physical or chemical form of movement, and biological systems cannot be fully described From the standpoint of any one of these forms of movement. These forms of movement can serve as models of the biological form of movement, that is, its simplified images.

To find out the basic principles for regulating the processes of a complex biological system using the construction of a first mechanical, physical or chemical model of the system, and then constructing their mathematical models, that is, the findings describing these models of mathematical functions, including equations (creating mathematical models). The lower the level of the hierarchy is the easier the model, the more factors of the real system are excluded from consideration.

Simulation is a method in which the study of a certain complex object (process, phenomena) is replaced by the study of its simplified analogue - models. Physical, chemical, biological and mathematical models are widely used in biophysics, biology and medicine. For example, the flow of blood according to the vessels is modeled by the movement of fluid on pipes (physical model). The biological model is simple biological objects that are convenient for experimental research, which are studied by the properties of real more complex biological systems. For example, the patterns of occurrence and dissemination of the potential of the nerve fiber action were studied on the biological model - the giant squid acone.

The mathematical model is a combination of mathematical objects and relations between them, reflecting the properties and characteristics of the real object that interests the researchers. An adequate mathematical model can be built only with the involvement of specific data and ideas about the mechanisms of complex processes. After construction, the mathematical model "lives" in its internal laws, the cognition of which allows you to identify the characteristic features of the system under study (see the scheme in Fig. 1.1.). The simulation results are the basis for managing the processes of any nature.

Biological systems are essentially extremely complex structural and functional units.


Fig. 1.1. Scheme of a systematic approach in modeling a biological object.

Most often, mathematical models of biological processes are defined in the form of differential or difference equations, but other types of model representations are also possible. After the model is built, the task is reduced to the study of its properties by methods of mathematical deduction or by machine modeling.

When studying a complex phenomenon, several alternative models are usually offered. Check the qualitative compliance of these models to the object. For example, establish the presence of stable stationary states in the model, the existence of oscillatory modes. The model, which is best relevant to the system under study, is chosen as the main one. The selected model is specified in relation to the specific system under study. Set the numeric values \u200b\u200bof parameters according to experimental data.

The process of finding a mathematical model of a complex phenomenon can be divided into steps, the sequence and relationship of which reflects the scheme nor Fig. 1.2.


Fig. 1. 2. Search diagram of the mathematical model.

Step 1 corresponds to the collection of data on the study of the object being studied.

At step 2, the choice of the base model (system of equations) from possible alternative models for quality features is performed.

At step 3, the model parameters are identified according to experimental data.

At step 4, the behavior of the model on independent experimental data is carried out. For this, it is often necessary to put additional experiments.

If the experimental data are taken to verify the model "do not fit" into the model, it is required to analyze the situation and put forward other models, investigate the properties of these new models, and then put experiments that make it possible to conclude about the preference of one of them (step 5).

Modern biology is widely used by mathematical and computer methods. Without the use of mathematical methods, it would be impossible to implement such global projects as a human genome, deciphering the spatial structure of complex biomacomolecules, remote diagnostics, computer simulation of new effective drugs ("Drag-Design"), planning measures to prevent epidemics, an analysis of the environmental consequences of industrial work Objects, biotechnology and much more.

The rapid introduction of mathematical methods into biology in recent decades is primarily due to the development of experimental physicochemical methods of biological research. X-ray structural and spectroscopic (NMR, EPR) methods, the analysis of the DNA sequence is impossible without mathematical processing of the experimental results.

On the other hand, the use of mathematical methods contributed to the understanding of the laws underlying many biological processes. Numerous examples are given in the recommended literature. Among them are the properties of cyclic fluctuations in population numbers, the principle of competitive exclusion of Gause for competing species, the threshold theorem in mathematical epidemiology, the conditions for the propagation of the nervous impulse, the conditions for the occurrence of various types of autowave processes in the active tissues, in particular in the heart muscle and many others.

Biological objectives initiated the creation of new mathematical theories that enriched the math itself. The first known mathematical model of the population of Leonardo rabbits from Pisa (13th century) is a series of fibonacci. Later examples of new mathematical productions provide the tasks of birth and death, diffusion processes, systems with cross-diffusion in equations with private derivatives, new types of boundary value problems for transfer equations, evolutionary game theory, replicator equations. The foundations of modern statistics were laid by R. Fisher, who also studied biological problems.

Mathematical models in biology

The first systematic studies dedicated to mathematical models in biology belong to A.D. Tray (1910-1920). His models have not lost importance. The founder of the modern mathematical theory of biological populations is fairly considered the Italian mathematician Vito Volterra, which developed a mathematical theory of biological communities, which serves as differential and integro-differential equations. (Vito Volterra. Lecons Sur La Theorie Mathematique De La Lutte Pour La Vie. Paris, 1931). In the following decades, population dynamics developed, mainly in line with ideas expressed in this book. V. Volterra owns the most famous "biological model" of the coexistence of types of types (1928), which is included in all textbooks on the theory of oscillations. The Russian translation of the Volterra book was published in 1976 entitled: "Mathematical theory of struggle for existence" edited and with the afterbirth Yu.M.Svirezhev, where the history of the development of mathematical ecology in the period 1931-1976 is considered. Starting from the forties of the 20th century, the mathematical models occupied a durable place in: the works of Mono (1942), Novika and Szyllard (1950) allowed to describe the patterns of growth of single-cell organisms.

The work of biological systems is fundamental to the development of mathematical models, the work of Alan Turing, the "chemical bases of morphogenesis" (Turing, 1952) laid the foundation of a dynamic approach to modeling distributed biological systems. It first shows the possibility of existence in the active kinetic environment of stationary and inhomogeneous structures. The fundamental results obtained in this work have formed the basis of a large number of models of morphogenesis describing the coloring of animal skins (Murray 1993; Murray, 2009), the formation of shells (Meinhardt 1995), marine stars and other living organisms.

Mathematical models played an important role in the study of the mechanisms for generating a nervous impulse. A. Hodgkin and E. Huxley, along with an experimental study, proposed a model describing the processes of ion transport through the membrane and the passage of the potential pulse along the membrane. The work of British scientists was awarded the Nobel Prize of 1963 (along with Sir John Ekls, Australia).

An explanation of the mechanism of cardiac arrhythmias with the help of axiomatic models of an excitable environment was devoted to the first work of N. Wiener and A. Rosenblut (Wiener and Rosenblueth 1946). Russian translation is published in the book: Cybernetic Collection. M..3. M. Il, 1961. In a more general form, similar ideas were developed by Soviet scientists with Gelfand and Zetlin (Gelfand et al., 1963; Gelfand, and others, 1966), and then other authors on cellular models. When constructing models, it was taken into account that the process of occurrence and distribution of excitement in biological objects, in particular, in the nerve tissues has a number of clearly pronounced properties, departing from which you can build a formal model of this phenomenon.

Russian scientific schools

Russian scientific schools have made a great contribution to the development of mathematical biology. A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov In 1937, in the work "Investigation of the diffusion equation connected to an increase in the substance, and its application to one biological problem" solved the problem of the limiting speed of moving the wave front and determined the limit form of the front. This work has become classic and laid the beginning of the development of theoretical and experimental study of autowave phenomena in systems of different nature.

Russian biophysics V.I. Krinsky, G.R. Ivanitsky et al. Belongs a series of brilliant work, which posted the beginning of an experimental study and theoretical description of excitable tissues (Ivanitsky, Creins, Selk. "Mathematical biophysics of cells. 1978). Currently, the direction for the study and computer modeling of the processes of nervous and distribution of waves in the heart muscle is intensively developing. The latest achievements in this area are presented in the book "Dynamic models of processes in cells and subcellular nanostructures", 2010. The most advanced models take into account the conjugation of electrical and mechanical chemical processes, structural and geometric heterogeneity of the heart.

Russian scientist B.P. Belousov (Belousov 1959, 1981) was opened by a class of chemical reactions, allowing to observe practically all the types of distributed systems' behavior currently known. A.M. Zhabotinsky with employees in detail investigated the properties of these reactions and the conditions for their flow, they also proposed the first mathematical model of the observed phenomenon (Zhabotinsky, 1975). In the future, Belousov-Zhabotinsky reaction (BZ reaction), as a distributed system model demonstrating various types of spatial-temporal organization, was investigated in hundreds of laboratories of the world (Field and Burger 1988; Vanag, 2008). A number of models were developed to describe the flowing processes, the most famous are the "Oregonator" model, proposed by researchers from the University of Oregon, USA (Field., Koros et al. 1972; field. And Noyes 1974), and the "Pressor" model proposed by researchers from Scientific Center for Biological Research G. Pushchino (Rovinsky and Zhabotinsky 1984).

Russian scientists have contributed a great contribution to the development of mathematical theory. This is, first of all, the work of the collections of the Institute of Mathematical Problems of Biology of the Russian Academy of Sciences (until 1992 - Scientific Computing Center of the Russian Academy of Sciences) under the leadership of A.M. Molchanova (A.D. Bizikin, F.S. Berezovskaya, A.I. Chibnik) and the team of employees of the Computational Center of the Russian Academy of Sciences under the leadership of Yu.M.Svirezhev (D.O., A. Tarko, V. Dezhevaykin, D. Saranscha, N. V. Belotelov, V. Pischik, V.V. Shakin et al.) . In the Central Bank of the Russian Academy of Sciences, under the leadership of Academician N.N.Miseiseev, in the 70-80 years of the 20th century, work was carried out on global and regional modeling. Here was created the famous model of "nuclear winter".

Significant contribution to the development of methods for modeling processes in energy-forming membranes was made by scientists of Moscow State University. The kinetic models of the primary processes of photosynthesis are designed by scientists of biological (A.B. Rubin, G.Yu. Riznichenko, N.E. Belyaeva) and physical (A.K. Kukushkin, A.N. Tikhonov, V.A. Karavayev, S. A. Kuznetsova). In recent years, at the Department of Biophysics of the Biological Faculty of Moscow State University, work is actively underway to develop a new method of direct multi-frequency computer simulation of processes in subcellular systems (A.Brubin, G.Yu.riznichenko, I.B. Kovalenko. D.M. Deligin)

A large role in the formation of the mathematical biology of Russia was played by scientific developments and books of the team of authors Yu.M. Romanovsky, N.V. Stepanova (Physical Faculty of Moscow State University) and D.S. Chernavsky (Fian): "Mathematical models in biophysics" M., 1976; "Mathematical biophysics" M., 1984; "Mathematical modeling in biophysics. Introduction to theoretical biophysics »M-Izhevsk, 2004. They consider the basics of biological kinetics, models of evolution and development in biology, cell populations growth models, autowaves in distributed kinetic systems, statistical aspects of biological kinetics. This direction continues to develop in Fiana (A. Polezhaev, V.I. Volkov, etc.)

Institutions where work on mathematical modeling in biology

In modern Russia, working on mathematical modeling in biology is carried out in a number of research institutes and universities. One of the leading places belongs to the Scientific Center in Pushchino, where in 1972 the Scientific Computing Center of the Russian Academy of Sciences (director - A.M. Molchanov) was organized, which in 1992 received the status of the Institute for Mathematical Problems of Biology RAS. The current director of the Imphiba is V.D. Lakhno, who is also the chairman of the Scientific Council of the Russian Academy of Sciences in mathematical biology and bioinformatics. The Imb RAS is a leading scientific institution on this issue and issues the electronic magazine "Mathematical Biology and Bioinformatics"

Work on mathematical modeling of biological processes is also conducted in other institutions of the Pushkin Scientific Center of the Russian Academy of Sciences: Institute of Biophysics Cells of the Russian Academy of Sciences. Director - ChL-Corr. RAS E.E.Feshenko (mainly on molecular dynamic and quantum-mechanical modeling of processes in biomacomolecules) and the Institute of Theoretical and Experimental Biophysics of the Russian Academy of Sciences, Director - Corr. RAS G.R.Ivanitsky (modeling of self-organization processes in active environments, busholnovna in living cells and biopolymers).

In the Scientific School of Academician G. and Marchuk, modeling methods are actively developing in relation to medicine, in particular, models of immunity and spread of epidemics are being developed.

Studies of biological systems using mathematical models are held at the Institute of Biophysics SB RAS (Krasnoyarsk, Institute of Genetics from RAS (Novosibirsk), at the universities of Nizhny Novgorod, Saratov, Rostov-on-Don, Yaroslavl, at the Moscow Physics and Technology State University, In the National Research Nuclear University "Miphy" and others.

Work on mathematical modeling in biology in Moscow State University is actively conducted at the Biological Faculty (models of primary photosynthesis processes and other processes in subcellular and cellular systems, molecular dynamics of proteins and biomembranes), Physical Faculty of Moscow State University (molecular machine models), Faculty of computing mathematics and cybernetics (population Dynamics, mathematical ecology, evolutionary models, control models), mechanics and mathematical faculty (vestibular machine models, model of plant communities).

Periodicals

Articles on mathematical models in biology are regularly published in journals:

  • Biophysics (M., 1956 -),
  • "Bulletin of Mathematical Biophysics" (1939 -1972); "Bulletin of Mathematical Biology" (1972-); Jurnal of theoretical Biology (1961 -),
  • Journal of Mathematical Biology (1974-);
  • Ecological Modeling (1975-),
  • Computer research and modeling (2009 -).

Separate articles on mathematical modeling are also printed in magazines:

  • Successes of Physical Sciences (1918 -)
  • Vestnik Moscow University
  • Biosystems (1967)
  • Journal of BioLogical Systems (1993)
  • Computational and Mathematical Methods in Medicine (1997)
  • Mathematical Biosciences (1967)
  • Mathematical Biosciences and Engineering
  • PNAS (1915)
  • Science Magazine (1880)
  • Journal Nature (1869)
  • Acta Biotheoretica (1935)
  • Comments on theoretical Biology
  • Rivista de Biologia / Biology Forum (1996)
  • Systema Naturae / Annali Di Biologia Teorica (1998)
  • TheORETICAL AND APPLIED GENETICS (1929)
  • TheORETICAL MEDICINE AND BIOETHICS (1980)
  • Theoretical Population Biology ()
  • Theory in BioSciences / Theorie in Den Biowissenschaften
  • Mathematical Modeling Of Natural Phenomena (2006)

Edition

Books on mathematical modeling in biologies are published by PCD-Iki publishing house in the Biophysics series. Mathematical biology, "Science, URSS and other publishers of scientific and educational literature.

Ivanitsky G.R., Krinskaya V.I., Selkov E.E. Mathematical biophysics cells. Science, 1978.

Murray D. Mathematical Biology. Volume 1. Introduction. Ed. IKI-RHD, M-Izhevsk, 2009

Matlev, V.D., Panchenko L.A., Risnichenko G.Yu., Terekhin A.T. Higher mathematics and its applications for biology. Theory of Probability and Mathematical Statistics. Mathematical models. Academy. M., 2009.

Risnichenko G.Yu. Lectures on mathematical models in biology. Ed. RHD, M-Izhevsk, 2003.

Risnichenko G.U., Rubin AB Biophysical dynamics of production processes. Ed. IKI-RCD, M-Izhevsk, 2004

Romanovsky Yu.M., Stepanova N.V., Chernavsky D.S. Mathematical modeling in biophysics. Ed. IKI-RHD, 2004

Rubin AB Biophysics. T. I. M., 2004. T. 2. M., 2004 (ed. 3rd)

Swingzhev Yu.M., Logohet D.O. Stability of biological communities. M., Science. 1978.

Swingzhev Yu.M. Nonlinear waves. Dissipative structures and disasters in ecology. M., Science, 1987

Smirnova O.A. Radiation and mammalian organism: model approach. Ed. RHD, M-Izhevsk, 2006

Course of lectures "Mathematical models in biology"

it is read by the author for students of the 2nd year of training of the bachelor of the biological faculty of Moscow State University named after M. V. Lomonosov. In parallel with lectures, seminars (practical classes) are held, during which students enshrine knowledge gained at lectures and get acquainted with software used for analyzing mathematical models and conduct computing experiments. After passing the course, students pass the exam. The course includes 14 lectures in 2 academic hours.

  • Textbook Risnichenko G. Yu. Lectures on mathematical models in biology (ed. 2nd, copy. and addition.) Publishing house RCD, 2011 560 p. ISBN 978-5-93972-847-8. Previous edition (much more short!) Is in free access on the Internet for the link http://www.library.biophys.msu.ru/lectMB/
  • Textbook Matlev, V.D., Panchenko L.A., Risnichenko G.Yu., Terekhin A.T. Theory of Probability and Mathematical Statistics. Mathematical models (ed. 2nd, copy. And addition.) M.: Publishing House Yurait, 2018. - 321 p. - (Series: Universities of Russia). - ISBN 978-5-534-01698-7.
  • Tutorial Plusinina T.Ya., Fursova P. V., Törlova L. D., Risnichenko G. Yu. Mathematical models in biology (Ed. 2-E extra. Tutorial. M.-Izhevsk: NIC: "Regular and chaotic dynamics", 2014. 136 p. ISBN: 978-5-4344-0224-8) - electronic version
  • Ask a question to teachers you can web.-Forum
  • You can send answers to lecture questions to the teacher using web.-Forum. Please read the forum rules

Lectures will be read in Big biological audience (BBA, 2nd floor) Biological Faculty of Moscow State University from September 7 to December 21, 2018 weekly on Fridays from 13 40 .

Those who missed the disease lecture control or electronic test can write them on December 24, 2018 at 15.35 and 17.10.

Part 1. Introduction. Concept of model. Objects, objectives and methods of modeling. Models in different sciences. Computer and mathematical models. The history of the first models in biology. Modern classification of biological processes models. Regression, imitation, high-quality models. Principles of simulation and examples of models. Specificity modeling of living systems.

  • Program: Integration of Data and Knowledge. Modeling goals. Basic concepts
  • Tutorial: Introduction (from the 1st Edition)
  • Introduction (from the 2nd publication)
  • Presentation (Download PDF)

Part 2. . Models leading to one differential equation. The concept of solving one autonomous differential equation. Stationary state (equilibrium condition). Stability of equilibrium state. Methods for assessing sustainability.

  • Program:
  • Tutorial: Models of biological systems described by one differential equation of first order
  • Presentation (download)

Continuous models: exponential growth, logistics growth, models with a smallest critical number. Model of human growth. Models with non-refining generations. Discrete logistics equation. Figure and ladder Lamerey. Types of solutions at different values \u200b\u200bof the parameter: monotonous and decaying solutions, cycles, quasisthastic behavior, numerical outbreaks. Matrix patterns of populations. Effect of delay. Probabilistic patterns of populations.

  • Program: Models described by an autonomous differential equation
  • Tutorial: Models of biological systems described by one differential equation of first order
  • Tutorial: populations growth models
  • Presentation (Download PDF)

September 21. Lecture 3.. Populations growth models.

Part 1. Models of population growth. Matrix patterns of populations. Effect of delay. Probabilistic patterns of populations.

Part 2. Models described by the systems of two autonomous differential equations. Phase plane. Phase portrait. Method isoclin. The main isoclines. Stability of the stationary state. Linear systems. Types of singular points: node, saddle, focus, center. Example: first-order chemical reactions.

  • Program: Models described by systems of two autonomous differential equations
  • Tutorial: Models described by systems of two autonomous differential equations
  • Tutorial: Investigation of the stability of stationary states of nonlinear second-order systems
  • Presentation: Matrix populations (Download PDF)
  • Presentation: Models described by systems of two autonomous differential equations (Download PDF)

September 28th. Lecture 4.. Investigation of the stability of stationary states of nonlinear second-order systems

Trigger. Examples of systems with two sustainable stationary states. Power and parametric trigger switching. Evolution. Selection of one of two and several equal species. Competition of two species in case of unlimited and limited growth. Genetic trigger jacob and mono. Bifurcation of dynamic systems. Types of bifurcations. Bifurcation diagrams and phase-permament of portraits. Catastrophe.

  • Program: Multi-Station Systems
  • Tutorial: multistationary systems
  • Tutorial: The problem of fast and slow variables. Tikhonov Theorem. Types of bifurcations. Catastrophe
  • Presentation: Stability and Asymptotic Sustainability (Download PDF)
  • Presentation: Biological triggers (Download PDF)
  • Materials on the theory of disasters:
    • Arnold V.I. Catastrophe Theory // Science and Life, 1989, № 10
    • Arnold V.I. The theory of disasters // Dynamic systems - 5, the results of science and tehn. Ser. Contemporary Probl. mat. Foundams. Directions, 5, Viniti, M., 1986, 219-277
    • Arnold V.I. Catastrophe theory. M., Science, 1990 - 128 p.

The concept of self-oscillations. An image of the behavior of an auto-oscillating system on the phase plane. Limit cycles. The conditions for the existence of limit cycles. Birth of the limit cycle. Andronova Bifurcation - Hopf. Soft and rigid excitation of oscillations. Model Brusseltor. Examples of self-oscillating models of processes in live systems. Oscillations in dark photosynthesis processes. Autocalbania in the glycolysis model. Intracellular oscillations of calcium concentration.

  • Program:
  • Tutorial: fluctuations in biological systems
  • Presentation (Download PDF)

The main concepts of the theory of dynamic systems. Limit sets. Attractors. Strange attractors. Dynamic chaos. Linear analysis of the stability of the trajectories. Dissipative systems. Stability of chaotic solutions. The dimension of strange attractors.

Stationary states and dynamic modes in the community of three species. Dynamic chaos in models of interaction of species. Trophic systems with a fixed amount of substance. Model of a system of four biological species.

Fractals and fractal dimension. Koha curve. Triangle and Napkin Serpinsky. Cantor set. Cantors rod, though staircase. Examples of fractal sets in live systems. Formation of crown trees. Alveoli lungs. Mitochondrial membranes.

  • Program: quasisthastic processes. Dynamic chaos
  • Textbook:
  • Presentation (Download PDF)

November 2. Lecture 9.. Models of interaction of two types. Simulation of microbial populations

Volterra hypothesis. Analogies with chemical kinetics. Volterrov models interactions. Classification of types of interactions. Competition. Predator sacrifice. Generalized models of interaction of species. Model Kolmogorov. Model of interaction of two types of insects MacArthur. Parametric and phase portraits of the Basykin system.

  • Program: models of interaction of species
  • Program: Models in Microbiology
  • Tutorial: Models of interaction of two types
  • Tutorial: Dynamic chaos. Models of biological communities
  • Presentation (Download PDF)

Equation reaction-diffusion. Why periodic structures and waves occur. Active kinetic environments in live systems. The problem of formation. Distribution of excitation waves. Spatial structures and autowave processes in chemical and biochemical reactions.

Diffusion equation. Primary and boundary conditions. Solving diffusion equation. Solving a homogeneous diffusion equation with zero boundary conditions. Method of separation of variables. Own values \u200b\u200band own functions of the assault-Liouville task. The solution of an inhomogeneous equation with zero initial conditions. Solving a common boundary value problem. Linear analysis of the stability of homogeneous stationary solutions of a single response type equation.

  • Program:
  • Textbook:
  • Textbook:
  • Textbook:
  • Presentation (Download PDF)

November 16th. Lecture 11.. Distributed biological systems. Distributed triggers and morphogenesis. Coloring Models Animal Skins

Stability of homogeneous stationary solutions of a system of two equations of the type of diffusion reaction. Duspative structures. Linear analysis of the stability of a homogeneous stationary state. The dependence of the type of instability from the wave number. Turing instability. Linear analysis of the stability of the homogeneous stationary state of the distributed Brusseltor. Duspative structures near the threshold of instability. Localized dissipative structures. Linear analysis of the electrodiffusion reaction system. Types of space-time modes.

Distributed triggers and morphogenesis. Models coloring animal skins. Differentiation and morphogenesis. Model of the genetic trigger with diffusion (Chernavsky et al.). The study of the stability of a homogeneous stationary state. Genetic trigger, taking into account the diffusion of substrates. Model Girera-Minerandt Girera. Simulation coloring animal skins. Models aggregation ameb.

  • Program: Live Systems and Active Kinetic Environments
  • Tutorial: distributed biological systems. Diffusion Equation
  • Tutorial: solving the diffusion equation. Stability of homogeneous stationary states
  • Tutorial: Distribution of the concentration wave in diffusion systems
  • Presentation (Download PDF)

November 23. Lecture 12.. Distribution of pulses, fronts and waves. Models of propagation of the nervous impulse. Automological processes and heart arrhythmias

Distribution of pulses, fronts and waves. Model spreading the front of the wave of Petrovsky-Kolmogorov-Piskunova-Fisher. The interaction of breeding and diffusion processes. Local reproduction functions. Automatic variable. Distribution of ambrosisian leaf.

Models of propagation of the nervous impulse. Automological processes and heart arrhythmias. The propagation of the nervous impulse. Experiments and model Herkkun-Huxley. The reduced model Fitzhu-Nagumo. Excitable element of the local system. Subject and outgoing arousal. Running pulses. Detailed models of cardiocytes. Axiomatic models of excitable environment. Automological processes and heart arrhythmias.

  • Program: Live Systems and Active Kinetic Environments
  • Program: models of interaction of species
  • Program: Models in Microbiology
  • Tutorial: distributed biological systems. Diffusion Equation
  • Tutorial: solving the diffusion equation. Stability of homogeneous stationary states
  • Textbook:

In this textbook, the main modern mathematical models for analyzing biophysical processes, living systems in ecology are well represented. The book consists of three sections that describe basic models in biophysics, dynamics of populations and ecology, and the corresponding descriptive examples are given, calculation methods and statistical data are presented. At the moment, some of the statistical data are outdated. However, this does not significantly affect the learning process with mathematical modeling of biological processes, and the changes that occurred can be taken into account by teachers.

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