title: Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ¸ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹ Ð´Ð»Ñ ÑÐ¾Ð¿ÑÐ¾Ð²Ð¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÐºÑÑÑÐ° Â«ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð² Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸Ð¸Â»
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Development of a library of mathematical models to accompany the course "Mathematical models in biology"
Creators: name: ÐÑÐº, ÐÐ½Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2024
Subjects: Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐºÐ¾Ð¼Ð¿ÑÑÑÐµÑÐ½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÑÑÐµÐ´Ñ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ
Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ° Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹
Ð½ÐµÐ¿ÑÐµÑÑÐ²Ð½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸
Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸
Ð¸Ð·Ð¾Ð»Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸
Ð¼Ð½Ð¾Ð³Ð¾ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½ÑÐ½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸
Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼
mathematical modeling
computer modeling
modeling environments
model library
continuous models
discrete models
isolated models
multicomponent models
models of biological systems

description: Ð¦ÐµÐ»ÑÑ Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ ÑÐ²Ð»ÑÐµÑÑÑ ÑÐ¾Ð·Ð´Ð°Ð½Ð¸Ðµ Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ¸ Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹ Ð´Ð»Ñ ÑÐ¾Ð¿ÑÐ¾Ð²Ð¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÐºÑÑÑÐ° Â«ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð² Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸Ð¸Â» Ð. Ð®. Ð Ð¸Ð·Ð½Ð¸ÑÐµÐ½ÐºÐ¾ Ð² ÑÑÐµÐ´Ðµ AnyDynamics. ÐÑÐµÐ´Ð¼ÐµÑ ÑÐ°Ð±Ð¾ÑÑ â Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼. Ð ÑÐ¾Ð´Ðµ Ð²ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð¸Ñ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ»Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ñ ÑÐ»ÐµÐ´ÑÑÑÐ¸Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸: Ð¼Ð¾Ð´ÐµÐ»Ñ ÑÐºÑÐ¿Ð¾Ð½ÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÐ¾ÑÑÐ° (ÐÐ°Ð»ÑÑÑÑ), Ð¼Ð¾Ð´ÐµÐ»Ñ Ð»Ð¾Ð³Ð¸ÑÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÐ¾ÑÑÐ° (Ð¤ÐµÑÑÑÐ»ÑÑÑ), Ð¼Ð¾Ð´ÐµÐ»Ñ Ð¿Ð¾Ð¿ÑÐ»ÑÑÐ¸Ð¸ Ñ Ð½Ð°Ð¸Ð¼ÐµÐ½ÑÑÐµÐ¹ ÐºÑÐ¸ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾ÑÑÑÑ, Ð´Ð¸ÑÐºÑÐµÑÐ½Ð°Ñ Ð»Ð¾Ð³Ð¸ÑÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ Ð¿Ð¾Ð¿ÑÐ»ÑÑÐ¸Ð¸, Ð¼Ð¾Ð´ÐµÐ»Ñ ÐÐ¾ÑÐºÐ¸, Ð¼Ð¾Ð´ÐµÐ»Ñ ÐÐ¾Ð»ÑÑÐµÑÑÐ°, Ð¼Ð¾Ð´ÐµÐ»Ñ ÐºÐ¾Ð½ÐºÑÑÐµÐ½ÑÐ¸Ð¸ Ð´Ð²ÑÑ Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð²Ð¸Ð´Ð¾Ð², Ð¼Ð¾Ð´ÐµÐ»Ñ Ð³ÐµÐ½ÐµÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÑÐ¸Ð³Ð³ÐµÑÐ° ÐÐ°ÐºÐ¾Ð±Ð° Ð¸ ÐÐ¾Ð½Ð¾, ÐÑÑÑÑÐµÐ»ÑÑÐ¾Ñ. ÐÐ»Ñ ÐºÐ°Ð¶Ð´Ð¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð±ÑÐ»Ð¾ ÑÐ¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð¾ Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ðµ, ÐºÐ¾ÑÐ¾ÑÐ¾Ðµ Ð²ÐºÐ»ÑÑÐ°ÐµÑ Ð² ÑÐµÐ±Ñ ÑÐµÐ¾ÑÐµÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ðµ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸, Ð¸Ð½ÑÑÑÑÐºÑÐ¸Ñ Ð¿Ð¾ ÑÐ¾Ð·Ð´Ð°Ð½Ð¸Ñ ÐºÐ¾Ð¼Ð¿ÑÑÑÐµÑÐ½Ð¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð² ÑÑÐµÐ´Ðµ AnyDynamics, ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð²ÐºÐ»ÑÑÐ°ÑÑ Ð² ÑÐµÐ±Ñ Ð³ÑÐ°ÑÐ¸ÐºÐ¸ Ð¸ ÑÐ°Ð·Ð¾Ð²ÑÐµ Ð¿Ð¾ÑÑÑÐµÑÑ. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð² ÑÑÐµÐ´Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ AnyDynamics Ð±ÑÐ»Ð° ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð° Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ° Ð±Ð¸Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð¼Ð¾Ð´ÐµÐ»ÐµÐ¹, ÐºÐ¾ÑÐ¾ÑÐ°Ñ Ð¸Ð»Ð»ÑÑÑÑÐ¸ÑÑÐµÑ Ð¼Ð¾Ð´ÐµÐ»Ð¸, ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼ÑÐµ Ð² ÐºÑÑÑÐµ Ð. Ð®. Ð Ð¸Ð·Ð½Ð¸ÑÐµÐ½ÐºÐ¾. ÐÐ¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ° Ð²ÐºÐ»ÑÑÐ°ÐµÑ Ð² ÑÐµÐ±Ñ Ð¸Ð·Ð¾Ð»Ð¸ÑÐ¾Ð²Ð°Ð½Ð½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ (Ð´Ð¸ÑÐºÑÐµÑÐ½ÑÐµ Ð¸ Ð½ÐµÐ¿ÑÐµÑÑÐ²Ð½ÑÐµ), Ð¼Ð½Ð¾Ð³Ð¾ÐºÐ¾Ð¼Ð¿Ð¾Ð½ÐµÐ½ÑÐ½ÑÐµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ñ Ð²ÑÐ¾Ð´Ð°Ð¼Ð¸-Ð²ÑÑÐ¾Ð´Ð°Ð¼Ð¸.
description: The aim of this work is to create a library of models to accompany the course "Mathematical models in Biology" by G. Y. Riznichenko in the AnyDynamics environment. The subject of the work is mathematical models of biological systems. In the course of the work, the following models were implemented: exponential growth model (Malthus), logistic growth model (Ferhulst), population model with the smallest critical number, discrete logistic population model, Lotka model, Volterra model, competition model of two biological species, Jacob and Mono genetic trigger model, Brusselator. A description was compiled for each model, which includes a theoretical description of the mathematical model, instructions for creating a computer model in the AnyDynamics environment, simulation results, which include graphs and phase portraits. As a result, a library of biological models was implemented in the AnyDynamics modeling environment, which illustrates the models studied in the course of G. Y. Riznichenko. The library includes isolated models (discrete and continuous), multicomponent models with inputs and outputs.