STRUCTURED MATHEMATICAL MODELS FOR DYNAMICS OF MICROBIAL GROWTH: FIRST ORDER AND DISCRETE TIME DELAYS

2000 ◽  
pp. 269-277
Author(s):  
PETER GÖTZ
Keyword(s):  
Mathematical Models ◽  
Discrete Time ◽  
Time Delays ◽  
Microbial Growth ◽  
First Order

scholarly journals SIMULATION BY FIRST-ORDER LINEAR DIFFERENCE EQUATIONS

2021 ◽  
Author(s):  
Valentyna Lisovska ◽  
Tetyana Kudyk ◽  
Dariia Vasylieva
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Discrete Time ◽  
Difference Equations ◽  
Dynamic Models ◽  
Discrete Models ◽  
Economic Research ◽  
Systems Of Equations ◽  
Linear Difference Equations ◽  
First Order

The article considers economic and mathematical models and studies the socio-economic processes that develop over time, as well as mathematical models that describe them. These are dynamic models. All variables in dynamic models generally depend on the time that acts as an independent variable. In economic research, there are often problems in which variables acquire discrete numerical values. For example, at the end of the month, quarter, year, etc., production results are optimized; accrual of interest on the bank deposit at the end of the month, six months, at the end of the year. In addition, because computers operate only with numbers, so when using computer technology, all continuous processes are reduced to discrete. In this case, from differential equations that describe certain economic processes, we move to difference equations. There are dynamic models with continuous and discrete time, ie continuous and discrete models. Therefore, depending on the type of dynamics of the system under study, dynamic models can be divided into discrete and continuous. In discrete dynamic models, difference equations or systems of difference equations are used; differential equations or systems of differential equations are used in continuous dynamic models. In addition, in some cases there may be systems with mixed dynamics, then differential-equation equations are used to describe them. Difference equations and systems of equations are used successfully in modeling dynamic processes (in economics, banking, etc.). It is when the change of process occurs abruptly, or discretely, that it is convenient and expedient to apply difference equations and systems of equations. The theory of dynamical systems with discrete time, which arose as a result of building mathematical models of real economic and physical processes at the junction of the theory of difference equations and discrete random processes, is currently experiencing a period of rapid development and widespread use in various spheres of human life. In this paper, we investigate the following equations, as well as show their application to solve economic problems. In particular, discrete models described by first-order difference equations are considered. Considerable attention is paid to the analysis of specific models that are meaningful and widely used in economic theory, banking, etc.

Nonlinear Dynamics of Two Discrete-Time Versions of the Continuous-Time Brusselator Model

2019 ◽  
Vol 29 (10) ◽  
pp. 1950142
Author(s):  
Paulo C. Rech
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Periodic Structures ◽  
Parameter Spaces ◽  
First Order ◽  
Brusselator Model ◽  
Dynamical Behaviors

In this paper, we report results related with the dynamics of two discrete-time mathematical models, which are obtained from a same continuous-time Brusselator model consisting of two nonlinear first-order ordinary differential equations. Both discrete-time mathematical models are derived by integrating the set of ordinary differential equations, but using different methods. Such results are related, in each case, with parameter-spaces of the two-dimensional map which results from the respective discretization process. The parameter-spaces obtained using both maps are then compared, and we show that the occurrence of organized periodic structures embedded in a quasiperiodic region is verified in only one of the two cases. Bifurcation diagrams, Lyapunov exponents plots, and phase-space portraits are also used, to illustrate different dynamical behaviors in both discrete-time mathematical models.

scholarly journals Agreement of Networks of Discrete-Time Agents with Mixed Dynamics and Time Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Xue Li ◽  
Xueer Chen ◽  
Yingchun Xie
Keyword(s):  
Discrete Time ◽  
Spanning Tree ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Model Transformations ◽  
Nonnegative Matrices ◽  
First Order ◽  
Fixed Topology ◽  
Simulation Results ◽  
Theoretical Results

This paper considers agreement problems of networks of discrete-time agents with mixed dynamics and arbitrary bounded time delays, and networks consist of first-order agents and second-order agents. By using the properties of nonnegative matrices and model transformations, we derive sufficient conditions for stationary agreement of networks with bounded time delays. It is shown that stationary agreement can be achieved with arbitrary bounded time delays, if and only if fixed topology has a spanning tree and the union of the dynamically changing topologies has a spanning tree. Simulation results are also given to demonstrate the effectiveness of our theoretical results.

Synchronization for discrete-time complex networks with probabilistic time delays

2019 ◽  
Vol 525 ◽  
pp. 1088-1101 ◽  
Author(s):  
Ranran Cheng ◽  
Mingshu Peng ◽  
Jinchen Yu ◽  
Haifen Li
Keyword(s):  
Complex Networks ◽  
Discrete Time ◽  
Time Delays
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