Construction of Dulac functions for mathematical models in population biology

2015 ◽  
Vol 08 (03) ◽  
pp. 1550035
Author(s):  
Osvaldo Osuna ◽  
Cruz Vargas-De-León
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Population Biology ◽  
Dulac Function

In this paper, we present a method for constructing a Dulac function for mathematical models in population biology, in the form of systems of ordinary differential equations in the plane.

scholarly journals On attracting sets in artificial networks: cross activation

2018 ◽  
Vol 16 ◽  
pp. 01005
Author(s):  
Felix Sadyrbaev
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Critical Points ◽  
Regulatory Networks ◽  
Main Diagonal ◽  
Sigmoidal Function ◽  
Genomic Regulatory Networks ◽  
Over Time

Mathematical models of artificial networks can be formulated in terms of dynamical systems describing the behaviour of a network over time. The interrelation between nodes (elements) of a network is encoded in the regulatory matrix. We consider a system of ordinary differential equations that describes in particular also genomic regulatory networks (GRN) and contains a sigmoidal function. The results are presented on attractors of such systems for a particular case of cross activation. The regulatory matrix is then of particular form consisting of unit entries everywhere except the main diagonal. We show that such a system can have not more than three critical points. At least n–1 eigenvalues corresponding to any of the critical points are negative. An example for a particular choice of sigmoidal function is considered.

Qualitative mathematical models of endocrine systems

1983 ◽  
Vol 245 (4) ◽  
pp. R473-R477 ◽  
Author(s):  
W. R. Smith
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Endocrine System ◽  
Nonlinear Ordinary Differential Equations ◽  
Dynamical Models ◽  
Rates Of Change ◽  
Reproductive Endocrine

Some qualitative dynamical models of endocrine systems are considered and analyzed, with the reproductive endocrine system as an example. The models considered are systems of nonlinear ordinary differential equations describing the rates of change of the hormonal concentrations with time. This type of general approach, which requires only the incorporation of the basic qualitative features of the interactions present in the underlying system into the model, is a potentially powerful tool for elucidating possible mechanisms for observed qualitative patterns of hormonal dynamics.

ESTIMATES OF SOLUTIONS OF IMPULSIVE PARABOLIC EQUATIONS AND APPLICATION

2008 ◽  
Vol 01 (02) ◽  
pp. 257-266
Author(s):  
GUOHUA SONG
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Population Biology ◽  
Model Problem ◽  
General Boundary ◽  
General Boundary Condition ◽  
Estimates Of Solutions

This paper is concerned with the estimates of solutions for an impulsive parabolic equations under general boundary condition. We prove that the solutions of impulsive parabolic equations can be controlled and estimated by the solutions of dominating impulsive ordinary differential equations. We also apply the above results to a model problem arising from population biology.

scholarly journals MATHEMATICAL MODELLING OF THE ANALYTICAL AND NUMERICAL SOLUTION OF NON-STATIONARY INITIAL-VALUE PROBLEMS BY MAPLE SOFTWARE.

2019 ◽  
pp. 87
Author(s):  
Iļja Sučkovs ◽  
Aleksandrs Pikurs ◽  
Ilmārs Kangro
Keyword(s):  
Differential Equations ◽  
Mathematical Modelling ◽  
Mathematical Models ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
General Information ◽  
Radioactive Substance ◽  
Initial Value

With the passage of time and the development of technology, humanity is exploring new unknown problems that require complex analytical and numerical mathematical solutions. Due to their complexity differential equations are often used for this purpose. The aim of this work is to solve mathematical models of initial value problems of ordinary differential equations using the analitical method and numerical solution using MAPLE software. Also authors have provided general information about differential equations and diferent ways how they can be solved. As a result have been created two mathematical models which describe process of Determination of the cooling time of a shot animal and decomposition of the radioactive substance. Similar methods are also used to determine the age of objects as well

scholarly journals THEORETICAL STUDY OF THE PROBLEM OF DEVELOPING A LOADER-LOADER WITH A TRANSFORMABLE WORKING BODY FOR THE MAINTENANCE, REPAIR OF HIGH ALTITUDE ROADS

2019 ◽  
pp. 586-593
Author(s):  
K. Isakov ◽  
K.T. Osmonov ◽  
T. Toktakunov
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Mathematical Models ◽  
High Altitude ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Theoretical Study ◽  
Initial Conditions ◽  
Multi Stage ◽  
Functional Mechanisms

The article examines the proposed modes of work of the project being developed to create a bulldozer-loader, adapted for more efficient maintenance, repair and construction of high-altitude and other roads. Mathematical models are proposed that describe multi-stage successive movements of one or more functional mechanisms corresponding to different modes of work. Nonlinear systems of ordinary differential equations of the second order with initial conditions, for numerical integration of which the Adams method is used, are presented.

scholarly journals Uncertainty Quantification in Simulations of Epidemics Using Polynomial Chaos

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
F. Santonja ◽  
B. Chen-Charpentier
Keyword(s):  
Sensitivity Analysis ◽  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Polynomial Chaos ◽  
Random Coefficients ◽  
System Of Differential Equations ◽  
Large Variability ◽  
Obesity Epidemic ◽  
Transmission Parameters

Mathematical models based on ordinary differential equations are a useful tool to study the processes involved in epidemiology. Many models consider that the parameters are deterministic variables. But in practice, the transmission parameters present large variability and it is not possible to determine them exactly, and it is necessary to introduce randomness. In this paper, we present an application of the polynomial chaos approach to epidemiological mathematical models based on ordinary differential equations with random coefficients. Taking into account the variability of the transmission parameters of the model, this approach allows us to obtain an auxiliary system of differential equations, which is then integrated numerically to obtain the first-and the second-order moments of the output stochastic processes. A sensitivity analysis based on the polynomial chaos approach is also performed to determine which parameters have the greatest influence on the results. As an example, we will apply the approach to an obesity epidemic model.

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.

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