Problems with Different Time Scales for Ordinary Differential Equations

1979 ◽  
Vol 16 (6) ◽  
pp. 980-998 ◽  
Author(s):  
Heinz-Otto Kreiss
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Scales ◽  
Different Time Scales

EMERGING PROPERTIES IN POPULATION DYNAMICS WITH DIFFERENT TIME SCALES

1995 ◽  
Vol 03 (02) ◽  
pp. 591-602 ◽  
Author(s):  
PIERRE AUGER ◽  
JEAN-CHRISTOPHE POGGIALE
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Time Scale ◽  
Population Level ◽  
Nonlinear Process ◽  
Linear Process ◽  
Fast Time ◽  
Slow Part ◽  
Spatial Patches ◽  
Different Time Scales

The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.

scholarly journals A Survey of Recent Results for the Generalizations of Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Daniel C. Biles ◽  
Márcia Federson ◽  
Rodrigo López Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Scales ◽  
Review Paper ◽  
Different Types ◽  
Discontinuous Equations ◽  
Generalized Ordinary Differential Equations

This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation.

Dynamics of Consumer-Resource Systems with Discrete Reproduction

2013 ◽  
Author(s):  
André M. de Roos ◽  
Lennart Persson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Time Scales ◽  
Discrete Event ◽  
Growth Season ◽  
Consumer Resource ◽  
Resource Systems

This chapter considers models where processes like foraging, metabolism, and mortality are continuous, whereas reproduction is assumed to take place as a discrete event at the start of the growth season. The assumption of discrete reproduction is relevant for many organisms living in seasonal environments (winter/summer, dry/wet seasons). The use of several time scales means that the analysis of the dynamics in this chapter will be restricted to simulations using the Escalator Boxcar Train (EBT) framework. The EBT is specifically designed to handle the numerical integration of the equations that occur in physiologically structured models using ordinary differential equations and is particularly well suited for systems with discrete reproduction.

Theory and applications of first-order systems of Stieltjes differential equations

2017 ◽  
Vol 6 (1) ◽  
pp. 13-36 ◽  
Author(s):  
Marlène Frigon ◽  
Rodrigo López Pouso
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Scales ◽  
Existence And Uniqueness ◽  
Basic Theory ◽  
Dynamic Equations ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
Systems Of Differential Equations ◽  
Set Up

AbstractWe set up the basic theory of existence and uniqueness of solutions for systems of differential equations with usual derivatives replaced by Stieltjes derivatives. This type of equations contains as particular cases dynamic equations on time scales and impulsive ordinary differential equations.

Dynamic Modeling of Core Processes

2014 ◽  
Author(s):  
Domitilla Del Vecchio ◽  
Richard M. Murray
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Time Scales ◽  
Dynamic Modeling ◽  
Feedback Systems ◽  
Biological Mechanisms ◽  
Dynamical Models ◽  
Deterministic Models ◽  
Biomolecular Systems ◽  
Modeling Formalisms

This chapter describes basic biological mechanisms in a way that can be represented by simple dynamical models. It begins with a discussion of the basic modeling formalisms that will be utilized to model biomolecular feedback systems. The focus in this chapter (as well as the next) is on deterministic models using ordinary differential equations. Here, the chapter proceeds to study a number of core processes within the cell, providing different model-based descriptions of the dynamics that will be used in later chapters to analyze and design biomolecular systems. In this chapter, emphasis is placed on dynamics with time scales measured in seconds to hours and mean behavior averaged across a large number of molecules.

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