scholarly journals The effect of domain growth on spatial correlations

2016 ◽  
Author(s):  
Robert JH Ross ◽  
Christian A Yates ◽  
Ruth E Baker
Keyword(s):  
Differential Equations ◽  
Cell Motility ◽  
Ordinary Differential Equations ◽  
Cell Movement ◽  
Domain Growth ◽  
Continuum Approximation ◽  
Spatial Correlations ◽  
One Dimensional ◽  
Growing Domain ◽  
Pair Density

Mathematical models describing cell movement and proliferation are important research tools for the understanding of many biological processes. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations between agent locations in a continuum approximation of a one-dimensional lattice-based model of cell motility and proliferation. This is important as the inclusion of spatial correlations in continuum models of cell motility and proliferation without domain growth has previously been shown to be essential for their accuracy in certain scenarios. We include the effect of spatial correlations by deriving a system of ordinary differential equations that describe the expected evolution of individual and pair density functions for agents on a growing domain. We then demonstrate how to simplify this system of ordinary differential equations by using an appropriate approximation. This simplification allows domain growth to be included in models describing the evolution of spatial correlations between agents in a tractable manner.

The Poincaré-Cartan forms of one-dimensional variational integrals

2020 ◽  
Vol 70 (6) ◽  
pp. 1381-1412
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Differential Equations ◽  
Variational Problem ◽  
Ordinary Differential Equations ◽  
Differential Forms ◽  
Point Of View ◽  
Full Generality ◽  
One Dimensional ◽  
Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

scholarly journals Exact solutions of linear reaction-diffusion processes on a uniformly growing domain: Criteria for successful colonization

2014 ◽  
Author(s):  
Matthew J Simpson
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Domain Growth ◽  
Initial Profile ◽  
One Dimensional ◽  
Growing Domain ◽  
Linear Reaction ◽  
Successful Colonization

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction?diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction?diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction?diffusion process on $0 \le x \le L(t)$, where $L(t)$ is the length of the growing domain. Comparing our exact solutions with numerical approximations confirms the accuracy of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near $x=0$, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the boundary where $x = L(t)$.

scholarly journals Quotients of Euler Equations on Space Curves

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 186
Author(s):  
Anna Duyunova ◽  
Valentin Lychagin ◽  
Sergey Tychkov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Euler Equations ◽  
Gas Flow ◽  
Euler System ◽  
Partial Differential ◽  
One Dimensional ◽  
Space Curves ◽  
Ode System

Quotients of partial differential equations are discussed. The quotient equation for the Euler system describing a one-dimensional gas flow on a space curve is found. An example of using the quotient to solve the Euler system is given. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations (ODEs). Possible solutions of the ODE system are discussed.

scholarly journals Chaos in an anharmonic oscillator

1995 ◽  
Vol 37 (2) ◽  
pp. 186-207 ◽  
Author(s):  
J. R. Christie ◽  
K. Gopalsamy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous System ◽  
Anharmonic Oscillator ◽  
Homoclinic Orbits ◽  
Chaotic Behaviour ◽  
Differential Systems ◽  
Melnikov's Method ◽  
One Dimensional ◽  
Melnikov’S Method

AbstractUsing Melnikov's method, the existence of chaotic behaviour in the sense of Smale in a particular time-periodically perturbed planar autonomous system of ordinary differential equations is established. Examples of planar autonomous differential systems with homoclinic orbits are provided, and an application to the dynamics of a one-dimensional anharmonic oscillator is given.

scholarly journals Automorphisms of Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-32 ◽  
Author(s):  
Václav Tryhuk ◽  
Veronika Chrastinová
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Local Theory ◽  
Full Generality ◽  
One Dimensional ◽  
Underdetermined Systems ◽  
Dependent Variables ◽  
Variational Integrals ◽  
Independent Variable ◽  
Internal Symmetries

The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.

CLUSTERING AND SYNCHRONIZATION IN A ONE-DIMENSIONAL MODEL FOR THE CA3 REGION OF THE HIPPOCAMPUS

2002 ◽  
Vol 12 (11) ◽  
pp. 2641-2653 ◽  
Author(s):  
N. MONTEJO ◽  
M. N. LORENZO ◽  
V. PÉREZ-MUÑUZURI ◽  
V. PÉREZ-VILLAR
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spatiotemporal Patterns ◽  
Transmission Time ◽  
Dimensional Model ◽  
One Dimensional ◽  
Dimensional Array ◽  
Ca3 Region

The behavior of a system of coupled ordinary differential equations is studied in order to characterize the CA3 region of the hippocampus. Clustering and synchronization behavior in a one-dimensional array of cells modeled by a modified Morris–Lecar model is analyzed in terms of a time delay included in the model. The random formation of phase dislocations whose number increases with the time delay seems to be responsible for complex spatiotemporal patterns that have been observed. Alterations to the transmission time between cells have been simulated by adding some noise to the system.

scholarly journals Ordinary differential equations for the dynamic characteristics of heating boilers

2018 ◽  
Vol 194 ◽  
pp. 01024
Author(s):  
Sergei Khaustov ◽  
Olga Guk ◽  
Igor Razov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Control Systems ◽  
Dynamic Characteristics ◽  
Solid Fuel ◽  
Computational Time ◽  
One Dimensional ◽  
Automatic Control Systems ◽  
Set Up

The paper presents ordinary differential equations for the dynamic characteristics of a solid fuel boiler that can be combined into one-dimensional nonstationary mathematical model for simulating long-term dynamics of a solid fuel boiler. This model requires less computational time for a qualitative simulation of boiler’s operation than known CFD-solutions. The long-term dynamic model presented can help to determine annual costs, to set up automatic control systems and to detect dangerous deviations in the project.

scholarly journals History dependence and the continuum approximation breakdown: the impact of domain growth on Turing’s instability

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160744 ◽  
Author(s):  
Václav Klika ◽  
Eamonn A. Gaffney
Keyword(s):  
Domain Size ◽  
Turing Instability ◽  
Domain Growth ◽  
Continuum Approximation ◽  
Growing Domain ◽  
Spatial Frequencies ◽  
History Of ◽  
The Stability ◽  
The Impact ◽  
The Continuum

A diffusively driven instability has been hypothesized as a mechanism to drive spatial self-organization in biological systems since the seminal work of Turing. Such systems are often considered on a growing domain, but traditional theoretical studies have only treated the domain size as a bifurcation parameter, neglecting the system non-autonomy. More recently, the conditions for a diffusively driven instability on a growing domain have been determined under stringent conditions, including slow growth, a restriction on the temporal interval over which the prospect of an instability can be considered and a neglect of the impact that time evolution has on the stability properties of the homogeneous reference state from which heterogeneity emerges. Here, we firstly relax this latter assumption and observe that the conditions for the Turing instability are much more complex and depend on the history of the system in general. We proceed to relax all the above constraints, making analytical progress by focusing on specific examples. With faster growth, instabilities can grow transiently and decay, making the prediction of a prospective Turing instability much more difficult. In addition, arbitrarily high spatial frequencies can destabilize, in which case the continuum approximation is predicted to break down.

scholarly journals ATTAINABLE ELECTRON ENERGY IN DIODE WITH PLASMA CATHODE AT THE GIVEN VOLTAGE

2019 ◽  
pp. 100-105
Author(s):  
V. Ostroushko ◽  
A. Pashchenko ◽  
I. Pashchenko
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Electron Energy ◽  
Time Dependent ◽  
Forward Motion ◽  
Plasma Cathode ◽  
One Dimensional ◽  
The Given

It is considered an acceleration of electrons in a one-dimensional interval, part of which is filled by ions initially compensated by electrons. For the stages of forward motion and backward motion of a part of electrons, the problem is reduced to a numerical solution of ordinary differential equations for some set of time-dependent quantities. The ratio of attainable energy to the energy corresponding to the voltage is maximum and near to 1.87475 for relatively small values of the width of the space without ions when applying a certain voltage, which, as the width is reduced, has to be reduced as a width cube.

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