Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach

2009 ◽  
Vol 11 (40) ◽  
pp. 9210 ◽  
Author(s):  
Arik Yochelis ◽  
Moshe Sheintuch
Keyword(s):  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Nonlinear Selection ◽  
Selection Of

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

Stripes or spots? Nonlinear effects in bifurcation of reaction—diffusion equations on the square

1991 ◽  
Vol 434 (1891) ◽  
pp. 413-417 ◽  
Keyword(s):  
Pattern Formation ◽  
General Pattern ◽  
Reaction Diffusion ◽  
Nonlinear Effects ◽  
Neural Nets ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Linearized Model ◽  
Selection Of

Bifurcation to spatial patterns in a two-dimensional reaction—diffusion medium is considered. The selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model. The absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns. Examples that include neural nets and more general pattern formation equations are considered.

How Information-Mapping Patterns Determine Foraging Behaviour of a Honey Bee Colony

2002 ◽  
Vol 09 (02) ◽  
pp. 181-193 ◽  
Author(s):  
Valery Tereshko ◽  
Troy Lee
Keyword(s):  
Food Source ◽  
Foraging Behaviour ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Common Information ◽  
Bee Colony ◽  
Nectar Sources ◽  
Nectar Source ◽  
Honeybee Colony ◽  
Selection Of

We have developed a model of foraging behaviour of a honeybee colony based on reaction-diffusion equations and have studied how mapping the information about the explored environment to the hive determines this behaviour. The model utilizes two dominant components of colony's foraging behaviour — the recruitment to the located nectar sources and the abandonment of them. The recruitment is based upon positive feedback, i.e autocatalytic replication of information about the located source. If every potential forager in the hive, the onlooker, acquires information about all located sources, a common information niche is formed, which leads to the rapid selection of the most profitable nectar source. If the onlookers acquire information about some parts of the environment and slowly learn about the other parts, different information niches where individuals are associated mainly with a particular food source are formed, and the correspondent foraging trails coexist for longer periods. When selected nectar source becomes depleted, the foragers switch over to another, more profitable source. The faster the onlookers learn about the entire environment, the faster that switching occurs.

scholarly journals An Information Theory-Based Approach to Assessing Spatial Patterns in Complex Systems

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 182
Author(s):  
Tarsha Eason ◽  
Wen-Ching Chuang ◽  
Shana Sundstrom ◽  
Heriberto Cabezas
Keyword(s):  
Complex Systems ◽  
Fisher Information ◽  
Spatial Data ◽  
A Priori ◽  
Geographical Area ◽  
Spatial Dynamics ◽  
Geographical Patterns ◽  
Social Ecological Systems ◽  
Natural Means ◽  
Selection Of

Given the intensity and frequency of environmental change, the linked and cross-scale nature of social-ecological systems, and the proliferation of big data, methods that can help synthesize complex system behavior over a geographical area are of great value. Fisher information evaluates order in data and has been established as a robust and effective tool for capturing changes in system dynamics, including the detection of regimes and regime shifts. The methods developed to compute Fisher information can accommodate multivariate data of various types and requires no a priori decisions about system drivers, making it a unique and powerful tool. However, the approach has primarily been used to evaluate temporal patterns. In its sole application to spatial data, Fisher information successfully detected regimes in terrestrial and aquatic systems over transects. Although the selection of adjacently positioned sampling stations provided a natural means of ordering the data, such an approach limits the types of questions that can be answered in a spatial context. Here, we expand the approach to develop a method for more fully capturing spatial dynamics. The results reflect changes in the index that correspond with geographical patterns and demonstrate the utility of the method in uncovering hidden spatial trends in complex systems.

scholarly journals On the Interplay of Geometrical Shapes and the Analysis of a Dispersal Model for Pattern Formations

2019 ◽  
pp. 1-10
Author(s):  
Zakir Hossine ◽  
Md. Kamrujjaman
Keyword(s):  
Reaction Mechanism ◽  
Finite Difference Method ◽  
Reaction Diffusion ◽  
Diffusion Kinetics ◽  
Dispersal Model ◽  
Reaction Diffusion Model ◽  
Chemical Reaction Mechanism ◽  
Geometrical Shapes ◽  
Pattern Formations ◽  
Selection Of

A reaction-diffusion model is put forward which is capable of generating chemical maps whose concentration contours are similar to the patterns seen on the flanks of zebras, leopards and other mammals. Initially, this type of reaction diffusion kinetics model was introduced by Turing and later Murray applied it to animal coat patterns. Among many chemical reaction mechanism, we consider Schnackenberg reaction mechanism in details and show that the geometry and scale of the domain, the relevant part of the integument, during the time of laying down plays a crucial role in the structural patterns which result. Patterns which exhibit a limited randomness are obtained for a selection of geometries. Finally the system was solved numerically using finite difference method.

scholarly journals Spatial Dynamics of A Reaction-Diffusion Model with Distributed Delay

2013 ◽  
Vol 8 (3) ◽  
pp. 60-77 ◽  
Author(s):  
Y. Zhang ◽  
X.-Q. Zhao
Keyword(s):  
Diffusion Model ◽  
Spatial Dynamics ◽  
Distributed Delay ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model

scholarly journals Emergence of form in embryogenesis

2018 ◽  
Vol 15 (148) ◽  
pp. 20180454
Author(s):  
Murat Erkurt
Keyword(s):  
Mirror Symmetry ◽  
Rotational Symmetry ◽  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Symmetry Operator ◽  
Three Dimensions ◽  
Dirac Delta ◽  
Integrated Simulation ◽  
Delta Type ◽  
Finite State

The development of form in an embryo is the result of a series of topological and informational symmetry breakings. We introduce the vector–reaction–diffusion–drift (VRDD) system where the limit cycle of spatial dynamics is morphogen concentrations with Dirac delta-type distributions. This is fundamentally different from the Turing reaction–diffusion system, as VRDD generates system-wide broken symmetry. We developed ‘fundamental forms’ from spherical blastula with a single organizing axis (rotational symmetry), double axis (mirror symmetry) and triple axis (no symmetry operator in three dimensions). We then introduced dynamics for cell differentiation, where genetic regulatory states are modelled as a finite-state machine (FSM). The state switching of an FSM is based on local morphogen concentrations as epigenetic information that changes dynamically. We grow complicated forms hierarchically in spatial subdomains using the FSM model coupled with the VRDD system. Using our integrated simulation model with four layers (topological, physical, chemical and regulatory), we generated life-like forms such as hydra. Genotype–phenotype mapping was investigated with continuous and jump mutations. Our study can have applications in morphogenetic engineering, soft robotics and biomimetic design.

A periodic reaction–diffusion system modelling man–environment–man epidemics

2017 ◽  
Vol 10 (05) ◽  
pp. 1750074 ◽  
Author(s):  
Zhenguo Bai
Keyword(s):  
Wave Speed ◽  
Global Attractivity ◽  
Travelling Waves ◽  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Positive Periodic Solution ◽  
System Modelling ◽  
Periodic Reaction ◽  
Minimal Wave Speed ◽  
Disease Spreading

The purpose of this work is to study the spatial dynamics of a periodic reaction–diffusion epidemic model arising from the spread of oral–faecal transmitted diseases. We first show that the disease spreading speed is coincident with the minimal wave speed for monotone periodic travelling waves. Then we obtain a threshold result on the global attractivity of either zero or the positive periodic solution in a bounded spatial domain.

scholarly journals Spatial dynamics of a nonlocal and time-delayed reaction–diffusion system

2008 ◽  
Vol 245 (10) ◽  
pp. 2749-2770 ◽  
Author(s):  
Jian Fang ◽  
Junjie Wei ◽  
Xiao-Qiang Zhao
Keyword(s):  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Delayed Reaction

scholarly journals The Role of Subnivean Access in Winter Habitat Selection of Marten in Yellowstone National Park

1989 ◽  
Vol 13 ◽  
pp. 215-216
Author(s):  
Stuart Sherburne ◽  
John Bissonette
Keyword(s):  
Factor Analysis ◽  
Habitat Selection ◽  
National Park ◽  
Spatial Dynamics ◽  
Habitat Type ◽  
Winter Habitat ◽  
Proximate Factor ◽  
Selection Of ◽  
Habitat Variables

This research project has two primary goals. The first is to determine home range spatial dynamics of marten (Maxes americana) in Yellowstone relative to habitat type. Results of this analysis will aid in the understanding of marten habitat selection. The study's second goal is aimed at identifying the habitat variables that influence subnivean access. A proximate factor analysis of subnivean access behavior will be conducted to determine the components that make old growth suitable for marten. Results from both objectives will allow assessment of the effects of the 1988 fires in Yellowstone on marten habitat.

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