scholarly journals Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

2005 ◽  
Author(s):  
William A. Brock ◽  
Anastasios Xepapadeas
Keyword(s):  
Optimal Control ◽  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Ecological Models

scholarly journals Ecological Models, Permanence and Spatial Heterogeneity

1996 ◽  
Vol 26 (1) ◽  
pp. 1-35 ◽  
Author(s):  
R.S. Cantrell ◽  
C. Cosner ◽  
V. Hutson
Keyword(s):  
Spatial Heterogeneity ◽  
Ecological Models

scholarly journals HIERARCHICAL MODELS FOR SPATIAL PATTERN FORMATION IN BIOLOGY

1995 ◽  
Vol 03 (04) ◽  
pp. 987-997 ◽  
Author(s):  
P. K. MAINI
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Spatial Pattern ◽  
Diffusion Model ◽  
Hierarchical Models ◽  
Reaction Diffusion ◽  
Skeletal Patterning ◽  
Reaction Diffusion Model ◽  
Spatial Pattern Formation ◽  
Set Up

We review some recent work investigating a hierarchy of patterning processes in which a reaction-diffusion model forms the top level. In one such hierarchy, it is assumed that the boundary is differentiated, and it is shown that this can greatly enhance the robustness of the patterns subsequently formed by the reaction-diffusion model. In the second, a spatial heterogeneity in background environment is first set-up by a simple gradient model. The resulting patterns produced by the reaction-diffusion system may be isolated to specific parts of the domain. The application of such hierarchical models to skeletal patterning in the tetrapod limb is considered.

DIFFUSION-DRIVEN INSTABILITIES AND SPATIO-TEMPORAL PATTERNS IN AN AQUATIC PREDATOR–PREY SYSTEM WITH BEDDINGTON–DEANGELIS TYPE FUNCTIONAL RESPONSE

2011 ◽  
Vol 21 (03) ◽  
pp. 663-684 ◽  
Author(s):  
RANJIT KUMAR UPADHYAY ◽  
N. K. THAKUR ◽  
V. RAI
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Collective Motion ◽  
Temporal Dynamics ◽  
Fish Predation ◽  
Building Blocks ◽  
Aquatic System ◽  
Predator Species ◽  
Predator Prey ◽  
Spatio Temporal

Predator–prey communities are building blocks of an ecosystem. Feeding rates reflect interference between predators in several situations, e.g. when predators form a dense colony or perform collective motion in a school, encounter prey in a region of limited size, etc. We perform spatio-temporal dynamics and pattern formation in a model aquatic system in both homogeneous and heterogeneous environments. Zooplanktons are predated by fishes and interfere with individuals of their own community. Numerical simulations are carried out to explore Turing and non-Turing spatial patterns. We also examine the effect of spatial heterogeneity on the spatio-temporal dynamics of the phytoplankton–zooplankton system. The phytoplankton specific growth rate is assumed to be a linear function of the depth of the water body. It is found that the spatio-temporal dynamics of an aquatic system is governed by three important factors: (i) intensity of interference between the zooplankton, (ii) rate of fish predation and (iii) the spatial heterogeneity. In an homogeneous environment, the temporal dynamics of prey and predator species are drastically different. While prey species density evolves chaotically, predator densities execute a regular motion irrespective of the intensity of fish predation. When the spatial heterogeneity is included, the two species oscillate in unison. It has been found that the instability observed in the model aquatic system is diffusion driven and fish predation acts as a regularizing factor. We also observed that spatial heterogeneity stabilizes the system. The idea contained in the paper provides a better understanding of the pattern formation in aquatic systems.

scholarly journals Fast solvers for optimal control problems from pattern formation

2016 ◽  
Vol 304 ◽  
pp. 27-45 ◽  
Author(s):  
Martin Stoll ◽  
John W. Pearson ◽  
Philip K. Maini
Keyword(s):  
Optimal Control ◽  
Pattern Formation ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Fast Solvers

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Spatio-temporal pattern formation, fractals, and chaos in conceptual ecological models as applied to coupled plankton-fish dynamics

2002 ◽  
Vol 172 (1) ◽  
pp. 31 ◽  
Author(s):  
Aleksandr B. Medvinskii ◽  
Sergei V. Petrovskii ◽  
Irina A. Tikhonova ◽  
D.A. Tikhonov ◽  
B.L. Li ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Temporal Pattern ◽  
Ecological Models ◽  
Spatio Temporal

scholarly journals Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

2019 ◽  
Vol 24 (2) ◽  
pp. 467-486
Author(s):  
Qingyan Shi ◽  
◽  
Junping Shi ◽  
Yongli Song ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Logistic Model

Spatio-temporal pattern formation, fractals, and chaos in conceptual ecological models as applied to coupled plankton-fish dynamics

2002 ◽  
Vol 45 (1) ◽  
pp. 27-57 ◽  
Author(s):  
Aleksandr B Medvinskii ◽  
Sergei V Petrovskii ◽  
Irina A Tikhonova ◽  
D A Tikhonov ◽  
B.-L Li ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Temporal Pattern ◽  
Ecological Models ◽  
Spatio Temporal
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