Stripes, spots, or reversed spots in two-dimensional Turing systems
ABSTRACT Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.
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ABSTRACT: In this paper, first we consider the global dynamics of a ratio-dependent predator–prey model with density dependent death rate for the predator species. Analytical conditions for local bifurcation and numerical investigations to identify the global bifurcations help us to prepare a complete bifurcation diagram for the concerned model. All possible phase portraits related to the stability and instability of the coexisting equilibria are also presented which are helpful to understand the global behaviour of the system around the coexisting steady-states. Next we extend the temporal model to a spatiotemporal model by incorporating diffusion terms in order to investigate the varieties of stationary and non-stationary spatial patterns generated to understand the effect of random movement of both the species within their two-dimensional habitat. We present the analytical results for the existence of globally stable homogeneous steady-state and non-existence of non-constant stationary states. Turing bifurcation diagram is prepared in two dimensional parametric space along with the identification of various spatial patterns produced by the model for parameter values inside the Turing domain. Extensive numerical simulations are performed for better understanding of the spatiotemporal dynamics. This work is an attempt to make a bridge between the theoretical results for the spatiotemporal model of interacting population and the spatial patterns obtained through numerical simulations for parameters within Turing and Turing–Hopf domain.Ecological Complexity 06/2014; 21. DOI:10.1016/j.ecocom.2014.05.005 · 2.00 Impact Factor
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ABSTRACT: The development of multicellular organisms involves cells to decide their fate upon the action of biochemical signals. This decision is often spatiotemporally coordinated such that a spatial pattern arises. The dynamics that drive pattern formation usually involve genetic nonlinear interactions and positive feedback loops. These complex dynamics may enable multiple stable patterns for the same conditions. Under these circumstances, pattern formation in a developing tissue involves a selection process: why is a certain pattern formed and not another stable one? Herein we computationally address this issue in the context of the Notch signaling pathway. We characterize a dynamical mechanism for developmental selection of a specific pattern through spatiotemporal changes of the control parameters of the dynamics, in contrast to commonly studied situations in which initial conditions and noise determine which pattern is selected among multiple stable ones. This mechanism can be understood as a path along the parameter space driven by a sequence of biochemical signals. We characterize the selection process for three different scenarios of this dynamical mechanism that can take place during development: the signal either 1) acts in all the cells at the same time, 2) acts only within a cluster of cells, or 3) propagates along the tissue. We found that key elements for pattern selection are the destabilization of the initial pattern, the subsequent exploration of other patterns determined by the spatiotemporal symmetry of the parameter changes, and the speeds of the path compared to the timescales of the pattern formation process itself. Each scenario enables the selection of different types of patterns and creates these elements in distinct ways, resulting in different features. Our approach extends the concept of selection involved in cellular decision-making, usually applied to cell-autonomous decisions, to systems that collectively make decisions through cell-to-cell interactions. Copyright © 2015 Biophysical Society. Published by Elsevier Inc. All rights reserved.Biophysical Journal 03/2015; 108(6):1555-65. DOI:10.1016/j.bpj.2014.12.058 · 3.83 Impact Factor
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ABSTRACT: Species in the suborder Serpentes present a powerful model for understanding processes involved in visual signal design. Although vision is generally poor in snakes, they are often both predators and prey of visually oriented species. We examined how ecological and behavioral factors have driven the evolution of snake patterning using a phylogenetic comparative approach. The appearances of 171 species of Australian and North American snakes were classified using a reaction-diffusion model of pattern development, the parameters of which allow parametric quantification of various aspects of coloration. The main findings include associations between plain color and an active hunting strategy, longitudinal stripes and rapid escape speed, blotched patterns with ambush hunting, slow movement and pungent cloacal defense, and spotted patterns with close proximity to cover. Expected associations between bright colors, aggressive behavior, and venom potency were not observed. The mechanisms through which plain and longitudinally striped patterns might support camouflage during movement are discussed. The flicker-fusion hypothesis for transverse striped patterns being perceived as uniform color during movement is evaluated as theoretically possible but unlikely. Snake pattern evolution is generally phylogenetically conservative, but by sampling densely in a wide variety of snake lineages, we have demonstrated that similar pattern phenotypes have evolved repeatedly in response to similar ecological demands.Behavioral Ecology 08/2013; 24(5-5):1237-1250. DOI:10.1093/beheco/art058 · 3.16 Impact Factor