1,084 research outputs found
Spikes for the Gierer-Meinhardt system with discontinuous diffusion coefficients
The original publication is available at http://www.springerlink.com/content/vw2m382276u4g814/We rigorously prove results on spiky patterns for the Gierer-Meinhardt system with a jump discontinuity in the diffusion coefficient of the inhibitor. Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the
position of the spike. In particular we show
that the spike is located in one-and-only-one
of the two subintervals created by the jump
discontinuity of the inhibitor diffusivity.
This localisation principle for a spike
is a new effect which does not occur for
homogeneous diffusion coefficients. Further, we show that this interior spike is stable.
Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.
To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov-Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work.
Finally, we confirm our results by numerical
computations for the dynamical behavior of the system. We observe a moving spike which
converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity
Heterogeneity induces spatiotemporal oscillations in reaction-diffusions systems
We report on a novel instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable, or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity, and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity
Using mathematical models to help understand biological pattern formation
One of the characteristics of biological systems is their ability to produce and sustain spatial and spatio-temporal pattern. Elucidating the underlying mechanisms responsible for this phenomenon has been the goal of much experimental and theoretical research. This paper illustrates this area of research by presenting some of the mathematical models that have been proposed to account for pattern formation in biology and considering their implications.To cite this article: P.K. Maini, C. R. Biologies 327 (2004)
On the Gierer-Meinhardt system with precursors
We consider the Gierer-Meinhardt system with a for the activator. Such an equation exhibits a typical Turing bifurcation of the second kind, i.e., homogeneous uniform steady states do not exist in the system. We establish the existence and stability of N-peaked steady-states in terms of the precursor and the inhibitor diffusivity. It is shown that the precursor plays an essential role for both existence and stability of spiky patterns. In particular, we show that precursors can give rise to
instability. This is a new effect which is not present in the homogeneous case
Stationary localized structures and the effect of the delayed feedback in the Brusselator model
The Brusselator reaction-diffusion model is a paradigm for the understanding
of dissipative structures in systems out of equilibrium. In the first part of
this paper, we investigate the formation of stationary localized structures in
the Brusselator model. By using numerical continuation methods in two spatial
dimensions, we establish a bifurcation diagram showing the emergence of
localized spots. We characterize the transition from a single spot to an
extended pattern in the form of squares. In the second part, we incorporate
delayed feedback control and show that delayed feedback can induce a
spontaneous motion of both localized and periodic dissipative structures. We
characterize this motion by estimating the threshold and the velocity of the
moving dissipative structures.Comment: 18 pages, 11 figure
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