Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations

Abstract

Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally develop in bistable and excitable systems, but may also appear far beyond Hopf and Turing bifurcations. The global behavior of domain patterns strongly depends on the fronts' inner structures. In this paper we study a symmetry breaking front bifurcation expected to occur in a wide class of reaction-diffusion systems, and the effects it has on pattern formation and pattern dynamics. We extend previous works on this type of front bifurcation and clarify the relations among them. We show that the appearance of front multiplicity beyond the bifurcation point allows the formation of persistent patterns rather than transient ones. In a different parameter regime, we find that the front bifurcation outlines a transition from oscillating (or breathing) patterns to traveling ones. Near a boundary we find that fronts beyond the bifurcation can reflect, while those below it either bind to the boundary or disappear.-2-

Gradual regime shifts in spatially extended ecosystems

“…Pattern dynamics associated with the bistability of uniform states has been the subject of intense study, both in potential and non-potential systems (Fife 1988;Hagberg and Meron 1994c;Pismen 2006;Cross and Greenside 2009). The building blocks of these patterns are fronts that separate domains of different states, as Fig.…”

“…The significance of local domain formation is that it may induce front dynamics (Hagberg and Meron 1994c), i.e., dynamics of the transition zones that separate adjacent domains of the two alternative states. The dynamics may act to reduce the newly formed domains, or to extend them.…”

Order parameter equations for front transitions: Planar and circular fronts

“…These motions can be driven by curvature [1,2], front interactions [3,4], convective instabilities [5,6], and external fields [7][8][9]. In some cases fronts reverse their direction of propagation, as for example in breathing pulses [10][11][12][13][14][15][16][17][18], where the reversal is periodic in time, and nucleation of spiral-vortex pairs, where the reversal is local along the extended front line [19][20][21][22][23].…”

Nonlocal Boundary Dynamics of Traveling Spots in a Reaction-Diffusion System

“…Existence of traveling spots in sharp-interface models is indicated by translational instability of a stationary spot [11]. This instability is a manifestation of a general phenomenon of parity breaking (Ising-Bloch) bifurcation [15,16] which takes a single stable front into a pair of counter-propagating fronts forming the front and the back of a traveling pulse. Numerical simulations, however, failed to produce stable traveling spots in the basic activator-inhibitor model, and the tendency of moving spots to spread out laterally had to be suppressed either by global interaction in a finite region [11] or by adding an extra inhibitor with specially designed properties [17].…”