Competitively coupled maps and spatial pattern formation

Abstract

Spatial pattern formation is a key feature of many natural systems in physics, chemistry and biology. The essential theoretical issue in understanding pattern formation is to explain how a spatially homogeneous initial state can undergo spontaneous symmetry breaking leading to a stable spatial pattern. This problem is most commonly studied using partial differential equations to model a reaction-diffusion system of the type introduced by Turing. We report here on a much simpler and more robust model of spatial pattern formation, which is formulated as a novel type of coupled map lattice. In our model, the local site dynamics are coupled through a competitive, rather than diffusive, interaction. Depending only on the strength of the interaction, this competitive coupling results in spontaneous symmetry breaking of a homogeneous initial configuration and the formation of stable spatial patterns. This mechanism is very robust and produces stable pattern formation for a wide variety of spatial geometries, even when the local site dynamics is trivial.

Discovering the effect of nonlocal payoff calculation on the stabilty of ESS: Spatial patterns of Hawk–Dove game in metapopulations

“…The emergence of spatial inhomogeneity is related to the nonlocal (or quasi-local) interactions that are shown to be a mechanism for spatial pattern formation for a number of ecolgical processes [7,14,3,4,25,12,30,26,47]. In these earlier works it was shown that the spatial inhomegeniety is due to the fluctuating density of a species.…”

Breaking reflection symmetry: evolving long dynamical cycles in Boolean systems

“…Future work could also seek to generalize and test the theory of dynamical reflection symmetry for more general Boolean networks and multiple updating schemes and to random Boolean networks [ 19 , 48 ]. Finally, various other deterministic network-based systems exhibit cyclical behavior, and it would be intriguing to investigate whether subclasses of motifs also exist in systems such as coupled maps on networks [ 49 , 50 ] or evolutionary games on networks [ 51 ].…”