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Noise and Correlations in a Spatial Population Model with Cyclic Competition
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LMU-ASC 66/07
Noise and Correlations in a Spatial Population Model with Cyclic Competition
Tobias Reichenbach, Mauro Mobilia?, and Erwin Frey
Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS),
Department of Physics, Ludwig-Maximilians-Universita?t Mu?nchen,
Theresienstrasse 37, D-80333 Mu?nchen, Germany
Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influ-
ence on the coevolution of populations with cyclic interspecies competition have been demonstrated
in recent experiments [e.g. B. Kerr et al., Nature 418, 171 (2002)]. To reach a better theoretical
understanding of these phenomena, we consider a paradigmatic spatial model where three species
exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial dif-
ferential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and
spatial diffusion at different levels, and show how fascinating patterns of entangled spirals emerge.
We rationalize our analysis by computing the spatio-temporal correlation functions and provide
analytical expressions for the front velocity and the wavelength of the propagating spiral waves.
PACS numbers: 87.23.Cc,02.50.Ey,05.10.Gg,87.18.Hf
Understanding the combined influence of spatial de-
grees of freedom and noise on biodiversity is an impor-
tant issue in theoretical biology and ecology. This im-
plies to face the challenging problem of studying com-
plex nonequilibrium structures, which form in the course
of nonlinear evolution [1, 2, 3, 4, 5, 6]. More gener-
ally, self-organized nonequilibrium patterns and travel-
ing waves are ubiquitous in nature and appear, for in-
stance, in chemical reactions, biological systems, as well
as in epidemic outbreaks [7]. Among the most stud-
ied types of patterns are spiral waves, which are rel-
evant to autocatalytic chemical react
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