Qian He, Uwe C. Tauber, R. K. P. Zia
We aim to clarify the relationship between interacting three-species models
and the two-species Lotka-Volterra (LV) model. We utilize mean-field theory and
Monte Carlo simulations on two-dimensional square lattices to explore the
temporal evolution characteristics of two different interacting three-species
predator-prey systems: (1) a cyclic rock-paper-scissors (RPS) model with
conserved total particle number but strongly asymmetric reaction rates that
lets the system evolve towards one corner of configuration space; (2) a
hierarchical food chain where an additional intermediate species is inserted
between the predator and prey in the LV model. For model variant (1), we
demonstrate that the evolutionary properties of both minority species in the
steady state of this stochastic spatial three-species corner RPS model are well
approximated by the LV system, with its emerging characteristic features of
localized population clustering, persistent oscillatory dynamics, correlated
spatio-temporal patterns, and fitness enhancement through quenched spatial
disorder in the predation rates. In contrast, we could not identify any regime
where the hierarchical model (2) would reduce to the two-species LV system. In
the presence of pair exchange processes, the system remains essentially
well-mixed, and we generally find the Monte Carlo simulation results for the
spatially extended model (2) to be consistent with the predictions from the
corresponding mean-field rate equations. If spreading occurs only through
nearest-neighbor hopping, small population clusters emerge; yet the requirement
of an intermediate species cluster obviously disrupts spatio-temporal
correlations between predator and prey, and correspondingly eliminates many of
the intriguing fluctuation phenomena that characterize the stochastic spatial
LV system.
View original:
http://arxiv.org/abs/1111.1674
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