The effect of chaotic dispersion on cyclic competition

Durrett and Levin (4998) considered a simple lattice model occupied by three species in cyclic competition and observed that the behavior of the system in a spatially extended system with short range local interaction is different from the corresponding mean-field model. In general, cyclic competition in spatially extended systems produces a dynamical equilibrium in which all species coexist, while the mean-field model leads to either periodically oscillating total populations, or extinction of all except one of the species. 

The different behavior of the spatially distributed and of the homogeneously well mixed systems was also confirmed by the experiments of Kerr et al. (2002), who studied cyclic competition of three strains of E. coli bacteria in a well mixed liquid environment and on a solid agar surface. In these experiments a killer strain (R), with a relatively low fitness, i.e. low reproduction rate, can eliminate the sensitive type cells (S) by producing a toxic substance. A third resistant strain (R), however, is immune to the toxin and has higher fitness than the killer cells, but is less fit than the sensitive ones, due to the metabolic cost of producing proteins for protection from the toxin. 

To connect the two markedly different scenarios observed in the static and the well-mixed environments, it is natural to analyze the role of increasing mobility (Reichenbach et al., 2007). Karolyi et al. (2005) studied the above competition model combined with dispersion by a chaotic map that represents advection of fluid elements in the alternating sine-flow. By continuously changing the frequency of the chaotic dispersion as a control parameter, it is possible to follow the transitions between the two limiting situations. When the chaotic mixing is much faster than the local population dynamics, the killer and resistant cells gradually disappear from the population and only the sensitive cells survive. This is because the killer cells 

The first terms on the r.h.s. of these equations represent the reproduction rate of each species that is proportional to their population 

When the chaotic dispersion is slow the populations of the three strains oscillates periodically in time, each strain becoming temporarily dominant in a cyclic fashion (Fig. 8.8). The chaotic dispersion also affects the spatial structure, by stretching the patches occupied by different strains into elongated filaments (Fig. 8.9). Thus the effect of chaotic dispersion is to synchronize the local oscillations over the whole system. The amplitude of the oscillations increases with the dispersion rate and eventually leads to the extinction of two of 

---