Directed Percolation has Colors and Flavors

Abstract

A model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented. We use renormalized field theory and demonstrate that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the one-species Gribov process (Reggeon field theory). In addition this universal model shows an instability that generically leads to a total asymmetry between each pair of species of a cooperative society, and finally to unidirectionality of the interspecies couplings. It is shown that in general the universal multicritical properties of unidirectionally coupled directed percolation processes with linear coupling can also be described by the model. Consequently the crossover exponent describing the scaling of the linear coupling parameters is given by $Φ=1$ to all orders of the perturbation expansion. As an example of unidirectionally coupled directed percolation, we discuss the population dynamics of the tournaments of three colors.