Cooperative mutualism is a major force driving evolution and sustaining
ecosystems. Although the importance of spatial degrees of freedom and number
fluctuations is well-known, their effects on mutualism are not fully
understood. With range expansions of microbes in mind, we show that, even when
mutualism confers a distinct selective advantage, it persists only in
populations with high density and frequent migrations. When these parameters
are reduced, mutualism is generically lost via a directed percolation process,
with a phase diagram strongly influenced by an exceptional DP2 transition.Comment: 8 pages, 4 figure
