Allelopathy of bacteria in a lattice population: Competition between colicin-sensitive and colicin-producing strains

We analyse the population dynamics of two strains of bacteria (Escherichia coli): one strain produces a toxin (called colicin) that increases the mortality of a colicin-sensitive strain in the neighbourhood, but does not harm the colicin-producing strain itself. On the other hand, in the absence of colicin in the environment, the colicin-sensitive strain enjoys a higher population growth rate. The model is closely related to the evolutionary dynamics of social interaction. It has been established previously that a perfectly mixing population shows bistability; that is, whichever strain dominates initially tends to defeat the other. On the other hand, empirical and computer simulation results in lattice structured populations show neither co-existence nor bistability of the two strains. In this paper, we analyse the lattice model based on pair approximation (forming a system of ordinary differential equations of global densities and local densities), and by the direct computer simulation of the spatial stochastic model. Both the pair approximation dynamics and the computer simulation show that, for most regions of parameter values, one strain defeats the other, irrespective of the initial abundance. However, the pair approximation analysis also suggests a relatively narrow parameter region of bistability, which should disappear when the model is considered on a lattice of infinitely large size. The biological implications of the results and the relationship of the present model with other models of the evolution of spite or altruistic behaviours are discussed. This suggests that the dynamics reflected by the spatially explicit lattice model may be sufficient, perhaps even necessary, to support the evolution of colicin.