Evolutionary classification of toxin mediated interactions in microorganisms

A trade-off between the parameters of Lotka–Volterra systems is used to give verifications of relations between intrinsic growth rate and limiting capacity and the stability type of the resulting dynamical system. The well known rock–paper–scissors game serves as a template for toxin mediated interactions, which is best represented by the bacteriocin producing Escherichia coli bacteria. There, we have three strains of the same species. The producer produces a toxin lethal to the sensitive, while the resistant is able to protect itself from that toxin. Due to the fact that there are costs for production and for resistance, a dynamics similar to the rock–paper–scissors game results. By using an adaptive dynamics approach for competitive Lotka–Volterra systems and assuming an inverse relation (trade-off) between intrinsic growth rate (IGR) and limiting capacity (LC) we obtain evolutionary and convergence stable relations between the IGR’s and the LC’s. Furthermore this evolutionary process leads to a phase topology of the population dynamics with a globally stable interior fixed point by leaving the interaction parameters constant. While the inverse trade-off stabilizes coexistence and does not allow branching, toxicity itself can promote diversification. The results are discussed in view of several biological examples indicating that the above results are structurally valid.