Evolution and stability of altruist strategies in microbial games

When microbes compete for limited resources, they often engage in chemical warfare using bacterial toxins. This competition can be understood in terms of evolutionary game theory (EGT). We study the predictions of EGT for the bacterial “suicide bomber” game in terms of the phase portraits of population dynamics, for parameter combinations that cover all interesting games for two-players, and seven of the 38 possible phase portraits of the three-player game. We compare these predictions to simulations of these competitions in finite well-mixed populations, but also allowing for probabilistic rather than pure strategies, as well as Darwinian adaptation over tens of thousands of generations. We find that Darwinian evolution of probabilistic strategies stabilizes games of the rock-paper-scissors type that emerge for parameters describing realistic bacterial populations, and point to ways in which the population fixed point can be selected by changing those parameters.

Figure 1

Fixed points and flow for different two-player games as a function of game parameters, for $\omega \in (0,1)$ and $\varepsilon \in (0,2)$. Region I, prisoner's dilemma; region II, “harmony” game; region III, “snowdrift” game; region IV, “coordination” game (see, e.g., [10]).

Figure 2

ESS as a function of the parameters $\varepsilon$ and $\omega$. (a) Numerical simulation of the population frequencies based on replicator equations, with both types at equal frequency initially. (b) Simulation of a finite well-mixed population of agents that encode the strategies 00 and RT deterministically, with the same initial conditions as in (a) (50 independent simulations for each of $21\times 21$ different parameter combinations). In (a) and (b), red and green intensity indicates the frequency of strategy 00 and RT in the population, respectively. The separation between red and green in region IV is due to the choice of initial condition, and does not represent a phase boundary.

Figure 3

Phase portraits of stable dynamics for the three-player game using payoffs (12). The shaded parameter region $\omega <\varepsilon /\left(\varepsilon +1\right)$ has an interior fixed point that can be repulsive or attractive. In Zeeman's phase portrait pictograms [30], arrows denote the flow on the boundary of the simplex, solid circles are attractors, and open circles are repellers. All fixed points on the boundary and the interior are indicated. The Zeeman classes for each region are indicated in Table .

Figure 4

(a) Left: Dominating strategy as determined by replicator equations as a function of cost $\omega$ and benefit $\varepsilon$, where the density of each pure strategy is indicated by the intensity of the three colors red (00), green (RT), and blue (R0), given the starting condition where each strategy has 1/3 of the population. Regions as in Fig. 3. Right: Population fraction trajectories obtained from iterating the replicator equations for four parameter combinations $(\varepsilon ,\omega )$: $A=(0.75,0.125),B=(1.25,0.125),C=(1.75,0.375)$, and $D=(1.75,0.625)$, whose locations are shown on the left. Arrows indicate the location of the fixed points. (b) Dominating strategies in a finite-population-size agent-based simulation of competition between pure strategies in a mell-mixed population (left, 20 runs of 5000 updates per pair of parameters, for 21 $\times 21$ parameter combinations). Right: The trajectories for the four parameter combinations picked in (a) show stable coexistence for points C and D, but extinction of strategies for A and B. (c) Left: Darwinian evolution of probabilistic strategies in an agent-based simulation at fixed mutation rate (left, 20 replicates per pair of parameters, 21 $\times 21$ parameter combinations), where colors indicate average fixed point probabilities in the scheme of (a). The trajectories (right) represent the average probabilities to engage in the plays 00, R0, and RT on an (averaged) line of descent, not trajectories of population frequencies. Arrows indicate the predicted fixed point. Average of 50 replicate trajectories for 100 000 updates for each of the example parameters A–D.