Fig 1.
Examples of graphs with proper two-colorings: a, cycle; b, complete, symmetric bipartite graph; c, square lattice (von Neumann neighborhood); d, hexagonal lattice; e, complete, asymmetric bipartite graph; and f, star (an extreme case of e).
If the fitness of A and B are aG and bG on a green site and aR and bR on a red site, respectively, then on each of these graphs we can explicitly calculate the probability that one type replaces the other (Theorem 1).
Fig 2.
Selection condition when aG = r + σA, aR = r − σA, bG = 1 + σB, and bR = 1 − σB.
The graph is bipartite, regular, and properly two-colored with NG = NR = 50. The difference between fixation probabilities of the two types, ρA − ρB, is shown for several values of σA and σB when r = 1.5. Warmer colors represent greater differences between the fixation probabilities, ρA − ρB. A is neutral relative to B if and only if .
Fig 3.
Effects of background heterogeneity on a complete bipartite graph with NG = NR = 5.
Background heterogeneity increases the fixation probability of an advantageous mutant (black lines) and decreases that of a disadvantageous mutant (white lines). These effects are monotonic in background heterogeneity, σ. By Theorem 1, this behavior is identical to that of a regular (not necessarily complete) properly two-colored graph with equal numbers of green and red nodes.
Fig 4.
Effects of background heterogeneity for alternative colorings on two-colorable graphs.
In each panel, we plot fixation probability against background heterogeneity for all (non-isomorphic) permutations of the proper two-coloring shown at the top. The proper two-coloring in each case is depicted in black, which is given by Eq 2. a, On the cycle, the proper two-coloring is “optimal” in the sense that it gives the maximum fixation probability for an advantageous mutant and the minimum fixation probability for a disadvantageous mutant. b, On the star, we observe the opposite behavior, with the proper two-coloring giving the minimum fixation probability for an advantageous mutant and the maximum for a disadvantageous mutant. c, On a complete bipartite graph with NG ≠ NR (shown here with NG = 3 and NR = 7), a mixture of these two results is possible. In particular, there need not be a coloring that is “optimal” for all levels of background heterogeneity. The fixation probabilities in all panels were approximated by building transition matrices for each process and looking at the exact distribution after 107 steps.
Fig 5.
Fixation probability in the presence of dynamic resources on bipartite graphs.
ρA is shown here as a function of resource heterogeneity, σ, for several values of the resource redistribution rate, p, and mutant mean fitness, r. The population initially has a proper two-coloring. At each time step, with probability p the colors are shuffled according to permutation chosen uniformly at random. With probability 1 − p, the coloring is not changed in that time step. In a and b, high environmental fluctuations (i.e. large p) attenuate the effects of background heterogeneity relative to the initial distribution of resources at p = 0 (corresponding to a proper two-coloring, shown in black and given by Eq 2). In c, this behavior holds for all but the highest levels of background heterogeneity. The fixation probabilities in all panels were approximated by building transition matrices for each process and looking at the exact distribution after 107 steps.