Fixation probability of quorum sensing (QS) mutants. Note the sensitivity of fixation probability in all cases to relatedness, *r*, and the frequency of QS expression, *ϕ. A*, Signal mutants under model 1 behave as deleterious mutants when *r* and/or *ϕ* are high but become advantageous when *r* and/or *ϕ* are sufficiently low, peaking at a maximum of *P* = 2*c*_{S}, which is twice the maximal benefit of cheating (see also ). *B*, Note the change in scale from model 1, where fixation probabilities are much higher in general. Unlike model 1, the sign of fitness does not change with model 2 (see ): cheating mutants always act as deleterious mutants in model 2. Infrequent expression and low relatedness relax selection against mutants, increasing their probability of fixation to a maximum at the neutral fixation probability of *P* = 1/*N. C*, Fixation probability of mutations at the response-regulator locus. As with signal mutants under model 1, response-regulator mutants can act as either beneficial or deleterious, depending on relatedness. Note that the effect of conditional expression reverses when relatedness reaches the Hamiltonian threshold at neutrality (vertical dashed line), so that rarely expressed mutants (small *ϕ*) have a lower probability of fixation than frequently expressed mutants (large *ϕ*) when relatedness is low but a greater fixation probability when relatedness is high. This reversal corresponds to the transition from beneficial (at low relatedness) to deleterious (at high relatedness). The maximum fixation probability occurs at *r* = 0 and is equal to the standard probability of fixation for a beneficial, nonsocial mutation, *P* = 2*c*_{R}. Parameter values are as follows: *c*= 0.01, *b*_{R} = 0.02, *N* = *N*_{e} = 1,000, and *ϕ* = 1.

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