How public can public goods be? Environmental context shapes the evolutionary ecology of partially private goods
Fig 6
Adaptive dynamics in the colimitation model.
(a) Pairwise-invasibility plot (PIP; see Box 2) using α = 0.5 and γ = 0.35 (marked by dots in panels c, d, and e) with gray indicating regions of successful invasion and arrows indicating the direction of evolution. The ESS, which here represents loss of fixation, is marked with a dot. (b) PIP using α = 0.62 and γ = 0.25 (marked by dots in panels c, d, and e) showing that evolution leads to an intermediate ESS maximum fixation rate. (c) ESS fixation b as a function of privatization α (horizontal axis) and cost of fixation γ (vertical axis) with lighter colors corresponding to higher ESS b. Note that low fixation costs result in unbounded growth (hence the “no stable equilibrium” region). (d) Geometrical envelope (see Box 2) shown by thick gray line, with ESS b values (black numerals) and their corresponding impact vectors (dashed lines) shown for points along the envelope. PIPs qualitatively the same as a) and b) are mapped onto the impact vectors. (e) ESS fixation rate (black line, left axis) and N:S ratio at equilibrium (purple line, right axis) plotted as functions of position along the geometrical envelope from d), moving from the siderophore-limited to N-limited regimes. To the left of the dashed line (i.e. for N:S ratios greater than the starred value), there is no fixation at equilibrium, as in a); for lower N:S ratios, there is an intermediate ESS fixation rate as in b). (f) Critical N:S ratio where fixation transitions from 0 to positive, marked by star on e). Parameters as shown in Table 1.