Table 1.
Summary of model parameters and functions for both models (SR = single resource model, Co = colimitation model).
Numerical exploration of the colimitation model was done with randomly generated parameter combinations over the ranges given that satisfied inequality Eq (S2.2) (about 50% of all combinations). Default values used for adaptive dynamics are given in the final column.
Fig 1.
Overview of results from the single-resource model.
(a) Graphical representation of single-strain equilibrium . (b) The strains’ per capita growth rates excluding external mortality gi as a function of resource concentration R. Note that when growth gi(R) balances mortality δ, the cell population is at equilibrium. (b)
is defined as the resource concentration at which the producer halts production (S1 Appendix). As long as the producer actually produces resources at equilibrium (see S1 Appendix for case where this does not hold), competitive exclusion occurs. i) If privatization is greater than production costs (α > γ), then the producer outcompetes the non-producer and resource production is maintained. Note that unbounded growth may occur in this case (see S1 Appendix). ii) If production costs are greater than privatization (γ > α), then the non-producer outcompetes the producer and resource production is lost from the community.
Fig 2.
A broad overview of the colimitation model system.
(a) We model cell populations that uptake, and may facultatively produce, nitrogen and siderophores at rates determined by the extracellular concentrations of these resources. The effects of increasing each production/uptake parameter, relative to the thickest curve, is shown. (b) We consider four different strains (functions performed by these strains marked with checks in (a), functions not performed marked with x’s): Full (fixes nitrogen and produces siderophores), LOFN (a LOF fixation mutant that only produces siderophores), LOFS (a LOF siderophore mutant that only fixes nitrogen), and LOFB (a LOF mutant that neither fixes nitrogen nor produces siderophores). (c) Because extracellular resource concentrations govern resource uptake and, when applicable, production, the cell population growth rate is also a function of extracellular resource concentrations. According to our colimitation model, availability of siderophores mediates the effect of fixed nitrogen on growth.
Fig 3.
Graphical analysis of consumer-resource models (see Box 1 for details).
(a) Illustration of a zero net growth isocline (ZNGI, gray curve) for a cell population as a function of resource availabilities. The shaded region marked with a “+” denotes the resource levels for which the strain can grow. Whether S, N, or both resources are limiting depends on the resource availabilities relative to the minimum individual concentrations required for growth (S*, N*). Impact vectors (the thick line with arrows is one example) map resource supplies onto equilibrium resource concentrations. (b) Drawing ZNGIs for two different strains allows for an assessment of the conditions in which both (purple) or only one (red or blue) strain can grow. (c) The outcome of competition between two strains also requires knowledge of how they influence the environment (their impact vectors; dashed lines). Here, coexistence (necessarily at the intersection of the two ZNGIs) is unstable (a priority effect) because each strain draws down the resource that its competitor most needs to grow more than the resource that it itself most needs to grow. (d) Stable coexistence happens when each strain draws down the resource that it most needs to grow more than the resource that its competitor most needs to grow.
Table 2.
Viability and competitive ability of each of our four strains (each column corresponds to a different resident).
The first row of the body (‘% Viable’) shows the percentage of parameter combinations for which each resident type was viable. The remaining rows show the percentage of parameter combinations for which the invasion was successful (conditioned on resident viability; left-hand column indicates which strain was the invader). Abbreviations in Fig 2b.
Fig 4.
Sensitivity analysis and ZNGIs for the role of privatization α and cost of fixation γ for three pairwise competition scenarios (rows).
The first column of each row (i) shows the probability that the second strain listed ((a) LOFS, (b) LOFN, (c) LOFS) successfully invades the first strain listed ((a,b) Full, (c) LOFN). Subsequent columns of each row plot outcomes of competition against the external environment (see Box 1 for explanation) when: (ii) fixation is beneficial even when N is in excess such that siderophores are limiting, i.e. ; (iii) fixation is beneficial when N is limiting but costly when siderophores are limiting, i.e.
; (iv) fixation is costly even when N is limiting, i.e.
. (a) Full vs. LOFS. i) Regardless of parameters, LOFS can successfully invade Full’s resident equilibrium ii-iv) because the ZNGI for LOFS always falls below the ZNGI for Full (siderophore production is always a net cost). (b) Full vs. LOFN. i) Full can resist invasion from LOFN if fixation is private (high α) and not very costly to produce (low γ). ii) If fixation is always a net benefit, then Full always wins. iii) If the benefit of fixation depends upon the environment, then either strain could win or they could coexist (with the outcome depending on environmental resource supplies). iv) If fixation is always a net cost, then LOFN always wins. (c) LOFN vs. LOFS. i) LOFS can invade LOFN if fixation is private (high α) and not very costly to produce (low γ). ii) If fixation is a benefit when siderophores are limiting, then LOFS outcompetes LOFN regardless of environmental conditions. iii) If the benefit of fixation depends upon the environment, then LOFS either always outcompetes LOFN (shown here) or two coexistence equilibria arise (shown in S5 Fig), with the latter case arising if fixation is more costly than siderophore production moving away from S*. iv) If fixation is costly when nitrogen is limiting, then LOFN wins in N-limited regimes and LOFS wins in siderophore-limited regimes, with a priority effect in regions of overlap.
Fig 5.
Overview of adaptive dynamics (see Box 2 for details).
(a) Example pairwise invasibility plot (PIP). Shaded regions indicate pairs of resident (x-axis) and mutant (y-axis) trait values that result in a positive invasion growth rate for the mutant and thus predict a successful invasion; white regions indicate a negative invasion growth rate and loss of the mutation. An ESS is any resident trait value at which a vertical line passes through only white regions. (b) ZNGIs for different resident trait values (colors) as a function of the resource environment. The thick gray line marks the geometrical envelope, defined by the ESS strategy at any supply point (i.e., following the outermost ZNGI). Numbers on the envelope list select ESS values along this envelope and the dashed lines are their impact vectors (see Box 1).
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.