Box 2: Adaptive dynamics analysis.
Interaction between resident and mutant strains. To model the competition effect of a resident phenotype, φ res, on the population growth of a mutant phenotype,φ mut, we extend the ecosystem model written for a single type (equation (1c) in Box 1). To account for the local nature of the interaction between rare mutant and common resident cells, we introduce a function (hereafter denoted by c ) of the difference between φ res and φ mut to measure how local decomposition by mutant and resident cells differ from ‘mean field’ (average) decomposition by resident cells. Thus, for givenC , D , Z , the growth of the mutant population is governed by
(5)\(\frac{dM_{\text{mut}}}{\text{dt}}=\left(1-\varphi\right)\gamma_{M}\frac{v_{\max}^{U}\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}{K_{m}^{U}+\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}M_{\text{mut}}-d_{M}M_{\text{mut}}\)
where D res is the equilibrium D predicted by the ecosystem model for the resident phenotypeφ res. Here function c satisfiesc (0) = 0, c (z ) > 0 if z> 0 and c (z ) < 0 if z< 0.
The underlying assumption is that each microbe has access to DOC partly as a public good and partly as a private good (Driscoll & Pepper 2010). The public good part results from the diffusion of exoenzymes. The private good part results from local decomposition at the microscopic scale of cells and exoenzymes that they produce themselves. A mutant cell that invests more (resp. less) in exoenzyme has access to more (less) DOC than the average resident cell because the cell’s private good is greater (smaller) whereas all cells share the same public good. In a spatially implicit model like ours, diffusion is not directly modeled, but its effect on the accessibility of DOC to a mutant strain can be phenomenologically accounted for by a parameterization that puts mutant cells at a competitive advantage for DOC if the mutant phenotype invests more in exoenzyme production than the resident phenotype, or at a competitive disadvantage if the mutant phenotype invests less. This parameterization is achieved with the function c in equation (5), where c < 1 when φ mut< φ res and c > 1 when φ mut >φ res. This phenomenological approach is consistent with the mathematical construction and numerical analysis of a spatially explicit model of resident-mutant local interaction that accounts for soil diffusion, which will be reported elsewhere.
Invasion fitness and selection gradient. Mutant fitness\(s\left(\varphi_{\text{mut}},\varphi_{\text{res}}\right)\) is given by the mutant population growth rate per unit biomass:
(6)\(s\left(\varphi_{\text{mut}},\varphi_{\text{res}}\right)=\left(1-\varphi\right)\gamma_{M}\frac{v_{\max}^{U}\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}{K_{m}^{U}+\left(1+c\left(\varphi_{\text{mut}}-\varphi_{\text{res}}\right)\right)D_{\text{res}}}-d_{M}\)
The selection gradient then obtains by taking the first order derivative of the invasion fitness with respect to the mutant trait:
(7)\(\nabla s\left(\varphi\right)=\frac{d_{M}}{1-\varphi}\left(\left(1-\varphi-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}\right)c_{0}-1\right)\)
where \(c_{0}=c^{\prime}(0)\) measures the local competitive advantage to stronger exoenzyme producers, which we call ‘competition asymmetry’. Note that by definition of function c , we always have\(c_{0}>0\). Variation in \(c_{0}\) may be caused by different soil diffusion properties, due to e.g. physical texture or moisture.
Evolutionary singularity. Trait values that nullify the selection gradient are called ‘evolutionary singularities’. An evolutionary singularity can be attractive or repelling, and invadable or non-invadable. Evolutionary singularities that are attractive and non-invadable represent potential end-points of evolutionary adaptation. Evolutionary singularities that are attractive and invadable can lead to evolutionary branching (Geritz et al.1998).
In a given environment (fixed parameters, constant temperature) there is at most one evolutionary singularity given by defining \(\varphi\)* as the value of \(\varphi\)that makes\(\nabla s\left(\varphi\right)\)= 0 in equation (7):
(8)\(\varphi^{*}=1-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}-\frac{1}{c_{0}}\).
Existence of φ * > 0 requires\(\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}<1\) and\(c_{0}>\frac{1}{\left(1-\frac{d_{M}}{v_{\max}^{U}\gamma_{M}}\right)}\). Thus, the (cooperative) trait φ can evolve above zero only if the local competition advantage to stronger enzyme producers is large enough. The condition for φ * to be evolutionarily stable is\(c^{\prime\prime}\left(0\right)<2\ c_{0}^{2}\) and no other condition than existence is required for φ * to be always convergent. Here we assume that function c is such that φ * is evolutionarily stable and attractive.
Equation (8) shows that more cooperation (larger φ *) should evolve in microbial populations with lower mortality, greater nutrient uptake, and/or higher MGE. When comparing microbial populations with similar life-history traits γ M, \(v_{\max}^{U}\)and d M, stronger competitive advantage to exoenzyme producers (i.e. higher c 0) selects for larger φ *.