Inoculum size $${\mathbf {n}}= (n_1,n_2,\ldots , n_i, \dots )$$ Total inoculum size $$n = n_1 + n_2 + \cdots + n_i + \cdots$$ Composition of inoculum $$x_i = n_i/n$$ $${\mathbf {x}}= (x_1,x_2,\ldots ,x_i,\ldots )$$ Seeding probabilities $${\mathbb {P}}\bigl [{\mathbf {n}} \bigl \vert \bigr .\, {\overline{n}},{\overline{{\mathbf {x}}}}\bigr ] = \prod _i\frac{({\overline{n}}\,{\overline{x}}_i)^{n_i}}{n_i!}e^{-{\overline{n}}\,{\overline{x}}_i}$$ Averages over seeding $$\bigl \langle F(n,{\mathbf {x}}) \bigr \rangle = \sum _{{\mathbf {n}}}{\mathbb {P}}\bigl [{\mathbf {n}} \bigl \vert \bigr .\, {\overline{n}},{\overline{{\mathbf {x}}}}\bigr ]F(n,{\mathbf {x}})$$ Cycle index $$(\tau )$$ Mixing time $$T_\text{mix}$$ Depletion time $$T_\text{depl}\bigl ({\mathbf {n}},\text{environment}\bigr )$$ Within-deme observables Population sizes $${\mathbf {N}} = (N_1(t),N_2(t),\ldots )$$ Population composition $${\mathbf {X}} = (X_1(t),X_2(t),\ldots )$$ $$X_i = N_i/N$$ Growth rate $$\alpha _i(t) = \alpha (1+\delta \alpha _i)A(t)$$ Yield $$\varphi _i(t) = \varphi (1+\delta \varphi _i)Y(t)$$ Resources $$S(t)$$, $$S(0) = S_0$$ $$\varphi \sim 1\Rightarrow N(T_\text{depl})\approx {\mathcal {O}}\bigl (S_0\bigr )$$ Depletion $$\alpha (t>T_\text{depl}) = 0$$ Public good dynamics Production rates $$\varvec{\rho }= (\rho _1,\rho _2,\ldots ,\rho _i,\dots )$$ usually $$\rho _1 > 0, \rho _i \approx 0, i\ge 2$$ Antibiotics parameters B(t); $$\kappa$$, $$\gamma$$, $$\mu$$$$\Rightarrow$$$$\alpha (t)$$ see “Collective reduction of antibiotics” section Pyoverdine parameters P(t); $$\sigma$$$$\Rightarrow$$$$\varphi (t)$$ see “Iron extraction via siderophores” section