Table 1 Notation used throughout the main text

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