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Table 1 Notation used throughout the main text

From: Coexistence and cooperation in structured habitats

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 parametersB(t); \(\kappa\), \(\gamma\), \(\mu\)\(\Rightarrow\)\(\alpha (t)\)
  see “Collective reduction of antibiotics” section
Pyoverdine parametersP(t); \(\sigma\)\(\Rightarrow\)\(\varphi (t)\)
  see “Iron extraction via siderophores” section