sources/growth-curve-intro.md

Multiplex growth curves

How do strains grow when competing with each other?

That's given by the Lotka–Volterra competition model. For two strains:

$$\frac{d{n}_{wt}}{dt}=w_{wt}{n}_{wt}(1-\frac{n_{wt}+n_1}{K})\backslash frac\{dn\_\{wt\}\}\{dt\}\; =\; w\_\{wt\}\; n\_\{wt\}\; \backslash left(\; 1\; -\; \backslash frac\{n\_\{wt\}\; +\; n\_\{1\}\}\{K\}\; \backslash right)$$

$$\frac{d{n}_1}{dt}=w_1{n}_1(1-\frac{n_{wt}+n_1}{K})\backslash frac\{dn\_1\}\{dt\}\; =\; w\_\{1\}\; n\_1\; \backslash left(\; 1\; -\; \backslash frac\{n\_\{wt\}\; +\; n\_\{1\}\}\{K\}\; \backslash right)$$

• $n_i(t)$: abundance of species (or strain) $i$ at time $t$.
• $w_i$: intrinsic (exponential) growth rate of species $i$.
• $K$: carrying capacity.

We can generalize to many strains. For each one:

$$\frac{d{n}_i}{dt}=w_i{n}_i(1-\frac{{\mathrm{\Sigma }}_j{n}_j}{K})\backslash frac\{dn\_i\}\{dt\}\; =\; w\_\{i\}\; n\_i\; \backslash left(\; 1\; -\; \backslash frac\{\backslash Sigma\_j\; n\_j\}\{K\}\; \backslash right)$$

It's not possible to algebraically integrate these equations, since they are circularly dependent on each other. But we can numerically integrate, to simulate multiplexed growth curves.