Grow a culture in a flask, watch the lag, exponential and stationary phases, then switch to a chemostat and find the washout point.
| Lag adaptation α (0 → 1) | - |
| Specific growth rate μ(S)·α | - |
| Net rate d(ln X)/dt | - |
| Doubling time now | - |
| Optical density OD600 (illustrative 2.5 per g/L) | - |
| Critical dilution rate D_c (washout) | - |
| Chemostat S* (theory) | - |
| Exponential slope vs μ | - |
| Chemostat steady state | - |
| Washout threshold | - |
| Batch mass balance | - |
Monod growth. The specific growth rate depends on the limiting nutrient: μ(S) = μmax·S/(Ks + S). Ks is the concentration at which growth is half of maximal. Biomass and substrate are linked by the yield Y: dS/dt = −(1/Y)·dX/dt (growth part).
Growth phases. The lag phase is modelled with the Baranyi–Roberts adaptation factor α = q/(1+q), dq/dt = μmax·q, with q₀ set from the lag time λ (h₀ = μmax·λ). Exponential growth has ln X rising with slope μ (doubling time ln2/μ). When S falls, growth decelerates to a stationary phase; the loss term kd·X produces a decline or death phase.
Chemostat. With dilution rate D and feed concentration S₀ the steady state is S* = Ks (D + kd)/(μmax − D − kd) and X* = Y (S₀ − S*) (for kd = 0). The population is washed out when D exceeds D_c = μmax·S₀/(Ks + S₀).
Checks. The four checks integrate the ODEs with RK4 and compare with these closed forms (slope of ln X, steady state, bisection of the washout threshold, conservation of X + Y·S).
Honest limits. One limiting substrate and one species; no pH, oxygen, product inhibition or diauxie; deterministic (no cell-to-cell noise). Default constants are illustrative (real glucose Ks for E. coli is much smaller, μmax depends strongly on temperature and medium). The visible bacteria are a display scaled with the biomass, not a count.