### Logistic growth: More examples of logistic growth

### Modellling microbial growth

We have already seen some examples of the logistic growth model in the assignments. Here we infer that this model is applicable to bacterial growth under different assumptions.

Assumptions and modelling of microbial growth

For the mathematical model of bacterial growth, we assume that

- the microbial growth model is "exponential", but with a non-constant growth rate factor;
- the microbial growth rate factor depends on the concentration of a growth-limiting substrate (nutrient) in the growth medium;
- the growth rate instantaneously adapts to changes in substrate concentration (i.e., without any delay);
- the rate of decrease of the substrate is proportional to the growth rate of the bacteria.

Suppose that \(N\) is the number of bacteria and \(S\) is the substrate concentration. The above assumptions can be translated into mathematical formulas:

- \(N'=r(S)\cdot N\);
- A special choice: \(r(S)=\mu\cdot S\) for some constant \(\mu\);
- There is no need to build in delays into the model and we can just set up differential equations;
- \(\gamma\cdot S'=-N'\) for some constant \(\gamma\).

From \[\gamma\cdot S'=-N'\] follows \[(\gamma\cdot S+N)'=0\] In other words \[\gamma\cdot S+N=a\] for some constant \(a\). So: \[S=\frac{1}{\gamma}\cdot(a-N)\] But then it follows from the first two relations: \[\begin{aligned} N' & =r(S)\cdot N =\mu\cdot S\cdot N\\&= \frac{\mu\cdot a}{\gamma}\cdot N\cdot \left(1-\frac{N}{a}\right) \end{aligned}\] This is a logistic growth equation.

Growth model of Jacques Monod The second assumption in the derivation of the logistic model of microbial growth was a special choice. The Nobel laureate Jacques Monod has introduced a more realistic assumption: \[r(S)=\frac{\mu\cdot S}{K_h+S}\] for some constants \(\mu\) and \(K_h\).

If \(S\) is large, \(r(S)\approx \mu\) and there is exponential growth.

If \(S\) is small, \(r(S)\approx \frac{\mu}{K_h}\cdot S\) and there is logistic growth.