The model of population dynamics

Background

First, we establish the model of population dynamics to study the variation of E. coli population density. Here, we use the Logistic equation to build our model.

The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used.

Theory

Let $N$ $N$ be the population density of E. coli. With the Logistic equation, we know that

where $r$ $r$ and $K$ $K$ are the growth rate and the environmental capacity of E. coli respectively.

Parameter

The parameters are shown in the table below.

Result

Let the initial value of population density be $0.01\mathrm{\%}$ $0.01\%$ of the environmental capacity. The result is shown in the figure below.

Conclusion

Equation $(1.1)$ $(1.1)$ and result show that:

• When $N<{\displaystyle \frac{K}{2}}$ $N<\dfrac{K}{2}$, the population density grows exponentially;
• When $N>{\displaystyle \frac{K}{2}}$ $N>\dfrac{K}{2}$, the environmental resources have a restrictive effect on E. coli;
• Finally the population density approaches $K$ $K$;
• The population density reach balance at about $33\mathrm{h}$ $33\mathrm{h}$.

Reference

Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, 1845.

Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, 1847.