BioIT 08-09: Comparison of models

Here's a simpler comparison of three different growth models to describe algal growth.

1. Constant growth rate (A). The same number of algae are added to the population every time. That is, the rate at which N grows is the same at all times, or

dN/dt = rate

where "rate" is a constant.

2. Exponential growth (B). The rate at which N grows (dN/dt) is not constant, and depends at every time on the size of the population able to reproduce. The amount of new algae being added to the population represents, however, a constant proportion of the population. We call that constant proportion "rate", the equivalent of "interest rate" in banking.

That is,

dN/dt = N x rate

3. Sigmoid or density-dependent growth (C). As in the exponential model, the new population is a certain fraction of the old (dN/dt = N x rate). But now this proportion "rate" is not constant and depends on the N. That is, algae are reproducing maximally (rate = ratemax) when N is very low, and no longer reproducing (rate = 0) when the population reaches a certain size called the "carrying capacity", cc. (You might say the algae are "having the most fun when no one else is around" then become more and more "shy" as their neighbors increase. In reality, it has to do with less and less food and space as population grows: the culture has become too crowded for growth to be sustainable.) For simplicity, the relation between "rate" and N may be described by the linear curve in D.

Thus, as in the exponential

dN/dt = N x rate

but

rate = ratemax - (ratemax/cc) x N, the equation of the line in D.

Thus,

dN/dt = N x (ratemax - (ratemax/cc) x N), or factoring out

dN/dt = N x ratemax x (1 - N/cc)

What can make this confusing is the word "rate". In one context it refers to a proportion of the population (the variable "rate"), while in another context the same word refers to the number of new algae per unit time (dN/dt). Thus, if you are confused, change "rate" (proportion) into "interest" (as in banking), and use the word exclusively to refer to dN/dt.