## 5. An alternative logistic model

In considering exponential growth by microorganisms reproducing by binary division, we introduced the equation:

Pt = Pstart. exp(k.t)

Where:

• Pstart is the initial population, and Pt is the population at time t.
• The expression exp(k.t) is termed the 'exponent' of the time (t) multiplied by a growth constant (k), which has the units of time-1.

As already established, this model predicts population change that accelerates as more individuals enter the population and in turn produce more successor organisms. The reality for most environments is obviously that resources will be used up as the population increases, in turn limiting growth of both individual organisms and the population as a whole. Other controls, such as density-dependent mortality due to predation, may also operate. As a result, the absolute rate of population increase does not continue to increase, but rather declines towards zero as the population approaches a maximum value, akin to the carrying capacity, K, in the model derived from the Lotka-Volterra equations (see section 2).

Here, we use a formula that is related directly to the exponential growth model at the start of this section. It is based on a starting population, an equilibrium population and a growth constant:

Pt = Pequil/(Pstart + [(Pequil - Pstart ).exp(1-kt)])

In this model:

• Pstart is the initial population, and Pt is the population at time t.
• Pstart is the starting population size.
• Pequil is the equilibrium population size (equivalent to K – the carrying capacity - in the Lotka-Volterra equation)
• k is a growth constant, which is has the same relation to doubling time as the growth constant in an exponential growth model (k = ln(2)/doubling time)

The model is similar in form to that derived from the Lotka-Volterra equations, and looks like this:

We can compare the population growth predicted by the logistic model with that resulting from exponential growth with the same values of Pstart and k (ie doubling time).

It is clear that for very low population size (that is, within the first few days of population increase), the two models predict identical numbers. Gradually, however, the numbers predicted by the logistic model fall below those from the exponential model. When the population is about 10% of carrying capacity, the value predicted by the logistic model is about 90% of that predicted for purely exponential growth, whilst at 50% of carrying capacity the ratio is 50%, and at 90% of the carrying capacity it is only 10% of the exponentially-increasing population.