## 2. The logistic growth model explored

In order to model a population growing in an environment where resource availability limits population growth, the model needs to have a variable growth rate, that decreases as the population size increases towards a notional maximum, often termed **carrying capacity** in population ecology.

For population ecologists, the most common type of model derives from a series of equations ascribed to two researchers. The Lotka-Volterra equations include an expression for population change where the key parameters are a growth term (broadly the balance between fecundity and mortality) and a maximum population size (carrying capacity). The relationship is normally expressed as a differential equation:

dN/dt = rN . (1-N/K)

In this equation, the expression dN/dt represents the rate of change of number of organisms, N, with time, t, and r is a growth term (units time^{-1}) and K is the carrying capacity (same units as N). It is clear that for small values of N, N/K is very small so that (1-N/K) is approximately equal to one, and the equation is effectively:

dN/dt = rN

As N increases, (1-N/K) becomes smaller, effectively reducing the value of r. This is termed **density-dependence**, and is an example of negative feedback because the larger the population, the lower the growth rate.

This equation can be re-written in a form that can be evaluated for any value of N at two times t and (t+1), giving the equation:

N_{(t+1)} = (N_{t}.R)/(1 + aN_{t})

Notice that this equation contains new parameters R and a. These are related to r and K by:

R = 1/(fecundity . mortality) = exp(r)

and

a = (R - 1)/K