Question

The logistic equation represents the idea that populations first increase exponentially but then their growth decreases as they approach carrying capacity.

Answer 1

logistic equation is a model of population growth that gives an idea through the population curves.

The continuous version of the logistic model is described by the differential equation

d t

(1)

where is the Malthusian parameter (rate of maximum population growth) and is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by and defining $x=N/K$ then gives the differential equation

dx=rx(1-x),

(2)

which is known as the logistic equation

so, dN/dT= r max x (dN/dT) = rmax x N x (KN)/K

The formula we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. The expression "K – N" is indicative of how many individuals may be added to a population at a given stage, and "K – N" divided by "K" is the fraction of the carrying capacity available for further growth. Thus, the exponential growth model is restricted by this factor to generate the logistic growth equation.

Notice that when N is very small, (K-N)/K becomes close to K/K or 1; the right side of the equation reduces to r_maxN, which means the population is growing exponentially and is not influenced by carrying capacity. On the other hand, when N is large, (K-N)/K come close to zero, which means that population growth will be slowed greatly or even stopped. Thus, population growth is greatly slowed in large populations by the carrying capacity K.