Logistic Growth Maximum Population Dynamics

The provided equation represents logistic population growth:

$\frac{dN}{dt} = 0.8N - 0.01N^{2}$

This equation describes the rate of change of population size (N) over time (t).

Identifying Carrying Capacity

The standard form of the logistic growth equation is:

$\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$

where 'r' is the intrinsic rate of increase and 'K' is the carrying capacity.

To find 'K' from the given equation, we can factor it:

$\frac{dN}{dt} = N(0.8 - 0.01N)$

Factor out 0.8:

$\frac{dN}{dt} = 0.8N\left(1 - \frac{0.01N}{0.8}\right)$

$\frac{dN}{dt} = 0.8N\left(1 - \frac{N}{80}\right)$

By comparing this with the standard form, we identify the carrying capacity:

$K = 80$

Calculating Population for Maximum Growth

In logistic growth, the population growth rate ($\frac{dN}{dt}$) is maximum when the population size (N) is exactly half of the carrying capacity (K).

Maximum growth occurs at:

$N = \frac{K}{2}$

Substitute the value of K:

$N = \frac{80}{2}$

$N = 40$

Therefore, the population exhibits maximum growth at N = 40.