Making Population Models More Realistic –
Incorporating K
The populations you have looked at in “getting a feel for population parameters” were an
introduction to population modeling, but those models were unrealistic. For example the
wolf example ended up with far more wolves than exist in a relatively short time. As we
have discussed, populations do not grow unchecked. More realistic models incorporate
carrying capacity (K) as a population regulation mechanism. The simplest such model is:
(1)
𝑑𝑁
𝑑𝑇
= Nr (
Where:
𝑑𝑁
𝑑𝑇
𝐾−𝑁
𝐾
)
= population change over time
N = population size
r = rate of growth (λ – 1)
K = carrying capacity
Because
from:
𝑑𝑁
𝑑𝑇
is the change of population, the population after a time increment follows
(2) Nt+1 = Nt +
𝑑𝑁
𝑑𝑇
So, revisiting the wolves from “getting a feel for population parameters”, that population
had a λ = 1.25. Thus, for this population, r = 0.25. Assuming a carrying capacity of 70
wolves and a starting population of 10 wolves, calculate the change in population year to
year with equation 1 and the next year’s population with equation 2. Do this for 30 years.
Graph the results.
After doing this, explain how carrying capacity affects population growth rate and how
𝐾−𝑁
the 𝐾 piece of the equation works.