Population modeling – Threshold values

(Uses rational functions and roots of polynomials.)

For low population sizes, the search for a mate can be the limiting factor on population growth. If we assume that for small populations, the probability of finding a mate is roughly proportional to the population size then we want per capita growth nearly linear. If we model our population by generations, where is the population in the nth generation, then we expect

for small P.

For large populations, the limiting factor is the carrying capacity of the environment, so we expect, for large P, to have

Now we build a rational function that behaves that way… and end up with

We can graph this:

The population will settle down where , or where the curve and the y=x line intersect:

Solve this:

We see that we have three roots for some values of K and c, but as K gets smaller, we only have one real root, P=0. Now consider what happens as the carrying capacity changes graphically

The lower, threshold value hits the upper steady state value, and our population now crashes!