3-2 Verhulst Equation

Verhulst's Population Equation

n Pn Pn+1 = 2 Pn
1 P1 = 100 P2 = 2 (100) = 200
2 P2 = 200 P3 = 2 (200) = 400
3 P3 = 400 P4 = 2 (400) = 800
4 P4 =800 P5 = 2 (800) = 1600
5 P5 = 1600 P6 = 2 (1600) = 3200
6 P6 = 3200 P7 = 2 (3200) = 6400

The easiest way to visualize how populations vary in time is to make a graph. The horizontal axis represents time, and the vertical axis represents the number of individuals in the population. If we start with 100 insects (P1 = 100) , in year two, there will be 200 insects (P2 = 200).

Make a graph of the equation Pn+1 = 2 Pn using the graph points tabulated in the table at right.

The factor 2 was put into the equation because we imagined that each year our insect population would be double the year before. But, said Verhulst, if we want to think about all kinds of increases we must recognize that they all don’t double each year. Some grow faster, some slower. Instead of the number 2 let’s put in a letter which will stand for the number, or factor, that we multiply by the previous year’s population to get this year’s population. Let’s call G the population growth factor. Our equation now looks like this:

Pn+1 = G * Pn

Now make a table and graph showing how the population would change over six years if G= 3. On the same graph show how the population would grow if G=1, and if G= 0.5.

Question 3.1.
How do these different growth factors affect the population?

Question 3.2.
What would the population be after ten or fifteen years for these different growth factors?

The method of calculating the population for each year based on the previous year is called iteration. The invention of calculators and computer spreadsheets has made the process of iteration a very popular and easy way to calculate population growth. If you have access to these tools, you may want to calculate the population for a wide variety of growth factors and lengths of time.

The result of our calculation is called a geometric progression. Geometric population growth can occur in special situations where the food supply is plentiful and there is space for organisms to grow. An example is the growth of bacteria in a petri dish. They can double their population in an hour. The population explosion of rabbits in Australia is another example. But no population can grow geometrically forever. Limitations always exist. The petri dish has just so much space. There are limits to the rabbit food supply. The first years on Goat Island were those of geometric growth for the population of goats because the small population of goats had not yet discovered the limits to their growth.