Some basic equations for population growth:
Your textbook uses two different sets of equations to model population growth. One set is built around the “per capita
rate of increase” (r) and the other uses the “finite rate” (l). To avoid the needless complexity of two sets of equations,
we will simply define r as the net rate of growth of a population per unit time, as described in this handout.
The increase (I) of a population of N individuals is equal to births minus deaths:
I = births - deaths
It's often more convenient, however, to determine birth rate (b ) and death rate (d)for a
population. Such a rate is simply the number of births or deaths over a certain period of
time divided by the size of the population:
b=
Births
N
d=
Deaths
N
When you take birth rate and death rate into consideration together, it's
possible to combine them to to produce a value that reflects the overall
net rate of population growth, (r). Remember: r is a rate!
r=b-d
Now life gets interesting. Suppose that we know the rate of population growth (r) and
we'd like to figure out how much that population will increase over a period of time. To
determine the increase to be expected we simply multiply r by the size of the population
(N). This gives us a value for the increase (I) of the population per unit time, which your
textbook describes as ΔΝ/ΔΤ (change in number per unit time).
ΔΝ/ΔΤ = I = rN
The population expected at the end of the growth period, of course, is equal to the starting
number (N) + the increase in number (I). If the period was one generation, we could
figure out the population expected at the end of that generation like this:
N = No (1 + r)
Note that N= the final population, No = the initial population, and r = rate of growth.
To determine how much a population will increase over many generations, we simply
place an exponent around the (1+r ) term:
N = No (1 + r)n
Note that n= time (expressed as the number of intervals. Years, generations, whatever)
This is a great equation, and has many uses. For example, let's suppose that a population
of 250 animals is increasing in size by 6% a year (rate = 0.06). At that growth rate, how
many of them will be around in 10 years?
N = 250 (1.06)10 = ?
Work it out. The population should be 448 animals. How many generations until the
population reaches 1 million?
Admittedly, these equations assume that the rate of growth does not change over time,
which is not a realistic situation. Therefore, ecologists have tried to model growth by
adjusting their predictions to account for the fact that as a population grows there are
fewer and fewer resources available for it to exploit. Therefore, they first try to
determine the growth rate that would prevail under ideal conditions (unlimited resources),
which is sometimes called the intrinsic growth rate (It's called that because it's
presumed that only factors intrinsic to the organism - like how fast the organisms can
reproduce - affect this rate) and it's denoted by the symbol ro. Therefore, the maximum
increase in numbers that we should expect for a population is simply the intrinsic rate
times population size:
I = roN
A population would grow at this rate only if it found unlimited resources. However, in
all real situations, resources are limited. Therefore, ecologists assume that each system
has a limit to the number of individuals that it can support, and this limit is called the
carrying capacity of the population, K. As a population approaches the carrying
capacity, population growth should slow.
We can model this effect by multiplying the equation in
K-N
which we used the intrinsic growth rate by an expression
I = roN K
that approaches zero as the population approaches the
carrying capacity:
• This equation is known as the Logistic Growth Equation, and is widely used to model
population growth and carrying capacity.
• Be sure to think about the implications of this equation. For example, are there any
circumstances in which you would expect the increase in a population (I) to be negative?
Think about how the Logistic Growth Equation predicts the form of growth curve seen
above (from Figure 52.7 on p. 1045 of Freeman 4/e [p. 1181 of Freeman 3/e]).