RE: Lack of clarity in population equations (aka "What am I missing?") (by Chana Kaufman)

I wrote this reply in another thread but I'll post it here in case
it might help anyone. There are two completely different B and D
variables. One is B and D = birth/death rate. For example, B= 20
births/yr and D = 5 deaths/yr which would give a population growth
rate (dN/dt) of 15/yr. This is what the collegeboard equation sheet
means in the equation dN/dt = B-D **and they are correct in
calling them rates**.

Usually it makes more sense to use the *per capita *birth
or death rate. This means the average births or death *per
individual*. The variables used for per capita birth/death rate
is *lowercase* b and d. For example, in a population
of 50 with a birth rate of 20, the per capita birth rate would be
20/50 or .4. If the death rate in the same population is 5, than
d=5/50 or .1. This is much more useful b/c we can compare birth and
death rates among different population sizes. The equation for b
would be b = B/n. Many times we are given b so to convert it to B you
would multiply by the population number (N). For example, if b=.2 in
a population of 100, the birth rate would be 20 (per year if that was
the time specified). To explain it to your students point out that
the decimals and percents are interchangeable so when we say a per
capita birth rate of .4 and per capita death rate of .1 it is the
same as saying a 40% birth rate and 10% death rate. Basically, if you
see decimals, you can safely assume that we are discussing per capita
birth and death rate, although the question should definitely have
made that clear. If we plug this in to our original equation we get
population growth rate (dN/dt) = (b -d) x N. So if you are dealing
with per capita rates, we need to multiply by population size to get
the population growth rate.

Population ecologists are concerned with the __difference__ between
the birth and death rate. This would show the per capita rate of
increase and is known as **r **with r = b
- d. [__Important__**: r = b - d NOT r = B
- D**] If b>d, r is greater than zero and population is
increasing. If d>b, r is less than zero and population is
decreasing. If b=d, then r = 0, and there is zero population growth.
Plugging this in to our original equation, we get dN/dt = rN. Since r
is a per capita rate, we multiply it by our population number to get
the actual number of the population growth rate. So if b=.4 and d=.1,
then r=.3 (we have a 30% increase) and in a population of 50, that
would translate into an increase of 15 per year (dN/dt). r_{max} is
really the same as r. When we calculate r_{max} we are
assuming that the maximum growth rate will continue with no
limitations.

So, you are not confusing r and rmax. They are calculated the same way and in essence mean the same thing (see above). And the reason for students' confusion about r max in the exponential growth equation is the incorrect formula of rmax = B - D when it really equals (B - D)/N or just plain b - d.

I hope this was helpful.

I am going to elaborate on what Kevin Karplus said b/c it is a very important point that I think many of us are confused about. There are two completely different B and D variables. One is B and D = birth/death rate. For example, B= 20 births/yr and D = 5 deaths/yr which would give a population growth rate (dN/dt) of 15/yr. This is what the collegeboard equation sheet means in the equation dN/dt = B-D.

Usually it makes more sense to use the *per capita *birth
or death rate. This means the average births or death *per
individual*. The variables used for per capita birth/death rate
is *lowercase* b and d. For example, in a population of 50
with a birth rate of 20, the per capita birth rate would be 20/50 or
.4. If the death rate in the same population is 5, than d=5/50 or .1.
This is much more useful b/c we can compare birth and death rates
among different population sizes. The equation for b would be b =
B/n. Many times we are given b so to convert it to B you would
multiply by the population number (N). For example, if b=.2 in a
population of 100, the birth rate would be 20 (per year if that was
the time specified). To explain it to your students point out that
the decimals and percents are interchangeable so when we say a per
capita birth rate of .4 and per capita death rate of .1 it is the
same as saying a 40% birth rate and 10% death rate. Basically, if you
see decimals, you can safely assume that we are discussing per capita
birth and death rate, although the question should definitely have
made that clear. If we plug this in to our original equation we get
population growth rate (dN/dt) = (b -d) x N. So if you are dealing
with per capita rates, we need to multiply by population size to get
the population growth rate.

Population ecologists are concerned with the __difference__
between the birth and death rate. This would show the per capita rate
of increase and is known as **r **with r = b - d.
If b>d, r is greater than zero and population is increasing. If
d>b, r is less than zero and population is decreasing. If b=d,
then r = 0, and there is zero population growth. Plugging this in to
our original equation, we get dN/dt = rN. Since r is a per capita
rate, we multiply it by our population number to get the actual
number of the population growth rate. So if b=.4 and d=.1, then r=.3
(we have a 30% increase) and in a population of 50, that would
translate into an increase of 15 per year (dN/dt). r_{max} is
really the same as r. When we calculate r_{max} we are
assuming that the maximum growth rate will continue with no
limitations.

This is the example you are asking about, **A
population of 265 swans are introduced to Circle Lake. The
population’s birth rate is 0.341 swans/year, and the death rate
is 0.296 swans/year. What is the rate of population growth, and
is it increasing or decreasing? **

Assuming .341 and .296 are per capita rates (based on the fact that they are given in decimals) we get dN/dt = (.341-.296) x 265 = .045 x 265 = ~12/yr. However, based on the answer, the question seems to be looking for per capita growth rate (r) which would be r = b - d = .341 - .296 =.045. This population is growing at a rate of 4.5% per yr which is obviously increasing since it is a positive number. If we needed to figure out the population growth rate using r, we have the equation dN/dt = rN which would be .045 x 265 = ~12 (the same number we got by solving for it immediately.

Your mistake was that you got .045 [really you got .018 but that
was a careless error from using .314 instead if .341 as your b] as
yout dN/dt instead of multiplying that number by N. Then you would
have gotten 11.9 which would have made sense in your second equation
of 11.9 = r_{max} x 265 so rmax would be .045. Of course,
this is a very backwards way of doing it and you only need to stick
with one equation!

I don't think the worksheet is very accurate since in one question
r_{max} is given as 80 dandelions per month. This makes no
sense since r_{max} should be expressed as a per capita rate
which in that question would be 80/40 or 2. This is a large r_{max}
value and means the population is increasing by 200% per month. The
answer given is 3200. What does this even mean? 3200 dandelions per
month? per year? The correct way to solve would be dN/dt = r_{max}N
= 2 x 40 = 80 dandelions per month. Basically the 80 per month is
given in the question which would be the answer to the population
growth rate. I don't mean to knock the worksheet. It was extremely
helpful to me and I am planning on adopting some of the questions for
my class. So thank you for putting it out there! Right now, it the
only population equation resource available and we all really
appreciate it! I hope my answer was helpful and please feel free to
ask any more questions if I wasn't clear.