Exponential Growth Model Video Lecture | Science for ACT

hi it's mr. Andersen and in this video
I'm going to talk about exponential
growth which is how populations can
explode most students understand
exponential growth but the math
sometimes it gets a little tricky and so
I'm going to step you through that in a
couple of ways and so let's start with
this rabbit right here let's say it's
part of a population we don't only have
one rabbit but we have a number of
rabbits in our population we refer to
that in all of the equations as n n is
going to be the population size now
that's going to change as we go through
time but this is going to be our
original population which is going to be
n let's stack those rabbits up so we can
count them so our n to start is going to
be 10 so we have 10 rabbits at time 0
now that population is here they're
going to increase it's going to decrease
or it's going to stay the same and what
things are determining that it's going
to be our growth rate and so this is the
second letter you should remember and
that's our R is going to refer to how
much it's changing over time there's
really only two things that are going to
change that population we're going to
have new rabbits that's going to be
births and then we're going to have dead
rabbits and that's going to be deaths
and so those two things are going to
contribute to the change in the
population but it's a per capita in
other words we have to divide by the N
which is going to be the original
population size and so let me make some
baby rabbits so if I click here we've
got five baby rabbits and so our births
would be five and then let's say I want
to kill a couple of rabbits let's kill
that guy don't worry they're okay
they're just virtual rabbits and so I
kill that one as well and so we've got
births of five we've now got deaths of
two and what was our end to begin with
it was 10 and so if we figure out our R
value that's going to be 5 minus 2
divided by 10 which is 3 over 10 which
is going to be 0.3 and so our R value is
0.3 or our growth rate is 0.3 what does
that really mean it's the factor at
which our population is increasing and
so if I take 10 times 0.3 I'm going to
get 3 and that's how much our population
increased and one thing you should know
about that growth rate is that if the
ecosystem is stable the growth rate is
essentially going to stay the same it's
not going to
change overtime and so you might think
well if the growth rate stays the same
is in the population just going to
increase along a consistent amount not
really and so let's watch what happens
now we're going to take point three
growth rate for the next generation and
we're going to instead of multiplying at
times ten we now have to multiply it
times thirteen and if we do that and
we'll use this equation right here this
is the change in n or the change in T
we're going to take our growth rate
which is 0.3 and now multiply it times
13 well we don't get 3 anymore we get
3.9 and so I'm going to round that
sounds a lot like for rabbits so I'm
going to add 4 rabbits and now our
population is up to 17 so even though
our stayed the same since we multiplied
it times a larger value we're going to
get more rabbits in the next generation
so let's do 10 eration 3 we're now
taking point 3 times 17 and I get 5.1
which is a lot like five rabbits and I'm
going to add those two rabbits or if we
now multiply that growth rate times 22 I
get 6.6 which is pretty close to 7
rabbits so we're going to add those 7
rabbits and so we now have got a
population of 29 and so you can see that
the population is increasing but if I
were to ask you a question I could ask
you some hard questions the first one is
not so hard what's the population going
to be in year five well to do that you
take 29 times point three and then we'd
add that to 29 but what if I asked you
10 or even 30 well this probably gets
pretty hard and so you're quickly going
to want a little bit of help and for me
when I want help the first place I go to
is to a spreadsheet and let's go to the
spreadsheet so we're going to go to
excel kind of remember those numbers
there and so um let's kind of rebuild
that chart so on the left side we're
going to have 0 as our first time and
one as our second time if you didn't
know how to do this in Excel I can
select both of those grab this little
corner here and I can increase and it'll
do the counting for me and so let's go
up to population time 30 okay so I
filled that in now what's our original
population that's going to be 10 so I'm
going to just put that in to start with
now what's the next population so I'm
going to put a formula in this box and
to do that I'm just going to put an
equal sign so I'm going to put equal
sign you can see the formula right here
and so what did we do remember we're
going to take the original population so
let me click on that
so that's going to be ten and then we're
going to add that to that we're going to
add our growth rate which was point
three times and then we're going to
click on that again and so let's see
what we did so we get 13 and so what we
did is again we took what was in this
cell and we added it to what was in this
cell times 0.3 and so again through the
magic of a spreadsheet I can simply grab
this and it's just going to use that
same thing over and over it's going to
iterate on that and so what we're going
to get is it's going to do all the math
for us and so what did we have in the
first one 13 17 22 the sounds familiar
now it's obviously not rounded off and
so this isn't the correct number of
rabbits but we see now this exponential
curve here or this J shaped curve or
sometimes we call it like a hockey stick
curve because it's quickly turning up
like that and so now we could quickly
answer those questions at time 5 we
should have around 37 rabbits what was
the next one I think 10 we should around
138 rabbits and if we go all the way
down here to time 30 we're going to have
226,000 199 rabbits so that's a lot of
rabbits really really quickly and so you
can see how exponential growth takes off
but what's fun about a spreadsheet is I
could play around with it a little bit
let's say instead of 0.3 if I change my
growth rate to 0.1 so if I do that what
are we going to get well if I move this
all the way down again we're going to
get another J shaped curve now it's not
going to be as steep as that one was and
we didn't have as many rabbits at the
end but we're still going to be
exponential growth we refer to go edit
that variable again let's make it zero
so let's say we take it times zero what
would we get then well it's ten in other
words if I go all the way down here what
are we going to get for a value well
we're not increasing at all and so it's
going to be ten it's going to stay ten
the whole time a really hard question
that I asked the students is this let's
say we get a negative growth rate so
let's make it negative 0.3 what are we
going to get for a value there well we
get seven here but if I go all the way
down what do we get that's weird we're
going to approach kind of a limit we're
going to approach zero um and that's
because we're going to take off larger
amounts
to begin with and then we're going to
take off the less amounts as the
population gets smaller and smaller and
smaller so again that's spreadsheets but
you don't have these have a spreadsheet
with you sometimes you just need a
calculator and a little bit of algebra
so let's go to the algebra this is going
to be the algebraic solution to this and
so we've got an equation for exponential
growth and so change in n over T is
going to be equal to n which is going to
be our population size times 1 plus R
where R is going to be the growth rate
and then we're going to raise that to T
where T is going to be equal to time and
so let's make sure that this works so so
right here with zero and so let's plug
in our numbers so we're going to put 10
in here for n that's at original
population 1 and then R is going to go
right here it's going to be 0.3 and then
we raise it to the zero power because
our time is going to be zero well if we
simplify that a little bit anything
raised to the 0 power is always 1 and so
that's going to be 10 and so that works
out so far but don't trust me let's keep
going let's go to the next one let's go
to 1 so if we now put in 1 for time it's
going to be n is still 10 this is 1 plus
R again there R is not changing but
we're raising it to the 1 power anything
raised to the first power is going to be
itself so it's going to be 1.3 we get 13
let's try that again with 2 so if we go
to the second power again we plug in 2
here it's the only thing that we're
changing so we get 1 point 3 to the
second power so we're going to have to
square one point three which is one
point six nine and we get 16 point nine
which is a lot like 17 where I could
just throw out another time so let's say
we go time 30 so then we're going to
raise it to the 30th power and so I get
26,000 199 which is going to match up
exactly with our spreadsheet we're going
to have a lot of rabbits really really
quickly and so that's kind of an
algebraic solution um it's a quick way
to be given a time and then figure out
how much it's going to grow and so I'm a
good question I could ask you is this
let's look at bacteria rather than
rabbits okay and so let's say we have
one bacteria e.coli can reproduce in
about 20 minutes in other words one can
make an exact copy of itself in about 20
minutes and so let's say that none of
them died let's say we get rid of the
death rate so we've got our births over
N and so how many births would we have
we're just dividing in half we're going
to have one new ecoli what was our
original population it was going to be
one as well and so now we're going to
get an R of one and so instead of
increasing by 0.3 we're now increasing
by 1 which is really increasing by a
hundred percent so we had one bacteria
and now we have a hundred percent as
many bacteria twice as many or in 100%
in addition to that original and so now
we have two and so what are we going to
have on the next round we're going to
have 4 and 8 and 16 and then 32 and you
can see how exponential growth gives us
a huge amount of bacteria really really
quickly and so the question I might ask
is is the sky the limit and so if we're
looking at your exponential growth you
know after 20 rounds like this we're
going to be way up in here I can't even
read this five million something like
that and so does it just keep going and
going and going no because what I told
you is kind of a lie R is not going to
stay the same forever as it starts to
grow they're going to run out of food
resources space and so our R is going to
start to change and so then we're going
to start to move into what's called
logistic growth and that's going to be a
totally set a different set of equations
and I'm going to include that in another
video and so that's exponential growth
and I hope that was hell