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# Exponential and logistic growth in populations

## Video transcript

- [Voiceover] Let's say
that we're starting with a population of 1,000
rabbits, and we know that this population is
growing at 10% per month. What I wanna do is explore
how that population will grow, if it's growing 10% per month. So let's set up a little bit of a, let's set up a little
table here, a little table. And on this left column,
let's just say this is the number of months that have gone by, and on the right column, let's
say this is the population. So we know from the
information given to us, that at zero months, we're
starting off with 1,000 rabbits. Now let's think about what's
gonna happen after one month. Well, our population's gonna grow by 10%, so we can take our population
at the beginning of the month, and growing by 10%,
that's the same thing as multiplying by one point one. You have your original population, and then you grow it by 10%,
one plus 10% is one point one. So we can multiply it by one point one, and that math we can do in our head, it is 11 hundred, or 1,100, but let's just write this
as 1,000 times one point, not one point five, times one point one. Now let's think about what
happens as we go to month two. What's gonna have is
gonna be the population that we started at the
beginning of the month times one point one again, so it's gonna be the population at the beginning of the
month, which was that, which we have right over there, but then we're gonna multiply
by one point one again, or we can just say this
is one point one squared. I think you see a pattern emerging. After another month, the
population's gonna be 1,000 times one point
one to the third power, we're just gonna multiply
by one point one again. And so if you were to go
n months into the future, well you can see what's going to be. It's gonna be 1,000
times, or being multiplied by one point one n times, or 1,000 times one point
one to the nth power. And so we can set up an expression here, we can say look the population, let's say that the population is P. The population as a function
of n, as a function of n, is gonna be equal to
our initial population, our initial population,
times one point one to the nth power. And you might say, "Okay this makes sense, "it doesn't look like we're
getting crazy numbers", but just for kicks, let's just think about what's gonna happen in 10 years. So 10 years would be 120 months. So the population at the end
of 120 months is gonna be 1,000 times one point
one to the 120th power, and so let's, let me get a
calculator out to do that. I can not calculate one point one to the 120th power in my head. One point one to the 120th
power is equal to that, times our original
population, so times 1,000, one two three, is going
to be equal to roughly 93 million rabbits,
let me write that down. So we started with 1,000,
and we're gonna have approximately 93 million
rabbits, 93 million, million rabbits. And so we grew by a factor
of 93,000 over 10 years, so over another 10 years, we're gonna grow by 93,000 times this. And so you quickly realize
10% per month is quite fast, and this might seem extremely fast, but it's actually not outlandish
for a population of rabbits that are not limited by
space or predators or food, and if you were to plot
something like this out, if you were to plot the rabbit population with respect to time, you would see a graph that
looks, let me draw it. So this axis it is time,
let's say in months, and this axis you have your population, you have your population. This type of function,
or this type of equation, let me see population I, population, this is an exponential function, and so your population
as a function of time is gonna look like this,
it's gonna have this kinda hockey stick j shape right over here. And if you let these rabbits
reproduce long enough, they would frankly take over the planet, if they had enough food and
they had enough space to do it. But if you notice I keep
saying if they have enough food and if they have enough space. The reality in the world
is that there is not infinite food and infinite space, and it isn't the case that
there are no predators, or competition for resources. And so there is actually a
maximum carrying capacity for certain part of the environment for a certain type of species. And so what's more likely to happen, what we described right over
here, is exponential growth, exponential growth, and why is
it called exponential growth? Well you notice, we are
growing by the input, which is time, is being
thrown into our exponent. And so that is exponential
growth, but obviously you can't have an
infinite number of rabbits or you can't just grow forever. There is going to be some
natural maximum carrying capacity that the environment can actually sustain. And so the actual growth
that you would see, when the population is well
below that carrying capacity, is reasonable to model it
with exponential growth, but as it get closer and closer
to that carrying capacity, it is going to axiom up towards it, so it's gonna get up towards
it, but not cross it, and that's just a model. There are other situations
where maybe it goes up to it, and it crosses it, and
then it cycles around it, so these are all different
ways of thinking about it, but the general idea is you
wouldn't expect something to just grow unfettered forever. Now this blue curve,
which people often use to model population,
especially when they're thinking about the
population once they approach the environment's carrying capacity. This is, this kinda s shaped
curve, that is considered, that's called logistic growth, and there is a logistic
function that describes this, but you don't have to know it in the scope of a kinda introductory biology. There's a logistic, logistic growth, and it's described by
the logistic function. If you're curious about
it, we do have videos on Khan Academy about logistic growth and also about exponential growth, and we go into a lot more detail on that. But the general idea
here is when populations are not limited by their
environment, by food, by resources, by space, they
tend to grow exponentially, but then once they get close, that exponential growth
no longer models it well, once they start to really
saturate their environment, or they start to get
close to that ceiling, and overall the logistic,
or logistic function, or logistic growth, is a better model for what is actually going to happen.