Intrinsic growth rate and exponential growth calculations are included along with a discussion of logistic growth. K-selected and r-selected species are explained along with survivorship curves.
Transcript provided by YouTube:
Hi. It’s Mr. Andersen and this is environmental science video 12. It is on population ecology.
One of the greatest conservation stories in biology is the story of the whooping crane.
They used to number 10,000 in the U.S. but by 1938 their numbers had dropped to only
15 individuals. So scientists had to figure out where are they, where are they breeding,
how do we protect those areas and you can see the population is starting to rebound.
But the health of the population is dependent upon the size of the population. How do we
increase the size of a population? Through births and immigration. New individuals coming
into the population. Likewise, how do we decrease it? Through deaths and emigration. These things
contribute to what is called the intrinsic growth rate. Is it increasing? Or is it decreasing?
It is not the only characteristic. We also have the density and distribution. We have
the sex ratio and the age structure as well. But what other factors, outside of this intrinsic
growth rate can affect their growth? Well we break that into two groups. Density dependent
and independent. Density dependent factors are factors that limit growth based on the
density of the population. So if you think about it as the population’s density increases,
if there is not enough food or water or shelter, we call those limiting resources. And what
happens to the population? It will eventually level off. It hits something called the carrying
capacity or K. It is the maximum number of individuals an area can support. We also have
density independent. And those are going to be things just related to chance. So a flood
or a fire could be examples that limit the size of a population. So in population ecology
we are studying these factors. And scientists come up with models that help to describe
what is going on in a population. So a famous model is the exponential growth model. What
we are looking at is this growth rate and how it is increasing the population over time.
And then we have a logistic model. It is also showing exponential growth but eventually
it is reaching what is called a carrying capacity or this limit of population growth. Scientists
also study strategies that species have. Some are what are called K selected. That means
their population size will increase until it gradually hits a carrying capacity. And
those who live more of a boom or bust cycle, that are r selected. And we can look at how
long individuals survive and that tells us a little bit about which strategy they are
using. And so the population size is incredibly important. So if we have these rabbits, so
we have 9 rabbits and their N value at this point would be 9. If we lose 2 of them our
N value is 7. If we gain 3 now our N value is going to be 10. It is the set number we
have. But also density is important. That is the number of individuals we have in a
given area. And so we could call this one density but we would call this greater density.
We could also look at their distribution. I would say that these rabbits are now randomly
distributed. But they could be distributed uniformly. Or they could be just clumped in
their distribution. And we could also look at their sex ratio. So how many are males
and how many of them are going to be females. And we could expand that to look at what is
called their age structure. Not only what is their gender but also how old are they.
So we could organize them like this where this is going to be our first year female
rabbits, second year and third year. And we can do the same thing with males. But when
it comes to the health, the population size is incredibly important. It is dictated by
births, deaths, immigration and emigration. And so we have a formula that allows us to
look at that. And the calculations are very simple. You can do them just in your head.
And so let’s say we have a population of 10. So our N naught is going to be 10. That
is our initial population. Here is our equation. So it is really simple. The change in N is
going to be the births minus the deaths plus the immigration minus the emigration. So let’s
look at this population over here and see what happens. So this rabbit gave birth to
3 other rabbits. And so if we write this out what is our births going to be? It is going
to be 3. Now let’s watch the population again. So you can see 1 of the rabbits died.
And so we are going to be put a 1 here in the deaths. We could look at immigration,
how many come in. It looks like just 1. So we would put a 1 right here. And then how
many emigrate? It looks like 2 left. And so we would put a 2 right here. And so the delta
N or the change in N is simply going 3 minus 1 plus 1 minus 2, or 1. That is the change.
Or we have seen an increase in 1. Now what is the growth rate? The growth rate is going
to be the change divided the initial population. So 1 divided by 10 gives us a 10 percent growth
rate of 0.1 is our growth rate. We call that the intrinsic growth rate. And as long as
we have no other factors outside that population, that will remain constant over time. And you
could solve a really hard problem. We could have a million people in an area. 100,000
are born. 10,000 die. If you are given the immigration and emigration you should be able
to calculate r for that population. So if we study a group of rabbits over time their
population will increase. But it will eventually level out at some point. Now that leveling
out point is called the carrying capacity or the K. Now why is a population going to
level out? It is because they are running out of something. They are running out of
food or water or shelter. And so we call all of those things limiting resources. Disease
could be another limiting resource. The more rabbits we have the more disease. And so it
is eventually going to level it off. Now it will not look perfect like that. The normal
population is going to have over shoots and it is going to have a lot of die off. But
we are going to have the average that we eventually hit. These are density dependent factors because
they are based on the density of the population. We can also have density independent. So imagine
that these rabbits over on this side are killed in a forest fire. That is just chance. It
is just chance taking over and so it is not based on the density of rabbits that we had.
So if we start to use models to explain how this works, a really important model is the
exponential growth model. And so the equation looks like this. It is a little scary but
it is really not that bad. N sub t is going to be the population at any time into the
future. N sub O is going to be the initial population. So let’s say we start with a
population of 10. r is going to be the growth rate. That is that intrinsic growth rate.
And t is going to be time. So the only thing that you really do not know in this equation
is e. e is going to be the mathematical constant. So it is a number. It is just like pi. It
is going to be 2.718. It just keeps going like that. So for our purposes we just think
of it as 2.71. And so let’s say we want to figure out what is going to happen to the
population in year 1. So if we want to figure out, we started at 10, what is going to be
the population probably at year 1? We just use this equation. So e is going to be the
same. So what is going to be our r value? Our r value will always be 0.5. That is that
intrinsic growth rate. What is our t value? Our t value is going to be time. What is our
initial population? It is going to be 10. So if I expand that a little bit or simply
multiply 1 times 0.5, 1 year times that growth rate. And so that is going to be 10 times
2.71, again that is e, raised to the 0.5 power. So that is really like taking the square root
of 2.71. And so that is 1.64. So if we work that out that is going to be around 16 rabbits
after 1 year. So let me graph that. And let’s go to year 2. So same thing. We are going
to plug in r value of 0.5 but now our t value is going to be 2. Still have that same initial
population. And so now it is going to be 2.71 raised to the 1 power. So what is that? That
is simply 2.71. So if we work this out now we are going to have 27 rabbits in that next
year. You can see the population is increasing. We are starting to see that exponential growth.
Let’s go for year 3. So if we figure out year 3, again our intrinsic growth rate is
still 0.5. 3 is going to be the year we are at. Still have that same initial. And so this
is going to be 2.71 raised to the 1.5 power. You probably need a calculator to do this.
We now get 44.6 or, let’s say 45 rabbits. So if we graph it, you can see that the population
is increasing like that. We have what is called a j-shaped curve. And it is going to increase
rapidly over time. We are going to, the whole world would be filled with rabbits if we keep
following this model. And so we know that is not what occurs. And so not only intrinsic
growth rate is important but K, that carrying capacity. So if you are given a problem like
this could you graph what is going to happen over time if K is 70? Well you are going to
get something that looks like this. It is going to be j for awhile but is eventually
going to curve off and we are going have a s shaped curve. This is a logistic growth
model. There is also a mathematical model we will not work through. I will put a link
to another video where I do that down below. And so scientists, now that they have models,
they can start to apply that to nature. So what we have found is that species kind of
fall into one of two camps. We have what are called K selected species. Those are going
to be species that their population increases and then it will eventually hit a carrying
capacity and it stays there. What are some characteristics of species like that? They
are going to give a lot of parental care to their offspring. They are just going to have
a few offspring. And so the whooping crane would be an example of that. Humans are an
example of that. We do not just go up and down in our population. r selected are going
to do that. So an arctic hare is an example of that. A famous study was looking at the
pelts that were collected by the Hudson Bay Company. And they found from 1850 to 1930
that the population of arctic hare just went up and down and up and down. And so hares
are going to be groups of individuals that have lots of offspring. They do not get tons
of parental care and their population is going to increase and then it will crash. So we
have this boom and bust cycle. Now what is interesting is that there is another species.
And so the arctic hare are fed on by the Canada Lynx. And if we look at their population,
their population goes through a boom and bust as well. We have what is called a predator
prey relations where as the arctic hare population increases then we can have more lynx feeding
on it. But as they crash then the lynx are going to crash as well. Now a way to look
at which strategy species are using is figuring out their survivorship. And so we have time
on the bottom and then we have the survivors on the side. So if we look at humans as a
type 1 survivorship curve, what that means is when we are born almost all of the humans
survive. And then throughout their lifetime they all die right at the end. And so we give
a lot of parental care to our offspring. Almost all of them survive and then when we get into
our 80s, 90s, then we all die off. We could also have a type 2 survivorship curve. Songbirds
are an example of that. From the moment they are born they are dying off at a constant
rate. Or we could look at type 3. Those are individuals like the acorns from a tree. Almost
all of them die but a few of those survive and those make up the plants that we have.
And so could you link that to K or r selected species? Well type 1 individuals are generally
going to be those K selected species. And the type 3 are generally going to be those
r selected species. But there are so many examples that are in the middle. So if you
think about a sea turtle for example, they have lots of offspring. They do not give them
much parental care, but they live a long time. And so it is not as simply as are you r or
are you K? It is somewhere in the middle. But they are applying these different strategies
in life. And so did you learn the following? Could you pause the video at this point and
fill in the blanks? If not, population size is determined by immigration and birth. That
increases it. Decreased by emigration and deaths. We have other characteristics, density,
distribution, sex ratio and age structure. There are density independent and dependent
factors. Density independent remember are related to chance. Density dependent lead
to what is called a carrying capacity or K. We use models to study it. Exponential models
are built on the growth rate. Logistic models, also built on the growth rate but include
carrying capacity. And then we have different strategies in species. K selected, r selected.
Remember we are K selected. And then we have survivorship curves that we can study to get
that. That is a lot. I hope it made sense. And I hope that was helpful.
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