Peopling the Americas

WEBVTT
Kind: captions
Language: en

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- [Voiceover] Hey Becca!

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- [Voiceover] Hey Kim!

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- [Voiceover] Alright, so
we're talking about how people

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got to the Americas today.

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So when did people first
arrive in North America?

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Was Columbus the first one?

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- [Voiceover] So, no he was not.

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You know, back in the day people believed

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that actually pre-Columbus
no one was here,

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but in this video I'm
not going to be talking

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about if, right?

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There definitely were people here,

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but I'll be talking about
how people got here,

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when, and who they were.

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And so these are the
questions that historians

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and archaeologists are
still debating today.

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So Kim who are archaeologists?

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(chuckling)

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- [Voiceover] Well
archaeologists are scholars

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who work in the lands.

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They will dig in the places
where people lived in the past,

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find remains of earlier civilizations,

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and use that evidence to build a picture

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of what life was like in a place

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before there was a written record.

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- [Voiceover] So a lot of
the history we'll talk about

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in this video has to do
with different evidence

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and competing evidence that
archaeologists have found

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in the lands that we now
call the United States.

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So as I mentioned, we're
not arguing if there

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were people here because
there definitely were.

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There were even big cities
and towns and people living

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together in societies that look a lot like

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what the colonists were
trying to make as well.

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- [Voiceover] Yeah I
think people often have

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the misconception that Native
Americans were primarily

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hunter-gatherers who pretty
much lived in very small bands

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that didn't leave much
of a record of buildings

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or living sites, they were more nomadic.

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But people were all over
North America and they had

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societies that were very
different from each other,

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some of which were very
complex politically,

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and some of which were quite large cities.

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This city here is Cahokia, which is--

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- [Voiceover] That's a
fun thing to say, Cahokia.

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- [Voiceover] (giggles) Yeah, I know.

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I love the name Cahokia.

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Close to modern-day St. Louis,

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this was the largest city in the Americas.

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About 40,000 to maybe 60,000
people lived in this city.

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It's gonna be the largest
city in the Americas

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until the 1780s when
Philadelphia has more people.

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- [Voiceover] Wow, that
is just really remarkable.

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So 40,000 people here in Cahokia,

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and there were still
historical records that said

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that there was no one here?

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- [Voiceover] (giggling) Well
you know, history is something

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that is highly influenced
by the kind of narratives

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that historians wanna tell.

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We know that humans as a
species today originated

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somewhere in Africa and then
spread throughout Europe

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and Asia to other land masses.

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So how did they get to the
North American continent?

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- [Voiceover] So we don't know for sure,

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but archaeologists and historians
have pretty much agreed

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that the Native Americans
came through the Bering Strait

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which is as you can see right here.

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This was like this big ice
sheet during the Ice Age

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that connected Asia and
North America through Alaska.

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So the Native Americans came from Asia,

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they were Asian really,

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and they walked across the Bering Strait

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and down into North America.

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- [Voiceover] What is
now water at that time

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was covered with ice.

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So I'm guessing they're
following herds of animals

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that they might like to eat.

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- [Voiceover] Yes, exactly,

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and these were called the Paleo-Indians,

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so they were the first Indians.

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Paleo-Indians really means ancient people.

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So these were the ancient
people and they were, again,

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following different animals.

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So historians estimate that
the Paleo-Indians came down

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through the Bering Strait
about 13,000 years ago, so--

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- [Voiceover] Wow!

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- [Voiceover] It was a long time ago.

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But now different archaeological
evidence may point

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to the Paleo-Indians coming
over much earlier than that,

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actually more like 16,000 years ago.

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- [Voiceover] Yeah, I mean 3,000 years is

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a significant difference.

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I mean think about 3,000
years earlier than today,

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we'd be talking 1000 B.C.E.

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- [Voiceover] So today
archaeologists and historians

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kind of argue about when exactly it was,

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and lots of different evidence and fossils

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and archaeological digs
are coming up with evidence

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that points us more
toward 16,000 than 13,000.

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So it was actually in
1925 when archaeologists

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found the first Clovis point.

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So they're called Clovis
points because in 1925

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archaeologists found
these kinds of arrowheads.

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They're very small,
probably nothing more than

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like two centimeters,

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but they found them in Clovis, New Mexico,

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so over here in New Mexico,

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and estimated that they
were about 13,000 old.

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So that's kind of our old hypothesis,

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that humans were hunting
with these Clovis points

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13,000 years ago,
probably buffalo or bison,

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or any other kind of larger
animal that they would need

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to carve out of stone
these arrowheads to shoot,

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with a bow and arrow most likely,

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these larger animals to eat.

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- [Voiceover] So this is
some of the first evidence

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that leads archaeologists
to an interpretation

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of when human habitation first
occurred in North America?

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- [Voiceover] Exactly and this
was kind of the overwhelming

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consensus for really long time.

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All throughout the 1900s
people are digging up

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different artifacts all
across North America,

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finding more Clovis points
and kind of all agree

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it was about 13,000 years
ago that they came, again,

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down through here and all around,

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and that's why we were
finding these Clovis points.

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But then in 1997, not too
long ago at all, actually,

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so a Vanderbilt archaeologist,
his name is Tom Dillehay,

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found evidence of humanity in
Chile, all the way down here.

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So it seemed like the humans were living

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about 14,000 years ago.

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So it would have only
taken them a thousand years

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to get all the way from up here down here.

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So now there's kind of
two competing hypotheses.

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Either they traveled by land
through the Bering Strait

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and got all the way down here
in just a thousand years,

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by foot, I mean they were really moving.

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Or they actually were using
some sort of maritime vessel

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to get down, they were in a boat.

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But there's just these
small different theories

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within this archaeological
movement about exactly how

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they moved about and exactly when.

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There's also this new
branch of DNA science

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which is enlightening
historians about who the first

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Native Americans were and who
their direct ancestors are.

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- [Voiceover] These Paleo-Indians,

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they're coming from Alaska,

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they're coming from
Siberia I guess up there,

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so are their direct
ancestors Russians, Asians?

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- [Voiceover] The DNA evidence
points that the Paleo-Indians

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were most closely related to the Asians.

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They moved through Asia and
down through the Bering Strait.

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- [Voiceover] Interesting,
so that kinda corraborrates

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the archaeological record, right?

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If we say that somewhere
between 10 and 16,000 years ago

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there was a splintering of this population

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from the Asian population.

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The Bering Strait became water again,

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and now in sort of classic
Darwin fashion we have

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two separated populations
evolving differently.

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- [Voiceover] Exactly,
and the DNA definitely

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corraborrates that evidence.

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So today archaeologists and DNA scientists

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are working together to
figure out really which

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of these hypotheses is more true.

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Did they move by land or was it by water?

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And they wanna learn more
about exactly who were they?

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Yeah, they were Asian but how
did they distance themselves

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from the rest of the
East Asian population?

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Really, what motivated
them to come down through

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the Bering Strait and
explore this territory?

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- [Voiceover] One thing that
I think is really important

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to take away from this is that
if you look at the history

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of the Americas, you
start 16,000 years ago,

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and the history of
Europeans in America is just

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a tiny fraction of that time,

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so the history of America
starts long before 1776,

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long before 1492.

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It starts when the first
people came to America

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15,000 years ago.

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- [Voiceover] Exactly, and
the Paleo-Indians started

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their own American history
much before Columbus

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sailed the ocean blue in 1492, right?

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(giggling)

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And archaeologists are
continuing to learn more

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about the first Americans
and really the beginning

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of American history.


Common denominators: 1/4 and 5/6

WEBVTT
Kind: captions
Language: en

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- [Voiceover] You have two
fractions, 1/4 and 5/6,

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and you want to rewrite them

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so they have the same denominator

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and have whole number numerators.

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What numbers could you
use for the denominator?

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So here's our fractions,
1/4 and 5/6, and we want

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to rewrite these fractions
to have new denominators.

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So we currently have a four
and a six as our denominator,

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and could we just pick any
new thing, like maybe five?

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Could we say, "Let's change
them both to have five

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"as the denominator?"

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The answer is no.

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We have to pick a multiple
of four and six, a multiple,

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some number that we can multiply four

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and get this number as an answer.

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So for example, for four,
some multiples of four

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would be four times one is four,

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four times two is eight,

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four times three is 12, and so on.

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Those are multiples of four.

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And let's just pause here
and look at why we have

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to pick a multiple of four and six,

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why we can't just pick any number

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but we have to pick a
multiple of our denominators.

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So the fraction we were
just talking about was 1/4.

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We could look at either
one, but let's look at 1/4.

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Here we have a picture showing fourths.

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And to show 1/4,

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we shade one of these
four equal-size pieces.

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Maybe I wanna change this and I wanna say,

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"I want two, I want two as my numerator."

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So to have a numerator of two,

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I'm gonna need to split this
fourth up here into two pieces.

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Now I have two shaded pieces.

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So can I say this is two, one, two,

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out of one, two, three, four, five pieces?

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It's not 2/5 because these
are not equal-size pieces.

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So if I split this fourth
right here in half,

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I need to split all of them in half.

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And what I'm doing is doubling

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the amount of pieces.

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So now this is two pieces,
this is two, and this is two,

00:02:14.281 --> 00:02:16.934
because we need equal-size pieces.

00:02:16.934 --> 00:02:20.561
So now, this is one, two
pieces out of one, two,

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three, four, five, six, seven,
eight equal-size pieces.

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So 2/8.

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And you can see, eight
is a multiple of four

00:02:30.186 --> 00:02:32.535
because we multiplied by
two, and that's what we did.

00:02:32.535 --> 00:02:36.609
We multiplied each of our pieces by two.

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We also multiplied our numerator by two

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because that was also doubled.

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The amount of shaded pieces doubled

00:02:42.689 --> 00:02:45.610
when the entire amount of pieces doubled.

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Now, we don't have to
do this just with two.

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We could do any multiple of four.

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For example, we can do one more here.

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If we, again, let's shade 1/4,

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one of the four pieces.

00:03:01.544 --> 00:03:03.241
And maybe this time we wanna split it

00:03:03.241 --> 00:03:05.658
into three equal-size pieces.

00:03:08.136 --> 00:03:10.516
So to have a new numerator of three,

00:03:10.516 --> 00:03:13.452
and these should be equal-sized,

00:03:13.452 --> 00:03:15.785
here's a numerator of three.

00:03:17.265 --> 00:03:20.680
But we can't do our denominator yet

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because we don't have equal-size pieces.

00:03:23.365 --> 00:03:25.751
So to get equal-size pieces we'll need

00:03:25.751 --> 00:03:29.251
to split each of these fourths into three.

00:03:31.065 --> 00:03:34.398
So we are tripling the amount of pieces.

00:03:36.902 --> 00:03:40.350
So now we have three shaded
pieces out of a total

00:03:40.350 --> 00:03:43.557
of one, two, three, four,
five, six, seven, eight,

00:03:43.557 --> 00:03:44.974
nine, 10, 11, 12,

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out of 12.

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And I could have figured that
out without even counting

00:03:50.117 --> 00:03:53.372
because I knew that we tripled.

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This time we multiplied

00:03:57.637 --> 00:04:00.741
our original denominator
times three, we tripled.

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And we also multiplied
our numerator times three.

00:04:05.086 --> 00:04:08.738
So these are the multiples,
eight, 12, and so on,

00:04:08.738 --> 00:04:10.957
and so those are the
denominators we can pick,

00:04:10.957 --> 00:04:15.008
something that we can
multiply our denominator by

00:04:15.008 --> 00:04:18.925
so we can multiply the
entire amount of pieces.

00:04:20.478 --> 00:04:24.571
And again, so this is super
clear, 1/4 and 2/8 and 3/12,

00:04:24.571 --> 00:04:27.234
they all represent the same amount,

00:04:27.234 --> 00:04:30.827
whether we had 1/4, the
original, here's 2/8, 3/12,

00:04:30.827 --> 00:04:32.716
they're all equivalent.

00:04:32.716 --> 00:04:34.900
They all represent the same amount.

00:04:34.900 --> 00:04:39.472
They're just different ways
of writing the same number.

00:04:39.472 --> 00:04:41.107
Back to our original question,

00:04:41.107 --> 00:04:45.274
what denominators can we use
for fourths and for sixths?

00:04:46.529 --> 00:04:48.600
We know we need to use multiples,

00:04:48.600 --> 00:04:50.401
so let's look at the multiples.

00:04:50.401 --> 00:04:53.764
For four, we've already
gone through some of these.

00:04:53.764 --> 00:04:55.929
The first multiple of
four is four times one,

00:04:55.929 --> 00:04:57.715
which is four.

00:04:57.715 --> 00:04:59.790
Second multiple of four is eight.

00:04:59.790 --> 00:05:02.041
Four times two is eight, so
we could split our fourths

00:05:02.041 --> 00:05:04.041
in half and get eighths.

00:05:04.934 --> 00:05:08.751
Or we could say four times three is 12,

00:05:08.751 --> 00:05:10.784
which we showed again
where we split our fourths,

00:05:10.784 --> 00:05:14.020
each fourth into three equal pieces.

00:05:14.020 --> 00:05:17.395
Or we could do four
times four which is 16.

00:05:17.395 --> 00:05:19.784
Four times five is 20,

00:05:19.784 --> 00:05:22.451
four times six is 24, and so on.

00:05:24.126 --> 00:05:26.711
The reason I'm stopping
at 24 is I've looked

00:05:26.711 --> 00:05:30.330
at my answer choices and I
can see the largest possible

00:05:30.330 --> 00:05:32.020
answer is 24, so I don't need

00:05:32.020 --> 00:05:34.161
to write any larger multiples.

00:05:34.161 --> 00:05:37.061
There are many, many, many
more multiples of four,

00:05:37.061 --> 00:05:38.701
but we don't need to list them all

00:05:38.701 --> 00:05:40.883
'cause the largest number we're gonna have

00:05:40.883 --> 00:05:42.383
to consider is 24.

00:05:43.404 --> 00:05:45.637
Let's do the same for sixths.

00:05:45.637 --> 00:05:48.783
We could leave our sixths
alone, six times one,

00:05:48.783 --> 00:05:50.485
and keep six pieces.

00:05:50.485 --> 00:05:52.196
Or we could double our sixths.

00:05:52.196 --> 00:05:53.669
Six times two would be 12,

00:05:53.669 --> 00:05:56.977
if we doubled the pieces
we would have 12 pieces.

00:05:56.977 --> 00:05:59.871
We could say six times three, which is 18,

00:05:59.871 --> 00:06:01.608
or six times four, we could divide each

00:06:01.608 --> 00:06:03.999
of our sixths into four pieces,

00:06:03.999 --> 00:06:08.166
and we'd have six times
four which is 24 or 24ths.

00:06:09.218 --> 00:06:11.751
And so on, again, I'll stop at 24

00:06:11.751 --> 00:06:15.878
since it's the largest
number we need to consider.

00:06:15.878 --> 00:06:17.919
So down to our answer choices,

00:06:17.919 --> 00:06:20.132
what numbers could we
use for the denominator?

00:06:20.132 --> 00:06:21.454
Could we use eight?

00:06:21.454 --> 00:06:23.570
Let's look at these lists.

00:06:23.570 --> 00:06:27.470
Eight is a multiple of four,
so we could definitely split

00:06:27.470 --> 00:06:31.730
fourths into eighths, but
eight is not a multiple of six,

00:06:31.730 --> 00:06:35.067
so we can not split sixths into eighths.

00:06:35.067 --> 00:06:36.484
So...

00:06:36.484 --> 00:06:40.651
Eight will not work as a
denominator for both fractions.

00:06:42.314 --> 00:06:43.397
How about 12?

00:06:44.803 --> 00:06:47.845
12 we can see is a multiple
of four, and we showed that,

00:06:47.845 --> 00:06:49.361
we drew that already.

00:06:49.361 --> 00:06:51.017
And 12 is a multiple of six,

00:06:51.017 --> 00:06:53.466
we could split our sixths
into two equal pieces each

00:06:53.466 --> 00:06:55.559
and we would have 12ths.

00:06:55.559 --> 00:06:59.264
So 12 does work, 12 is a denominator,

00:06:59.264 --> 00:07:02.931
a common denominator
for fourths and sixths.

00:07:03.843 --> 00:07:06.832
18, 18 is here on the sixths.

00:07:06.832 --> 00:07:08.795
We could split sixths into 18

00:07:08.795 --> 00:07:10.735
because 18 is a multiple of six,

00:07:10.735 --> 00:07:13.874
but it is not a multiple of four.

00:07:13.874 --> 00:07:17.883
So we can rule out 18, 18
is not a common denominator.

00:07:17.883 --> 00:07:21.121
And 24, you may remember,
was the last number

00:07:21.121 --> 00:07:22.858
we wrote on both of them.

00:07:22.858 --> 00:07:25.608
So yes, 24 could be a denominator

00:07:26.685 --> 00:07:28.696
for fourths and sixths.

00:07:28.696 --> 00:07:31.405
So we could use either 12 or 24,

00:07:31.405 --> 00:07:33.606
and there's a lot more
numbers we could use too

00:07:33.606 --> 00:07:36.365
as common denominators,
but from these choices

00:07:36.365 --> 00:07:40.069
we could use 12 or 24
as a common denominator

00:07:40.069 --> 00:07:41.986
for fourths and sixths.

00:07:43.071 --> 00:07:45.313
And just a note, lots of times people like

00:07:45.313 --> 00:07:48.565
to use the smallest one, the
least common denominator,

00:07:48.565 --> 00:07:50.264
which in this case is 12.

00:07:50.264 --> 00:07:52.517
And it makes a lot of
sense because it's easier

00:07:52.517 --> 00:07:54.753
to do computation with smaller numbers,

00:07:54.753 --> 00:07:58.453
but you don't absolutely
have to use the smallest one.

00:07:58.453 --> 00:08:02.359
You could use 12ths or 24ths
or lots of other options.

00:08:02.359 --> 00:08:05.634
But again, 12ths is
probably the simplest one

00:08:05.634 --> 00:08:07.935
to work with just 'cause
generally it's easier

00:08:07.935 --> 00:08:10.149
to work with smaller numbers.

00:08:10.149 --> 00:08:12.654
But for this question,
the common denominators

00:08:12.654 --> 00:08:14.369
we can use from these choices

00:08:14.369 --> 00:08:17.452
for fourths and sixths are 12 and 24.


Eulers sine wave

WEBVTT
Kind: captions
Language: en

00:00:00.068 --> 00:00:03.068
(upbeat jazz music)


Impedance vs frequency

WEBVTT
Kind: captions
Language: en

00:00:00.212 --> 00:00:01.692
- [Voiceover] In this
video we're gonna continue

00:00:01.692 --> 00:00:04.369
talking about AC analysis
and the concept of impedance

00:00:04.369 --> 00:00:08.536
as the ratio of voltage to
current and an AC situation,

00:00:09.944 --> 00:00:13.200
and just as a reminder of
the assumptions we've made

00:00:13.200 --> 00:00:16.842
for AC analysis we've assumed
that all of our signals

00:00:16.842 --> 00:00:21.009
are of the form of some kind
of a sinusoid like that,

00:00:22.063 --> 00:00:24.382
cosine is the typical one we pick,

00:00:24.382 --> 00:00:27.274
cosine omega T plus some offset angle.

00:00:27.274 --> 00:00:30.125
And we're also going
to use Euler's equation

00:00:30.125 --> 00:00:32.882
which says we can decompose our cosine

00:00:32.882 --> 00:00:35.775
into two complex exponentials,

00:00:35.775 --> 00:00:39.942
so one of the complex
exponentials is either the plus J

00:00:41.756 --> 00:00:43.006
omega T plus fi

00:00:46.119 --> 00:00:50.286
and J multiplies both of
them, plus e to the minus J

00:00:54.611 --> 00:00:55.944
omega T plus fi.

00:00:59.323 --> 00:01:01.351
And when we use Euler's
theorem what we do is

00:01:01.351 --> 00:01:05.058
we love putting these things
through differential equations

00:01:05.058 --> 00:01:07.608
so we track each of these
one at a time through

00:01:07.608 --> 00:01:10.034
our circuits and typically
what I'll focus on

00:01:10.034 --> 00:01:13.014
is the plus term, but the
minus term all the math

00:01:13.014 --> 00:01:15.769
is the same except there's
that one little minus sign.

00:01:15.769 --> 00:01:18.360
And so when we made up
the concept of impedance,

00:01:18.360 --> 00:01:22.488
we assumed that V or I was of this form,

00:01:22.488 --> 00:01:25.958
this complex exponential, and
that may, when we put that

00:01:25.958 --> 00:01:28.224
through our components
we came up with the idea

00:01:28.224 --> 00:01:31.974
of impedance, and as a
review for a resistor,

00:01:32.856 --> 00:01:36.485
the impedance of a resistor
turned out to be just R,

00:01:36.485 --> 00:01:40.161
that's the ratio of V to I in a resistor.

00:01:40.161 --> 00:01:41.828
And for an inductor,

00:01:45.810 --> 00:01:49.977
we decided Z of an inductor
was equal to J omega L.

00:01:53.012 --> 00:01:56.262
And finally for a capacitor of value C,

00:01:58.690 --> 00:02:01.190
Z C equals one over J omega C.

00:02:06.070 --> 00:02:07.966
So those are the three forms of,

00:02:07.966 --> 00:02:10.529
those are the three impedences
of our three favorite

00:02:10.529 --> 00:02:12.112
passive components.

00:02:13.134 --> 00:02:16.967
And the units of these,
the units are in ohms.

00:02:18.043 --> 00:02:21.319
R is measured in ohms,
that's the normal ohms law,

00:02:21.319 --> 00:02:24.700
and we use the same units
for impedance of inductors

00:02:24.700 --> 00:02:28.867
and capacitors so this is
in ohms and this is in ohms.

00:02:31.338 --> 00:02:33.852
That may seem a little funny
that because there's this

00:02:33.852 --> 00:02:37.852
frequency term in here,
but, this is what it is.

00:02:38.821 --> 00:02:41.042
So in the rest of this
video I want to look at

00:02:41.042 --> 00:02:44.551
qualitatively look at the
value of these impedences

00:02:44.551 --> 00:02:47.048
and specifically look at
what happens when there's

00:02:47.048 --> 00:02:49.726
a range of values of the omega term,

00:02:49.726 --> 00:02:52.716
what happens at zero
frequency and low frequency

00:02:52.716 --> 00:02:55.396
and high frequency and infinite frequency.

00:02:55.396 --> 00:02:57.896
So let's go ahead and do that.

00:03:02.039 --> 00:03:05.167
So we're gonna look at the
impedance terms at different

00:03:05.167 --> 00:03:07.728
frequencies and we'll
measure frequency at radiant

00:03:07.728 --> 00:03:11.150
frequency in this, so, let's
talk about zero frequency

00:03:11.150 --> 00:03:14.115
and we'll talk about low frequency,

00:03:14.115 --> 00:03:15.721
these are qualitative boundaries,

00:03:15.721 --> 00:03:18.360
we'll talk about high frequency,

00:03:18.360 --> 00:03:21.252
and let's talk about infinite frequency.

00:03:21.252 --> 00:03:25.290
So we're gonna build a little chart here.

00:03:25.290 --> 00:03:29.040
Alright so these are
the value of omega here.

00:03:30.370 --> 00:03:33.261
And now we'll do our
components, so I'm gonna do,

00:03:33.261 --> 00:03:36.152
first we'll do our resistor,

00:03:36.152 --> 00:03:40.021
and we're gonna fill in the table for ZR,

00:03:40.021 --> 00:03:42.003
and we know that equals R.

00:03:42.003 --> 00:03:46.753
So, at any frequency, R is
just R, couldn't be simpler.

00:03:46.753 --> 00:03:50.920
At zero frequency which is
just called DC or battery,

00:03:51.944 --> 00:03:52.777
R is R,

00:03:53.971 --> 00:03:57.842
at any low frequency, R is R, R is R,

00:03:57.842 --> 00:04:00.600
at infinite frequency, so there's
no dependence on frequency

00:04:00.600 --> 00:04:03.517
in R, so now let's do the inductor,

00:04:06.615 --> 00:04:10.782
and we decided that Z of
an inductor was J omega L.

00:04:15.318 --> 00:04:17.841
And let me do something
very specific, I'm gonna do,

00:04:17.841 --> 00:04:20.479
I'm gonna get rid of this
J, I'm gonna basically say

00:04:20.479 --> 00:04:25.105
I wanna just look at the
magnitude of the impedance.

00:04:25.105 --> 00:04:26.950
If we just look at the magnitudes,

00:04:26.950 --> 00:04:30.200
the magnitude of Z inductor is omega L.

00:04:36.610 --> 00:04:39.997
So now let's fill our
table in for omega L.

00:04:39.997 --> 00:04:41.830
So when omega is zero,

00:04:43.264 --> 00:04:46.485
the magnitude of the impedance
is zero for an inductor.

00:04:46.485 --> 00:04:50.568
And when the frequency is
low, when omega is low,

00:04:51.642 --> 00:04:55.933
the impedance is going
to be relatively low.

00:04:55.933 --> 00:04:58.516
And as the frequency gets high,

00:04:59.445 --> 00:05:03.612
then omega L becomes a larger
number, so it becomes high.

00:05:05.063 --> 00:05:08.939
And if the frequency, if we let
the frequency go to infinity

00:05:08.939 --> 00:05:13.106
then omega becomes infinity
and this becomes infinity.

00:05:14.686 --> 00:05:18.114
So we see an inductor from low
frequency to high frequency,

00:05:18.114 --> 00:05:22.281
the impedance of the inductor
goes from zero up to infinity.

00:05:23.605 --> 00:05:26.295
Alright let's fill in the last one here,

00:05:26.295 --> 00:05:28.045
here's our capacitor,

00:05:29.469 --> 00:05:33.636
and the Z of a capacitor
equals one over J omega C,

00:05:36.569 --> 00:05:40.736
and the magnitude of the
impedance is just 1 over omega C.

00:05:47.014 --> 00:05:50.816
Alright so let's plot that
out, what is 1 over omega C

00:05:50.816 --> 00:05:52.399
when omega is zero?

00:05:53.288 --> 00:05:54.871
Well it's infinity.

00:05:58.451 --> 00:06:01.636
And what if omega is a small number?

00:06:01.636 --> 00:06:05.803
If omega is a small number
then this is a large number.

00:06:07.976 --> 00:06:09.809
This is a high number.

00:06:11.444 --> 00:06:15.037
And if the frequency gets very
high, the higher omega gets

00:06:15.037 --> 00:06:18.461
the smaller the magnitude of Z gets,

00:06:18.461 --> 00:06:22.378
so we get low here, you
see it's anti symmetric

00:06:23.724 --> 00:06:27.872
with the inductor, and finally,
if we let the frequency

00:06:27.872 --> 00:06:32.156
go to infinity, 1 over
infinity times C is what?

00:06:32.156 --> 00:06:33.656
Is about, is zero.

00:06:36.358 --> 00:06:38.802
With this chart I can show you
some of the words that we use

00:06:38.802 --> 00:06:41.436
some of the sort of slang
words or jargon words

00:06:41.436 --> 00:06:43.625
that we use in electrical
engineering to describe

00:06:43.625 --> 00:06:48.050
the behavior of L and C
at different frequencies.

00:06:48.050 --> 00:06:51.029
So as we said, R is R at any frequency,

00:06:51.029 --> 00:06:53.089
so resistance is constant.

00:06:53.089 --> 00:06:55.852
An inductor it changes over frequency,

00:06:55.852 --> 00:06:58.658
over this big range of
frequency, and at zero frequency

00:06:58.658 --> 00:07:02.237
or low frequencies the
impedance is very low.

00:07:02.237 --> 00:07:05.041
So this leads to the
expression we say an inductor

00:07:05.041 --> 00:07:06.624
looks like a short.

00:07:11.580 --> 00:07:14.393
An inductor looks like a
short at low frequency,

00:07:14.393 --> 00:07:16.499
at zero frequency or low frequency.

00:07:16.499 --> 00:07:19.558
And now let's go to high
frequency, the impedance gets high

00:07:19.558 --> 00:07:22.073
and it eventually goes to infinity.

00:07:22.073 --> 00:07:24.463
That's the impedance of an open circuit.

00:07:24.463 --> 00:07:27.546
So we say L equals an open at high F,

00:07:31.578 --> 00:07:35.745
high frequency, and it's
a short at low frequency.

00:07:37.880 --> 00:07:40.266
Let's do the same sort of
jargon investigation here

00:07:40.266 --> 00:07:42.083
for our capacitor.

00:07:42.083 --> 00:07:45.420
A capacitor at zero frequency
has an infinite impedance.

00:07:45.420 --> 00:07:47.101
So it looks like what?

00:07:47.101 --> 00:07:50.351
It looks like an open at low frequency.

00:07:58.649 --> 00:08:00.659
So let's go look over at high frequencies,

00:08:00.659 --> 00:08:03.629
high frequencies, the
impedance becomes very low.

00:08:03.629 --> 00:08:06.676
And at infinite frequency
it becomes zero and that's

00:08:06.676 --> 00:08:09.187
the impedance of a perfect short.

00:08:09.187 --> 00:08:11.627
So the expression, the
slang expression you'll hear

00:08:11.627 --> 00:08:15.794
is that a capacitor is a short
circuit at high frequency.

00:08:23.503 --> 00:08:25.448
So the reason I'm telling you about this,

00:08:25.448 --> 00:08:29.794
very often experienced
engineers will talk to beginners

00:08:29.794 --> 00:08:32.142
and use these kinds of terms,
that oh an inductor looks

00:08:32.142 --> 00:08:34.003
like a short at low frequency.

00:08:34.003 --> 00:08:36.387
And I wanted to show you
where these things come from,

00:08:36.387 --> 00:08:38.403
where these terms come
from and what they mean.

00:08:38.403 --> 00:08:41.832
And this is not a simple idea,
you saw how many assumptions

00:08:41.832 --> 00:08:44.186
we made and how much work
we did to get to this idea

00:08:44.186 --> 00:08:48.103
of these sort of common
or familiar expressions

00:08:48.962 --> 00:08:51.844
for how our components work.

00:08:51.844 --> 00:08:55.633
I hope that helped for you
to see how this comes about.


Euler's cosine wave

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:03:03.040
(music playing)


Sine and cosine from rotating vector

WEBVTT
Kind: captions
Language: en

00:00:00.278 --> 00:00:01.699
- [Voiceover] Now, I'd like to demonstrate

00:00:01.699 --> 00:00:03.773
one way to construct a sine wave.

00:00:03.773 --> 00:00:05.791
What we're gonna do is
we're gonna construct

00:00:05.791 --> 00:00:09.353
something that looks like sine of omega t.

00:00:09.353 --> 00:00:13.239
So, we have our function of time here.

00:00:13.239 --> 00:00:15.489
And, we have our frequency.

00:00:19.843 --> 00:00:21.652
Now, this little animation
is gonna show us a way

00:00:21.652 --> 00:00:23.133
to construct a sine wave.

00:00:23.133 --> 00:00:24.728
So, what I have here, this green line

00:00:24.728 --> 00:00:26.818
is a rotating vector.

00:00:26.818 --> 00:00:30.601
And, let's just say that the
radius of this circle is one.

00:00:30.601 --> 00:00:34.379
So, here's a vector just rotating
slowly around and around.

00:00:34.379 --> 00:00:36.971
And, in the dotted line
here and that yellow dot

00:00:36.971 --> 00:00:40.709
going up and down, that's the projection.

00:00:40.709 --> 00:00:43.997
That's the projection of
the tip of the green arrow

00:00:43.997 --> 00:00:45.330
onto the y axis.

00:00:53.960 --> 00:00:55.954
And, as the vector goes around and around,

00:00:55.954 --> 00:00:58.770
you could see that the
projection on the y axis

00:00:58.770 --> 00:01:01.367
is bobbing up and down and up and down.

00:01:01.367 --> 00:01:02.812
And, that's actually going up and down

00:01:02.812 --> 00:01:04.275
in a sine wave pattern.

00:01:04.275 --> 00:01:06.375
So, now I'm gonna switch
to a new animation

00:01:06.375 --> 00:01:08.501
and we'll see what that dot looks like

00:01:08.501 --> 00:01:11.084
as it goes up and down in time.

00:01:15.552 --> 00:01:16.542
So, here's the plot.

00:01:16.542 --> 00:01:18.691
Here's what a sine wave looks like.

00:01:18.691 --> 00:01:20.980
As you notice, when the
green line goes through zero

00:01:20.980 --> 00:01:23.224
right there, let's wait
'til it comes around again,

00:01:23.224 --> 00:01:26.321
the value of the yellow line
when it goes through zero

00:01:26.321 --> 00:01:27.154
is zero.

00:01:29.203 --> 00:01:33.370
So, this yellow line here is
a plot of sine of omega t.

00:01:36.660 --> 00:01:38.581
Now, if I go to a projection,

00:01:38.581 --> 00:01:40.504
this projection was onto the y axis,

00:01:40.504 --> 00:01:42.455
and I can do the same
animation but this time

00:01:42.455 --> 00:01:44.522
project onto the x axis.

00:01:44.522 --> 00:01:46.810
And, that will produce
for us a cosine wave.

00:01:46.810 --> 00:01:49.393
Let's see what that looks like.

00:01:59.993 --> 00:02:01.279
Now, in this case that
we've switched over,

00:02:01.279 --> 00:02:02.953
you can see that the projection,

00:02:02.953 --> 00:02:06.009
that dotted green line, is onto the x axis

00:02:06.009 --> 00:02:07.656
and what this is doing is it's producing

00:02:07.656 --> 00:02:08.933
a cosine wave for us.

00:02:08.933 --> 00:02:12.183
So, this is gonna be cosine of omega t.

00:02:18.026 --> 00:02:21.765
Now, because we're tracking
the progress on the x axis,

00:02:21.765 --> 00:02:24.957
the cosine wave seems to
emerge going down on the page.

00:02:24.957 --> 00:02:27.540
So, the time axis is down here.

00:02:29.714 --> 00:02:31.691
When the green arrow
hits zero right there,

00:02:31.691 --> 00:02:33.882
the value of the cosine was one,

00:02:33.882 --> 00:02:35.747
and when it's minus 180 degrees

00:02:35.747 --> 00:02:37.959
it's minus one on the cosine.

00:02:37.959 --> 00:02:40.876
So, that's why this is cosine wave.

00:02:42.093 --> 00:02:44.192
And, it has the same
frequency as the sine wave

00:02:44.192 --> 00:02:45.275
we generated.

00:02:46.247 --> 00:02:47.781
And now I wanna show
you these two together

00:02:47.781 --> 00:02:51.614
because it's just sort
of a beautiful drawing.

00:03:09.638 --> 00:03:11.288
I'll leave our animation
here for a second.

00:03:11.288 --> 00:03:13.898
We see our sine wave
being generated in yellow

00:03:13.898 --> 00:03:16.719
and in orange we see the
cosine wave being generated

00:03:16.719 --> 00:03:21.595
and they're both coming from
this rotating green vector.

00:03:21.595 --> 00:03:24.125
So, this is a really simple
demonstration of a way

00:03:24.125 --> 00:03:25.977
to generate sines and cosines

00:03:25.977 --> 00:03:28.560
with this rotating vector idea.

00:03:29.847 --> 00:03:31.748
We're gonna be able to
generate this rotating vector

00:03:31.748 --> 00:03:34.143
using some ideas from complex arithmetic

00:03:34.143 --> 00:03:35.810
and Euler's formula.

00:03:38.753 --> 00:03:41.086
I find these to be a
really beautiful pattern

00:03:41.086 --> 00:03:45.253
and it emerges from such a
simple idea as a rotating vector.


Sine of time

WEBVTT
Kind: captions
Language: en

00:00:00.954 --> 00:00:03.655
- [Voiceover] Now I wanna
introduce a new idea,

00:00:03.655 --> 00:00:05.572
and that is the idea of

00:00:07.474 --> 00:00:11.423
voltage or current, some electrical signal

00:00:11.423 --> 00:00:14.200
being a function of time,

00:00:14.200 --> 00:00:15.700
cosine of omega t.

00:00:18.695 --> 00:00:21.219
So here what we're doing
is we're introducing time

00:00:21.219 --> 00:00:23.552
as the argument to a cosine.

00:00:26.710 --> 00:00:29.410
And time is that stuff
that always goes up,

00:00:29.410 --> 00:00:32.716
this is a number that increases forever.

00:00:32.716 --> 00:00:35.024
And we have another variable
in here called omega,

00:00:35.024 --> 00:00:38.143
this is the Greek lowercase omega,

00:00:38.143 --> 00:00:41.566
and omega has an important job in this.

00:00:41.566 --> 00:00:45.733
The argument to cosine,
whatever is inside the cosine,

00:00:46.971 --> 00:00:51.074
this has to be dimensionless,
this has to have no units.

00:00:51.074 --> 00:00:55.843
And so if we put in a unit
of seconds, that means omega

00:00:55.843 --> 00:00:58.592
is something that has the
units of one over seconds,

00:00:58.592 --> 00:01:00.389
or one over time.

00:01:00.389 --> 00:01:02.556
So omega is one over time.

00:01:06.537 --> 00:01:08.387
And when we multiply those
two numbers together,

00:01:08.387 --> 00:01:10.861
we get something that has no
units, and then we can take

00:01:10.861 --> 00:01:12.751
the cosine of it.

00:01:12.751 --> 00:01:16.204
So, this is referred to as a frequency.

00:01:16.204 --> 00:01:20.371
Something that has units of
one over time is a frequency.

00:01:23.203 --> 00:01:25.372
This is a constant number,
this is some number,

00:01:25.372 --> 00:01:28.021
this is a number, time is
a number that increases

00:01:28.021 --> 00:01:29.303
all the time,

00:01:29.303 --> 00:01:32.154
and so when we have that cosine,

00:01:32.154 --> 00:01:35.306
we now have something we call
a cosine wave or a sine wave,

00:01:35.306 --> 00:01:37.018
or a sinusoid.

00:01:37.018 --> 00:01:39.268
And that sine wave goes on.

00:01:40.708 --> 00:01:44.240
As time increases, it keeps
going and going and going.

00:01:44.240 --> 00:01:48.516
So now we've turned our
trigonometric cosine function,

00:01:48.516 --> 00:01:50.446
which is right here, which
is something that was

00:01:50.446 --> 00:01:54.660
well defined between
zero and two pi radians.

00:01:54.660 --> 00:01:57.820
Notice that I've changed the
axis, the axis is now in time

00:01:57.820 --> 00:02:00.891
over here, and now we're
counting off time in seconds,

00:02:00.891 --> 00:02:03.347
there's two seconds, three, four, five,

00:02:03.347 --> 00:02:06.412
and that dot there, that's at pi seconds,

00:02:06.412 --> 00:02:09.312
and this is at two pi seconds,

00:02:09.312 --> 00:02:11.450
right at that dot right there.

00:02:11.450 --> 00:02:15.617
And you can see that that is
the full cycle of one cosine

00:02:16.609 --> 00:02:18.865
before it starts repeating again,

00:02:18.865 --> 00:02:21.540
so that's 6.28 seconds.

00:02:21.540 --> 00:02:25.652
So for this image here,
omega has the value of one.

00:02:25.652 --> 00:02:28.819
So when time t reaches two pi seconds,

00:02:30.050 --> 00:02:33.148
we've gone through one full cycle.

00:02:33.148 --> 00:02:37.298
So this idea of this continuously changing

00:02:37.298 --> 00:02:39.731
cosine or sine wave going on forever,

00:02:39.731 --> 00:02:42.564
that gives us the term sine waves,

00:02:43.422 --> 00:02:45.384
and sine waves are a good
model for a lot of things

00:02:45.384 --> 00:02:46.582
that happen in nature.

00:02:46.582 --> 00:02:50.317
If you ever hear a pure
tone or a pure note,

00:02:50.317 --> 00:02:53.070
a bell being rang or a whistle,

00:02:53.070 --> 00:02:56.146
or if you sing a note,
the shape of those tones

00:02:56.146 --> 00:02:59.068
looks like a sine wave or a cosine wave.

00:02:59.068 --> 00:03:00.840
And these are often the
things that come into

00:03:00.840 --> 00:03:05.007
our electronic systems and
we wanna do things with them.

00:03:06.025 --> 00:03:09.043
So now I wanna talk a little
bit more about the details

00:03:09.043 --> 00:03:11.710
of this kind of a sinusoid wave.

00:03:12.860 --> 00:03:16.360
We're gonna learn some new words for this.

00:03:17.260 --> 00:03:19.640
So one important concept is the idea of

00:03:19.640 --> 00:03:23.655
any repeating waveform,
any repeating signal,

00:03:23.655 --> 00:03:25.140
is the idea of a period.

00:03:25.140 --> 00:03:27.403
Let's just do the zero crossings here.

00:03:27.403 --> 00:03:31.236
If I take the time change
from there to there,

00:03:32.459 --> 00:03:37.025
this is the repeating
interval of this function,

00:03:37.025 --> 00:03:41.337
and I'm gonna call that,
that distance right there,

00:03:41.337 --> 00:03:42.837
this is the period

00:03:46.001 --> 00:03:48.355
of this sine function.

00:03:48.355 --> 00:03:50.257
This is actually a cosine wave.

00:03:50.257 --> 00:03:54.121
The period of this sine is this
distance in time right here,

00:03:54.121 --> 00:03:57.628
and the symbol we use
is typically a capital T

00:03:57.628 --> 00:03:59.545
to indicate the period.

00:04:03.890 --> 00:04:07.465
So let's look at this
cosine wave, this sinusoid,

00:04:07.465 --> 00:04:09.859
and identify what its period is.

00:04:09.859 --> 00:04:13.026
I can do it easily if I go right here.

00:04:13.958 --> 00:04:17.293
It looks like it repeats on
this interval right here.

00:04:17.293 --> 00:04:20.424
Every time we hit one of those points,

00:04:20.424 --> 00:04:23.219
so this would be, if
this is time zero here,

00:04:23.219 --> 00:04:27.170
this is time big T, this is time two T,

00:04:27.170 --> 00:04:28.594
on and on like that.

00:04:28.594 --> 00:04:30.904
And I read off this graph,
and what I see right here

00:04:30.904 --> 00:04:32.487
is that the time is

00:04:34.444 --> 00:04:36.277
T equals 0.02 seconds.

00:04:41.473 --> 00:04:43.506
So that's how you find out what
the period of something is.

00:04:43.506 --> 00:04:44.604
You can take any two points.

00:04:44.604 --> 00:04:46.731
We could actually go right here

00:04:46.731 --> 00:04:48.887
and then go through one cycle,

00:04:48.887 --> 00:04:51.217
and go to right here

00:04:51.217 --> 00:04:54.169
and I can read off that
period there, there's T,

00:04:54.169 --> 00:04:58.013
and that's the same value
as that T right over there.

00:04:58.013 --> 00:05:01.763
So the time T, we can
also call that a cycle.

00:05:03.546 --> 00:05:05.804
That's the time it takes
to go through one period,

00:05:05.804 --> 00:05:07.221
that's one cycle.

00:05:09.052 --> 00:05:11.561
So one of the questions I can
ask about this waveform is,

00:05:11.561 --> 00:05:14.679
how many cycles fit in one second?

00:05:14.679 --> 00:05:18.617
How many cycles per second,
is another way to say that.

00:05:18.617 --> 00:05:20.950
So we can say that one cycle

00:05:22.525 --> 00:05:24.525
happens every T seconds,

00:05:28.934 --> 00:05:32.434
and in our particular case, it's one cycle

00:05:33.698 --> 00:05:35.115
per 0.02 seconds,

00:05:39.720 --> 00:05:43.284
and if take the reciprocal of 0.02,

00:05:43.284 --> 00:05:46.015
we get the answer to be,

00:05:46.015 --> 00:05:46.848
that's 50

00:05:48.004 --> 00:05:49.504
cycles per second.

00:05:55.258 --> 00:05:58.590
That's the speed, that's
the repetition rate

00:05:58.590 --> 00:06:01.805
of this sinusoid, 50 cycles per second.

00:06:01.805 --> 00:06:05.278
And this has another name,
it's named in honor of

00:06:05.278 --> 00:06:09.195
a German scientist, and
this is called a Hertz.

00:06:12.119 --> 00:06:14.886
Heinrich Hertz is the first
person to send a radio wave

00:06:14.886 --> 00:06:19.126
and receive it on purpose,
he knew what he was doing.

00:06:19.126 --> 00:06:21.983
We named the unit cycles
per second in his honor,

00:06:21.983 --> 00:06:24.316
and that's called the Hertz.

00:06:26.948 --> 00:06:28.695
So now we have two ways
to measure frequency.

00:06:28.695 --> 00:06:29.528
One is

00:06:30.365 --> 00:06:34.532
f, which is frequency,
which is measured in Hertz,

00:06:35.451 --> 00:06:37.868
and that's cycles per second.

00:06:43.442 --> 00:06:44.525
And one cycle

00:06:45.962 --> 00:06:47.045
equals two pi

00:06:49.154 --> 00:06:50.737
radians per second.

00:06:55.070 --> 00:06:58.206
So the two measures are cycles per second

00:06:58.206 --> 00:07:00.316
and radians per second, and
we'll flip back and forth

00:07:00.316 --> 00:07:01.483
between those.

00:07:03.385 --> 00:07:07.552
Okay, and radians per second,
or the variable is omega,

00:07:08.553 --> 00:07:11.553
and that's called angular frequency,

00:07:16.004 --> 00:07:17.671
or radian frequency,

00:07:22.125 --> 00:07:25.763
and you'll sometimes see the word rad

00:07:25.763 --> 00:07:28.526
used to indicate that
we're talking about angular

00:07:28.526 --> 00:07:32.443
or radian frequency, and
the variable is omega.

00:07:33.406 --> 00:07:35.938
So let's work out what,
the relationship between

00:07:35.938 --> 00:07:37.766
f and omega.

00:07:37.766 --> 00:07:40.064
It's actually sitting right here, okay.

00:07:40.064 --> 00:07:41.897
So if I give you an f,

00:07:43.334 --> 00:07:44.167
given f,

00:07:47.615 --> 00:07:48.782
what is omega?

00:07:53.370 --> 00:07:57.537
So I write down a number f,
and it's in cycles per second,

00:08:01.641 --> 00:08:03.681
and I'm gonna multiply
that by a conversion factor

00:08:03.681 --> 00:08:07.252
that I'm gonna make up, so
we're gonna multiply that by

00:08:07.252 --> 00:08:08.085
two pi

00:08:10.972 --> 00:08:12.472
radians per second

00:08:15.152 --> 00:08:18.152
is the same as one cycle per second.

00:08:28.796 --> 00:08:30.224
And that equals,

00:08:30.224 --> 00:08:33.298
cycles per second cancels
with cycles per second,

00:08:33.298 --> 00:08:34.465
so that equals

00:08:35.584 --> 00:08:36.417
two pi f

00:08:41.418 --> 00:08:43.001
radians per second.

00:08:45.525 --> 00:08:47.946
So the conversion factor is

00:08:47.946 --> 00:08:49.779
omega equals two pi f,

00:08:55.255 --> 00:08:57.672
and that's worth remembering.

00:08:59.748 --> 00:09:04.120
So if I have a sine wave, a
voltage sine wave, for instance,

00:09:04.120 --> 00:09:04.953
v of t

00:09:06.237 --> 00:09:07.070
equals

00:09:09.260 --> 00:09:10.510
cosine omega t,

00:09:12.126 --> 00:09:13.969
I can write that equivalently as

00:09:13.969 --> 00:09:14.802
v of t

00:09:16.940 --> 00:09:18.023
equals cosine

00:09:20.305 --> 00:09:21.722
two pi f times t.

00:09:28.848 --> 00:09:32.587
So one of the frequencies, this
one is in cycles per second,

00:09:32.587 --> 00:09:35.419
and this one is in radians per second,

00:09:35.419 --> 00:09:37.567
and we can interchange 'em that way

00:09:37.567 --> 00:09:39.984
using this conversion factor.

00:09:41.581 --> 00:09:43.708
So if we take the example
from earlier in the video,

00:09:43.708 --> 00:09:48.471
we had a signal that was 50
Hertz, or 50 cycles per second,

00:09:48.471 --> 00:09:50.892
so we would write that here like this.

00:09:50.892 --> 00:09:52.142
We'd say v of t

00:09:53.381 --> 00:09:54.464
equals cosine

00:09:57.707 --> 00:09:59.540
two pi f, and f is 50,

00:10:04.620 --> 00:10:05.453
times t,

00:10:08.638 --> 00:10:11.210
and that's the same as

00:10:11.210 --> 00:10:12.793
cosine of 100 pi t.

00:10:19.724 --> 00:10:24.262
So this number right here,
100 pi, that's omega,

00:10:24.262 --> 00:10:26.863
and this number right here

00:10:26.863 --> 00:10:27.696
is f.

00:10:30.151 --> 00:10:32.519
So that does it for our
review of trigonometry,

00:10:32.519 --> 00:10:34.664
and we've introduced
the idea of a sine wave

00:10:34.664 --> 00:10:37.174
where t is the argument
to the trig function.


Interactions between populations

WEBVTT
Kind: captions
Language: en

00:00:00.065 --> 00:00:01.565
- [Voiceover] In the
introduction to ecology,

00:00:01.565 --> 00:00:04.692
we introduce the idea of a community,

00:00:04.692 --> 00:00:07.758
which is all about different populations

00:00:07.758 --> 00:00:09.600
that are in the same habitat,

00:00:09.600 --> 00:00:12.632
that share the same area or
that are in the same area.

00:00:12.632 --> 00:00:15.091
So populations, and if we're thinking

00:00:15.091 --> 00:00:16.657
in terms of water or in the air,

00:00:16.657 --> 00:00:18.823
it could be to share
a similar volume even,

00:00:18.823 --> 00:00:21.323
populations sharing a habitat.

00:00:25.458 --> 00:00:26.958
Sharing a habitat.

00:00:28.260 --> 00:00:29.658
And in particular in this video,

00:00:29.658 --> 00:00:30.658
we're going to focus on

00:00:30.658 --> 00:00:33.524
the interactions between
those populations,

00:00:33.524 --> 00:00:36.758
the interactions between
the different species.

00:00:36.758 --> 00:00:40.390
The technical term for that
is interspecific interactions.

00:00:40.390 --> 00:00:44.257
I like to just say
interactions between species.

00:00:44.257 --> 00:00:46.557
Now the first one that
is often thought about

00:00:46.557 --> 00:00:48.974
is the notion of competition.

00:00:51.490 --> 00:00:54.192
And this is when different
populations, different species,

00:00:54.192 --> 00:00:57.603
are competing for the same resources.

00:00:57.603 --> 00:00:59.025
You can imagine a forest,

00:00:59.025 --> 00:01:01.491
where you have different
populations of plants

00:01:01.491 --> 00:01:03.457
that are competing for sunlight,

00:01:03.457 --> 00:01:05.057
that are competing for water,

00:01:05.057 --> 00:01:07.693
that are competing for
nutrients in the soil.

00:01:07.693 --> 00:01:09.359
Even in this picture right over here,

00:01:09.359 --> 00:01:10.925
this is a picture of a community.

00:01:10.925 --> 00:01:13.125
All of these different
populations of fish,

00:01:13.125 --> 00:01:15.948
and other things, sea anemones and coral,

00:01:15.948 --> 00:01:18.767
they are sharing this same region,

00:01:18.767 --> 00:01:22.236
and many of them could be in
competition with each other.

00:01:22.236 --> 00:01:23.924
They might be going after the same food,

00:01:23.924 --> 00:01:27.726
or they might be going after
the same shelter someplace.

00:01:27.726 --> 00:01:30.092
And oftentimes, when people
are talking about these

00:01:30.092 --> 00:01:33.158
inter-species or
interspecific interactions,

00:01:33.158 --> 00:01:34.957
you'll see something like this,

00:01:34.957 --> 00:01:37.257
a minus slash minus,

00:01:37.257 --> 00:01:38.992
or a negative sign slash a negative sign.

00:01:38.992 --> 00:01:41.858
And that means that this
type of interaction,

00:01:41.858 --> 00:01:43.492
when you have two species
or two populations

00:01:43.492 --> 00:01:45.624
that are in competition with each other,

00:01:45.624 --> 00:01:47.090
the more that you have of one,

00:01:47.090 --> 00:01:48.924
it's going to have a
negative effect on the other,

00:01:48.924 --> 00:01:50.824
and vice versa.

00:01:50.824 --> 00:01:53.290
If I'm a plant, and if I'm in competition

00:01:53.290 --> 00:01:54.424
with another plant,

00:01:54.424 --> 00:01:56.149
and that one's taking my light,

00:01:56.149 --> 00:01:58.202
and if there's more of it taking my light,

00:01:58.202 --> 00:02:00.691
that's gonna have a negative
impact on me and vice versa.

00:02:00.691 --> 00:02:02.325
If I'm in competition with you,

00:02:02.325 --> 00:02:03.957
and we eat the same thing,

00:02:03.957 --> 00:02:06.475
the more of me that there's
around eating your food,

00:02:06.475 --> 00:02:10.891
that's gonna have a negative
impact on you, and vice versa.

00:02:10.891 --> 00:02:14.157
So the next form of
interspecific interaction,

00:02:14.157 --> 00:02:17.907
or interaction between
species, is predation.

00:02:19.566 --> 00:02:23.723
This is when one population
likes to eat another population.

00:02:23.723 --> 00:02:26.192
And you might often associate predation

00:02:26.192 --> 00:02:29.817
with pictures like this that
you see on television shows,

00:02:29.817 --> 00:02:32.890
on documentaries, you
see a cheetah hunting,

00:02:32.890 --> 00:02:35.458
it looks like a cheetah hunting a gazelle

00:02:35.458 --> 00:02:37.299
or a deer of some kind.

00:02:37.299 --> 00:02:39.724
Actually it says right
here it's a young bushback.

00:02:39.724 --> 00:02:41.399
And this is predation.

00:02:41.399 --> 00:02:43.824
But this is not the
only form of predation.

00:02:43.824 --> 00:02:46.190
This picture here of
the goat eating grass,

00:02:46.190 --> 00:02:48.000
this is also predation.

00:02:48.000 --> 00:02:49.990
It's not quite as bloody and as violent,

00:02:49.990 --> 00:02:51.624
but it is still predation,

00:02:51.624 --> 00:02:54.832
because you have one species
eating another species.

00:02:54.832 --> 00:02:56.591
In this case you have
this animal, the goat,

00:02:56.591 --> 00:02:58.625
that is eating the grass.

00:02:58.625 --> 00:03:01.257
And this type of predation,
this specific type of predation,

00:03:01.257 --> 00:03:02.924
is called herbivory.

00:03:04.532 --> 00:03:07.623
But it is a type of predation.

00:03:07.623 --> 00:03:11.290
So we could say,
predation, slash herbivory.

00:03:12.402 --> 00:03:14.124
Let me do a little slash here.

00:03:14.124 --> 00:03:18.291
Slash herbivory, which is a
special case of predation.

00:03:19.398 --> 00:03:22.742
And you'll often see a plus slash minus.

00:03:22.742 --> 00:03:26.399
The more of, let's say
this species that you have,

00:03:26.399 --> 00:03:28.890
the species that is being eaten,

00:03:28.890 --> 00:03:30.458
it's going to benefit the predator,

00:03:30.458 --> 00:03:31.958
but the more of the
predator that you have,

00:03:31.958 --> 00:03:36.125
it's going to have a negative
effect on the actual prey.

00:03:37.333 --> 00:03:39.090
The next types of interactions

00:03:39.090 --> 00:03:41.158
are ones where you have long-term,

00:03:41.158 --> 00:03:43.249
fairly intimate interactions,

00:03:43.249 --> 00:03:46.132
where you have organisms that oftentime

00:03:46.132 --> 00:03:49.325
live with each other,
or often on each other.

00:03:49.325 --> 00:03:52.091
And this general term of organisms

00:03:52.091 --> 00:03:54.691
that have these long-term
intimate interactions

00:03:54.691 --> 00:03:55.774
is symbiosis.

00:03:59.467 --> 00:04:00.958
Now, in every day language,

00:04:00.958 --> 00:04:02.626
when people talk about symbiosis,

00:04:02.626 --> 00:04:04.292
they're often talking about organisms

00:04:04.292 --> 00:04:06.324
that really benefit each other.

00:04:06.324 --> 00:04:08.624
But technically, symbiosis isn't

00:04:08.624 --> 00:04:10.467
just about benefiting each other.

00:04:10.467 --> 00:04:11.767
It could be that they're even

00:04:11.767 --> 00:04:13.033
hurting each other in some way,

00:04:13.033 --> 00:04:14.824
or that maybe one benefits

00:04:14.824 --> 00:04:17.691
while the other one really doesn't care.

00:04:17.691 --> 00:04:20.125
And so there're several
types of symbiosis.

00:04:20.125 --> 00:04:24.208
The first that we could
talk about is parasitism.

00:04:25.267 --> 00:04:26.184
Parasitism.

00:04:29.567 --> 00:04:31.658
And this looks a lot like predation,

00:04:31.658 --> 00:04:35.158
where the more that the parasite benefits,

00:04:36.358 --> 00:04:38.258
the most of the host that there is,

00:04:38.258 --> 00:04:41.458
but the host is actually
hurt by the parasite.

00:04:41.458 --> 00:04:44.491
And there's all sorts of
examples of parasitism.

00:04:44.491 --> 00:04:48.658
We have, right over here, a
zoomed in picture of a louse.

00:04:50.498 --> 00:04:52.357
So why is this parasitism?

00:04:52.357 --> 00:04:55.824
Well if this lice, if
this louse, I should say,

00:04:55.824 --> 00:04:57.657
so this is parasitism.

00:04:59.998 --> 00:05:04.165
And we would call the louse
here a parasite, parasitism.

00:05:06.141 --> 00:05:08.925
And this benefits by living in your hair

00:05:08.925 --> 00:05:11.125
because that's where
it gets its food from,

00:05:11.125 --> 00:05:13.159
it can lay, or living on your scalp,

00:05:13.159 --> 00:05:14.625
it gets your food by sucking your blood,

00:05:14.625 --> 00:05:16.966
it can also lay eggs in your hair.

00:05:16.966 --> 00:05:19.025
In some ways you could
view it as almost a shelter

00:05:19.025 --> 00:05:21.291
from the rest of the environment.

00:05:21.291 --> 00:05:22.924
But it's negative for you.

00:05:22.924 --> 00:05:24.724
It will make you itchy,
it is taking your blood,

00:05:24.724 --> 00:05:26.899
it is uncomfortable.

00:05:26.899 --> 00:05:29.426
And so, parasitism, once again,

00:05:29.426 --> 00:05:31.190
it's good for the parasite,

00:05:31.190 --> 00:05:33.824
just like predation is
good for the predator,

00:05:33.824 --> 00:05:38.199
but not so good for the host
in the case of parasitism.

00:05:38.199 --> 00:05:39.790
Now you have another situation,

00:05:39.790 --> 00:05:43.057
where it is benefiting both sides.

00:05:43.057 --> 00:05:45.474
And that is called mutualism.

00:05:46.700 --> 00:05:50.091
Mutualism, let me do that
in a different color.

00:05:50.091 --> 00:05:51.174
So mutualism.

00:05:53.100 --> 00:05:56.098
This is where both sides benefit.

00:05:56.098 --> 00:05:58.191
And oftentimes when people
talk about symbiosis,

00:05:58.191 --> 00:05:59.959
they're really talking about mutualism,

00:05:59.959 --> 00:06:01.824
which is a specific type of symbiosis,

00:06:01.824 --> 00:06:06.099
where both species, or both
animals, organisms, benefit.

00:06:06.099 --> 00:06:08.291
They don't have to just be animals.

00:06:08.291 --> 00:06:11.185
This is an example of
mutualism right here.

00:06:11.185 --> 00:06:15.066
Let me do that in a color you can see.

00:06:15.066 --> 00:06:16.816
So this is mutualism,

00:06:18.549 --> 00:06:22.716
where you have a clownfish
living within a sea anemone.

00:06:24.332 --> 00:06:27.392
The sea anemone is providing
the clownfish shelter,

00:06:27.392 --> 00:06:30.592
while the clownfish is
keeping away other fish

00:06:30.592 --> 00:06:33.233
that might eat that sea anemone.

00:06:33.233 --> 00:06:35.490
So they are both benefiting
from that interaction,

00:06:35.490 --> 00:06:37.573
and so that is mutualism.

00:06:38.558 --> 00:06:40.183
Now you have another category,

00:06:40.183 --> 00:06:42.850
where one species is benefiting,

00:06:44.465 --> 00:06:47.800
and the other one is maybe a
little bit more indifferent.

00:06:47.800 --> 00:06:50.791
So one species is benefiting,
and then the other one,

00:06:50.791 --> 00:06:53.993
well, maybe it is a
little bit indifferent.

00:06:53.993 --> 00:06:56.966
And we call that commensalism.

00:06:56.966 --> 00:06:58.049
Commensalism.

00:06:59.066 --> 00:07:02.891
And once again, there's many
examples of commensalism.

00:07:02.891 --> 00:07:04.883
This right over here is a picture of

00:07:04.883 --> 00:07:07.058
bacteria living on your skin,

00:07:07.058 --> 00:07:10.690
and you do have bacteria
living on your skin right now.

00:07:10.690 --> 00:07:11.733
Accept it.

00:07:11.733 --> 00:07:13.599
And actually, well
oftentimes it's a good thing.

00:07:13.599 --> 00:07:15.024
Sometimes it's mutualism,

00:07:15.024 --> 00:07:17.758
that it's providing protection
from harmful bacteria.

00:07:17.758 --> 00:07:20.157
But sometimes it, the bacteria
are surely benefiting,

00:07:20.157 --> 00:07:23.191
it's living off of nutrients on your skin.

00:07:23.191 --> 00:07:25.384
The skin is its habitat.

00:07:25.384 --> 00:07:29.459
But oftentimes, it doesn't
really have an impact on you.

00:07:29.459 --> 00:07:32.408
Now, commensalism, let me write this down.

00:07:32.408 --> 00:07:35.291
Commensalism, oftentimes
the more that we study it

00:07:35.291 --> 00:07:36.659
and the more that we understand it,

00:07:36.659 --> 00:07:38.323
we realize that actually maybe

00:07:38.323 --> 00:07:40.658
the host actually is benefiting,

00:07:40.658 --> 00:07:42.258
which it is mutualism,

00:07:42.258 --> 00:07:44.525
or maybe the host
actually is getting hurt,

00:07:44.525 --> 00:07:46.968
in which case it is parasitism.

00:07:46.968 --> 00:07:48.425
So oftentimes commensalism

00:07:48.425 --> 00:07:50.757
isn't completely neutral for the host.

00:07:50.757 --> 00:07:53.257
It could go either way.

00:07:53.257 --> 00:07:56.566
And so these are all the
different types of interactions.

00:07:56.566 --> 00:07:58.060
So I encourage you to look around you,

00:07:58.060 --> 00:07:59.323
look at the world around you,

00:07:59.323 --> 00:08:00.658
and don't just limit yourself to animals,

00:08:00.658 --> 00:08:02.658
think about bacteria, think about plants.

00:08:02.658 --> 00:08:04.558
And think about within a habitat,

00:08:04.558 --> 00:08:07.724
what're all of the different
interspecies interactions,

00:08:07.724 --> 00:08:10.865
and how you might wanna classify them.


Limits at infinity of quotients with trig (limit undefined)

WEBVTT
Kind: captions
Language: en

00:00:00.485 --> 00:00:01.333
- [Voiceover] Let's see
if we can figure out

00:00:01.333 --> 00:00:04.363
what the limit of x squared
plus one over sine of x is

00:00:04.363 --> 00:00:07.358
as x approaches infinity.

00:00:07.358 --> 00:00:08.461
So, let's just think about what's going on

00:00:08.461 --> 00:00:10.040
in the numerator and then
think about what's going on

00:00:10.040 --> 00:00:11.481
in the denominator.

00:00:11.481 --> 00:00:14.220
So, the numerator, we
have x squared plus one.

00:00:14.220 --> 00:00:16.309
So, as x gets larger and larger and larger

00:00:16.309 --> 00:00:18.788
as it approaches infinity,
well, we're just squaring

00:00:18.788 --> 00:00:21.476
it here so this
numerator's gonna get even,

00:00:21.476 --> 00:00:23.430
approach infinity even faster.

00:00:23.430 --> 00:00:26.851
So, this thing is going to go to infinity

00:00:26.851 --> 00:00:28.767
as x approaches infinity.

00:00:28.767 --> 00:00:31.505
Now what's happening to
the denominator here?

00:00:31.505 --> 00:00:33.631
Well, sine of x, we've seen this before.

00:00:33.631 --> 00:00:35.852
Sine of x and cosine of x are bounded,

00:00:35.852 --> 00:00:37.300
they oscillate.

00:00:37.300 --> 00:00:39.611
They oscillate between
negative one and one,

00:00:39.611 --> 00:00:43.055
so negative one is gonna
be less than or equal to

00:00:43.055 --> 00:00:46.228
sine of x which is going to
be less than or equal to one.

00:00:46.228 --> 00:00:49.212
So, this denominator's going to oscillate.

00:00:49.212 --> 00:00:50.617
So, what does that tell us?

00:00:50.617 --> 00:00:52.540
Well, we might be tempted
to say, well the numerator's

00:00:52.540 --> 00:00:55.063
unbound and goes to infinity
and the denominator's

00:00:55.063 --> 00:00:57.740
just oscillating between
these values here.

00:00:57.740 --> 00:00:59.928
So maybe the whole thing goes to infinity.

00:00:59.928 --> 00:01:02.865
But we have to be careful because one,

00:01:02.865 --> 00:01:04.793
the denominator's going between positive

00:01:04.793 --> 00:01:06.627
and negative values.

00:01:06.627 --> 00:01:08.583
So, the numerator's just going to get more

00:01:08.583 --> 00:01:10.040
and more and more positive

00:01:10.040 --> 00:01:11.956
being divided sometimes
by a positive value,

00:01:11.956 --> 00:01:13.452
sometimes by a negative value.

00:01:13.452 --> 00:01:15.799
So, we're gonna jump between
positive and negative.

00:01:15.799 --> 00:01:17.157
Positive and negative.

00:01:17.157 --> 00:01:21.324
And then you also have all
these crazy asymptotes here.

00:01:22.226 --> 00:01:24.669
Every time sine of x becomes zero,

00:01:24.669 --> 00:01:27.653
well then, you're gonna
have a vertical asymptote.

00:01:27.653 --> 00:01:29.278
This thing will not be defined.

00:01:29.278 --> 00:01:30.474
So you have all these vertical asymptotes.

00:01:30.474 --> 00:01:32.643
You're gonna oscillate
between positive and negative

00:01:32.643 --> 00:01:35.037
just larger and larger values.

00:01:35.037 --> 00:01:37.254
So, this limit does not exist.

00:01:37.254 --> 00:01:38.833
So, it does not exist.

00:01:38.833 --> 00:01:40.083
Does not exist.

00:01:43.013 --> 00:01:44.551
And we can see that graphically.

00:01:44.551 --> 00:01:45.880
We described it in words,

00:01:45.880 --> 00:01:47.877
just inspecting this expression,

00:01:47.877 --> 00:01:49.805
but we can see it
graphically if we actually

00:01:49.805 --> 00:01:51.163
looked at a graph of this,

00:01:51.163 --> 00:01:52.835
which I have right here.

00:01:52.835 --> 00:01:56.967
And you can see that as x goes
towards positive infinity,

00:01:56.967 --> 00:01:59.139
as x goes to positive infinity,

00:01:59.139 --> 00:02:02.169
we, depending on which x we
are, we're kind of going,

00:02:02.169 --> 00:02:05.165
we go, we get really large, then we had a

00:02:05.165 --> 00:02:06.778
vertical asymptote than
we jump back down and

00:02:06.778 --> 00:02:08.450
go really negative, vertical asymptote,

00:02:08.450 --> 00:02:10.029
up, down, up, down, up, down,

00:02:10.029 --> 00:02:12.630
the oscillations just
get more and more extreme

00:02:12.630 --> 00:02:15.277
and we keep having these
vertical asymptotes

00:02:15.277 --> 00:02:16.325
on a periodic basis.

00:02:16.325 --> 00:02:19.873
So it's very clear that
this limit does not exist.


Limits at infinity of quotients with trig

WEBVTT
Kind: captions
Language: en

00:00:00.339 --> 00:00:01.500
- [Voiceover] So, let's
see if we can figure out

00:00:01.500 --> 00:00:03.868
what the limit as X approaches infinity

00:00:03.868 --> 00:00:07.320
of cosine of X over X
squared minus one is.

00:00:07.320 --> 00:00:08.736
And like always, pause this video

00:00:08.736 --> 00:00:11.879
and see if you can work
it out on your own.

00:00:11.879 --> 00:00:14.578
Well, there's a couple
of ways to tackle this.

00:00:14.578 --> 00:00:16.631
You could just reason
through this and say,

00:00:16.631 --> 00:00:19.777
"Well, look this numerator,
right over here, cosine of X,

00:00:19.777 --> 00:00:22.328
"that's just going to oscillate between

00:00:22.328 --> 00:00:23.497
"negative one and one."

00:00:23.497 --> 00:00:27.604
Cosine of X is going to
be greater than or equal

00:00:27.604 --> 00:00:29.793
to negative one, or negative
at one is less than or equal

00:00:29.793 --> 00:00:32.450
to cosine of X which is
less than or equal to one.

00:00:32.450 --> 00:00:34.107
So, this numerator just oscillates

00:00:34.107 --> 00:00:36.403
between negative one and one as X changes,

00:00:36.403 --> 00:00:38.454
as X increases in this case.

00:00:38.454 --> 00:00:40.710
While the denominator
here, we have an X squared,

00:00:40.710 --> 00:00:43.132
so as we get larger and larger X values,

00:00:43.132 --> 00:00:46.575
this is just going to become
very, very, very large.

00:00:46.575 --> 00:00:47.927
So, we're going to have something bounded

00:00:47.927 --> 00:00:50.141
between negative one and one divided

00:00:50.141 --> 00:00:53.173
by very, very infinitely large numbers.

00:00:53.173 --> 00:00:57.556
And so, if you take a, you
could say, bounded numerator

00:00:57.556 --> 00:00:59.977
and you divide that
infinitely large denominator,

00:00:59.977 --> 00:01:02.274
well, that's going to approach zero.

00:01:02.274 --> 00:01:04.200
So, that's one way you
could think about it.

00:01:04.200 --> 00:01:06.660
Another way is to make this same argument,

00:01:06.660 --> 00:01:09.534
but to do it in a little
bit more of a mathy way.

00:01:09.534 --> 00:01:13.473
Because cosine is bounded in this way,

00:01:13.473 --> 00:01:15.806
we can say that cosine of X

00:01:18.493 --> 00:01:20.493
over X squared minus one

00:01:21.588 --> 00:01:23.671
is less than or equal to.

00:01:25.075 --> 00:01:27.906
Well, the most that this
numerator can ever be is one,

00:01:27.906 --> 00:01:30.302
so it's going to be less
than or equal to one

00:01:30.302 --> 00:01:32.385
over X squared minus one.

00:01:33.383 --> 00:01:36.418
And it's going to be a
greater than or equal to,

00:01:36.418 --> 00:01:37.654
it's going to be greater than or equal to,

00:01:37.654 --> 00:01:40.319
well, the least that this
numerator can ever be

00:01:40.319 --> 00:01:42.055
is going to be negative one.

00:01:42.055 --> 00:01:45.555
So, negative one over X squared minus one.

00:01:47.632 --> 00:01:48.822
And once again, I'm just saying,

00:01:48.822 --> 00:01:51.198
look, cosine of X, at most, can be one

00:01:51.198 --> 00:01:53.621
and at least is going to be negative one.

00:01:53.621 --> 00:01:56.408
So, this is going to be true for all X.

00:01:56.408 --> 00:02:00.052
And so, we can say that also the limit,

00:02:00.052 --> 00:02:03.563
the limit as X approaches infinity of this

00:02:03.563 --> 00:02:06.228
is going to be true for all X.

00:02:06.228 --> 00:02:09.262
So, limit as X approaches infinity.

00:02:09.262 --> 00:02:11.845
Limit as X approaches infinity.

00:02:12.907 --> 00:02:15.201
Now, this here, you could
just make the argument,

00:02:15.201 --> 00:02:17.374
look the top is constant.

00:02:17.374 --> 00:02:19.956
The bottom just becomes infinitely large

00:02:19.956 --> 00:02:22.910
so that this is going to approach zero.

00:02:22.910 --> 00:02:26.054
So, this is going to be
zero is less than or equal

00:02:26.054 --> 00:02:29.137
to the limit as X approaches infinity

00:02:30.355 --> 00:02:33.355
of cosine X over X squared minus one

00:02:34.576 --> 00:02:36.545
which is less than or equal to.

00:02:36.545 --> 00:02:39.213
Well, this is also going to go to zero.

00:02:39.213 --> 00:02:42.534
You have a constant numerator,
an unbounded denominator.

00:02:42.534 --> 00:02:44.663
This denominator's
going to go to infinity,

00:02:44.663 --> 00:02:47.450
and so, this is going to be zero as well.

00:02:47.450 --> 00:02:50.363
So, if our limit is
going to be between zero.

00:02:50.363 --> 00:02:52.907
If zero is less than
or equal to our limit,

00:02:52.907 --> 00:02:54.425
is less than or equal to zero,

00:02:54.425 --> 00:02:56.151
well then, this right over here

00:02:56.151 --> 00:02:58.151
has to be equal to zero.


Predator prey cycle

WEBVTT
Kind: captions
Language: en

00:00:00.579 --> 00:00:02.279
- [Voiceover] What I
wanna do in this video

00:00:02.279 --> 00:00:04.265
is think about how different populations

00:00:04.265 --> 00:00:06.298
that share the same ecosystem can interact

00:00:06.298 --> 00:00:08.238
with each other and actually provide

00:00:08.238 --> 00:00:10.124
a feedback loop on each other.

00:00:10.124 --> 00:00:11.529
And there's many cases of this,

00:00:11.529 --> 00:00:13.663
but the most cited general example

00:00:13.663 --> 00:00:15.517
is the case when one population

00:00:15.517 --> 00:00:18.028
wants to eat another population.

00:00:18.028 --> 00:00:21.192
And so you have the predator population

00:00:21.192 --> 00:00:23.987
that likes to eat the prey.

00:00:23.987 --> 00:00:27.770
So you have the predator
and prey interactions.

00:00:27.770 --> 00:00:29.513
I'm doing the prey in I guess
a somewhat bloody color,

00:00:29.513 --> 00:00:31.758
I guess 'cause, well,
they're going to be eaten.

00:00:31.758 --> 00:00:33.610
So let's just think about how these

00:00:33.610 --> 00:00:35.267
populations could interact.

00:00:35.267 --> 00:00:37.615
Let me draw a little chart
here that you're probably

00:00:37.615 --> 00:00:39.328
familiar with by now where we show

00:00:39.328 --> 00:00:42.336
how a population can change over time.

00:00:42.336 --> 00:00:44.938
So the time, the horizontal axis is time.

00:00:44.938 --> 00:00:47.605
The vertical axis is population.

00:00:49.345 --> 00:00:50.262
Population.

00:00:51.549 --> 00:00:53.598
And so let's just, in our starting point,

00:00:53.598 --> 00:00:56.848
let's say that our prey is starting out

00:00:58.262 --> 00:01:00.405
at a relatively high point.

00:01:00.405 --> 00:01:02.174
Let's say we're right there in time,

00:01:02.174 --> 00:01:03.958
and let's say for whatever reason,

00:01:03.958 --> 00:01:07.476
our predator population is relatively low.

00:01:07.476 --> 00:01:09.817
So what do we think is
going to happen here?

00:01:09.817 --> 00:01:12.689
Well, at this point, with
a low density of predators,

00:01:12.689 --> 00:01:15.254
it's gonna be much easier
for them for find a meal,

00:01:15.254 --> 00:01:18.011
and it's gonna be much easier
for the prey to get caught.

00:01:18.011 --> 00:01:20.753
So since it's more easy,
it's easier for the predators

00:01:20.753 --> 00:01:23.071
to find a meal, you can
imagine their population

00:01:23.071 --> 00:01:25.603
starting to increase.

00:01:25.603 --> 00:01:26.747
But what's going to happen is

00:01:26.747 --> 00:01:28.204
their population is increasing.

00:01:28.204 --> 00:01:30.205
Well, it's gonna be more
likely that they're gonna,

00:01:30.205 --> 00:01:32.308
they prey is gonna get caught.

00:01:32.308 --> 00:01:34.163
There's gonna be more
of their hunters around,

00:01:34.163 --> 00:01:35.351
more of their predators around.

00:01:35.351 --> 00:01:39.046
So that population is
going to start decreasing

00:01:39.046 --> 00:01:42.399
all the way to a point
where if the population

00:01:42.399 --> 00:01:44.211
of the prey gets low enough, the predators

00:01:44.211 --> 00:01:45.921
are gonna have, they're gonna start having

00:01:45.921 --> 00:01:49.363
trouble finding food again,
and so that their population

00:01:49.363 --> 00:01:51.985
might start to decrease,
and as their population

00:01:51.985 --> 00:01:53.971
decreases, what's gonna
happen to the prey?

00:01:53.971 --> 00:01:56.467
Well, then, there's gonna
be less predators around,

00:01:56.467 --> 00:01:59.161
so they might be able to, their population

00:01:59.161 --> 00:02:00.798
might start to increase.

00:02:00.798 --> 00:02:02.717
And so I think you see what's happening.

00:02:02.717 --> 00:02:05.962
The predator and prey,
they can kind of form

00:02:05.962 --> 00:02:08.529
this cyclic interaction with each other.

00:02:08.529 --> 00:02:10.010
And what I've just drawn,
this is often known

00:02:10.010 --> 00:02:12.260
as the predator-prey cycle.

00:02:13.562 --> 00:02:16.098
And I just reasoned through
that you can imagine

00:02:16.098 --> 00:02:17.591
a world where you can
have the cycle between

00:02:17.591 --> 00:02:19.694
predator and prey populations.

00:02:19.694 --> 00:02:22.231
But you can also run computer simulations

00:02:22.231 --> 00:02:24.627
that will show this,
and even observational

00:02:24.627 --> 00:02:27.188
data out in the field also shows this.

00:02:27.188 --> 00:02:29.771
One of the often cited examples

00:02:30.946 --> 00:02:35.113
is interactions between,
between the snowshoe hare,

00:02:37.671 --> 00:02:41.093
which would be the prey in this situation,

00:02:41.093 --> 00:02:45.565
and the Canadian lynx,
which would be the predator,

00:02:45.565 --> 00:02:48.148
the predator in this situation.

00:02:49.620 --> 00:02:52.837
And you see a very similar
cycle to what I just drew,

00:02:52.837 --> 00:02:54.264
kind of just reasoning through it,

00:02:54.264 --> 00:02:56.499
and this, right here, is actual data.

00:02:56.499 --> 00:02:58.759
You see the passage of time here,

00:02:58.759 --> 00:03:00.553
and this is a long passage of time.

00:03:00.553 --> 00:03:02.298
We're starting in the early 1800's

00:03:02.298 --> 00:03:06.549
going all the way to the early-mid 1900's.

00:03:06.549 --> 00:03:08.720
So it's roughly 100 years
of data that we're showing,

00:03:08.720 --> 00:03:11.157
and in the vertical axis,
you have thousands of animals

00:03:11.157 --> 00:03:13.127
and we're plotting both the population

00:03:13.127 --> 00:03:16.679
of snowshoe hares and Canadian lynx

00:03:16.679 --> 00:03:19.253
in a certain area on this chart.

00:03:19.253 --> 00:03:22.586
And as you see, when the prey population

00:03:23.604 --> 00:03:26.437
is high, when the prey population,

00:03:27.503 --> 00:03:29.754
sorry, when the predator
population is high,

00:03:29.754 --> 00:03:32.337
when we have a lot of
the Canadian lynx around,

00:03:32.337 --> 00:03:35.587
that we see a lower, a lower population

00:03:37.705 --> 00:03:40.596
of the prey, of the hare.

00:03:40.596 --> 00:03:44.085
But then as, since you
have a low population

00:03:44.085 --> 00:03:46.476
of the food in this situation,

00:03:46.476 --> 00:03:49.682
the predator population
starts to decrease.

00:03:49.682 --> 00:03:51.148
So let me draw an arrow here.

00:03:51.148 --> 00:03:53.625
The predator population starts to decrease

00:03:53.625 --> 00:03:56.493
and, let me do that same blue color.

00:03:56.493 --> 00:03:59.538
And so the predator population decreases,

00:03:59.538 --> 00:04:01.816
and as that predator population decreases,

00:04:01.816 --> 00:04:04.540
well then the prey population increases

00:04:04.540 --> 00:04:08.327
'cause there's less folks
around to hunt them.

00:04:08.327 --> 00:04:10.520
So the prey population increases,

00:04:10.520 --> 00:04:12.144
and you see that the other way around.

00:04:12.144 --> 00:04:15.170
When the prey population
is really is, well maybe

00:04:15.170 --> 00:04:16.952
we'll show it right over
here, and this is real data.

00:04:16.952 --> 00:04:18.870
That's why it's not always super clean.

00:04:18.870 --> 00:04:22.901
But when the prey population
is really, really high

00:04:22.901 --> 00:04:26.831
and the predator population
is relatively low,

00:04:26.831 --> 00:04:29.412
well, then, the predators
say, "Hey, it's really

00:04:29.412 --> 00:04:31.559
"easy for us to find meals right now."

00:04:31.559 --> 00:04:33.166
That's kind of that
starting point in that,

00:04:33.166 --> 00:04:34.879
when I was just reasoning through it.

00:04:34.879 --> 00:04:37.430
And so their population starts to.

00:04:37.430 --> 00:04:38.545
Oh, oops, what did I do?

00:04:38.545 --> 00:04:39.628
There, there.

00:04:40.947 --> 00:04:42.268
Let me make sure.

00:04:42.268 --> 00:04:45.363
So their population starts to increase,

00:04:45.363 --> 00:04:48.044
and as the predator population increases,

00:04:48.044 --> 00:04:51.760
well the prey population,
the prey population

00:04:51.760 --> 00:04:53.510
is going to decrease.

00:04:54.465 --> 00:04:57.673
So this is real data
showing the snowshoe hare,

00:04:57.673 --> 00:05:00.585
the prey, and the Canadian
lynx, the predator,

00:05:00.585 --> 00:05:03.309
on over many, many decades to show

00:05:03.309 --> 00:05:05.392
this predator-prey cycle.