This week we're going to be discussing the mathematics and mathematical models used to
describe interactions between populations of species. In particular, we
will first discuss the interactions between predator, population of predators with
a population of prey. This is the type of setting in which one of these populations
serves as the food supply and resource that is needed for the other species
to survive. But before we rush off to start doing and start writing equations, I would
like to guide you to several examples of predator
prey interactions. So on this slide here, you're going to find some URLs with examples
about the Golden Eagle, the Golden Eagle, excuse me, the Golden
Eagle versus the Jack rabbit. This is a video by the National Geographic and then you'll
find other videos that also motivate these interactions.
This week we are interested on interactions between two species, two populations and specifically
in the context in which one species serves as the food supply
for another species. These type of interactions
are usually known as predator prey interactions and before we continue we're
going to take a look at a video
that was made by the National Geographic and this will serve as our motivation. Okay?
It is a king. The Golden Eagle. North America's largest raptor. This bird is an incredible
hunting machine. A seven foot wing span. Razor sharp vision.
Its eyes can spot prey from over 2 miles away. And a killer 150 mph dive. Which means
this jack rabbit doesn't stand a chance. Jack rabbits are a favorite prey of Golden
Eagles. And a hopping pack of protein. The rabbit can run as fast as 40 miles an hour
but that's no match for talons that descend from above.
The big bird has an incredible grip. It can squeeze over 750 pounds per square inch, so
most prey are killed instantly. In addition to jack rabbits, the Golden Eagle
mainly preys on other small mammals, reptiles, birds, fish and even large insects.
They're also known to eat carrion. These raptors are found all over the northern
hemisphere and can live for almost 20 years in the wild.
So if you're a jack rabbit, you might want to keep one eye and ear on the sky.
That video presentation can be accessed using this URL if you would like to replay it for
better resolution and better sound, you're more than welcome to do so. There are
other examples of predator prey interactions, other videos that you can find in these URLs
and I invite you to pause
this video at this time and visit each of each and every single one of these videos
on your own computer and before we continue this conversation.
So let's, let's take a, let's take a pause. Let's now talk about the Hudson's Bay Company,
the Hudson's Bay Company. If you visit the URL of this company the
their main, their homepage, this is what you will see. So, let's just
follow this link that describes, tells us a little bit of information about this company. This
company was founded in 1670. Okay? 1670. This company used to trade animal
furs in Canada and because they were well-established in the trading
business, they use to keep records of
and records that we're gonna see in just 1 minute. Basically
they had data for a specific period of time between 1848 and 1907
and they had observations or estimates on the numbers of two species, the population
sizes on two species. So the Hudson Bay's Company kept records on
the numbers, estimated numbers on a population of the Canadian lynx,
also formerly known as lynx canadiensis, and the population of snowshoe hare, formerly
known as lepus americanus. So if you have any doubt,
the Canadian lynx was the predator that feeds or used to feed on the snowshoe hare. Okay
so what we're gonna see is and what follows next is a visualization of this data.
It's an animation. These are year observations. These are records with
estimates per year on the number of hares
and the number of lynx in thousands. Okay so we're gonna see a display as how each of
these data points appears on a longitudinal in a longitudinal sense so as
time progresses then we will see the data points appear. So this is how the graph
of these observations looks like. We are gonna have on the x-axis the time,
the time points. The time points have units of years. The first observation was collected
in 1848 and on the vertical axis, the y-axis we're
gonna have the numbers. So in 1848 there were 21,000
hares and 44,000 lynx. Okay we can see how the next observation comes on 1849
and we're gonna see how this progresses in time. So here we are gonna press play and
you can see the progression in time as the data points, or the reports,
are coming in. So this will display observations every year
from the year 1848 through 1907.
So one thing to observe is that we can see some temporal patterns.
Right? And one thing would like to observe is where each species
attains approximately where each species attains its maximum value.
So we're looking there at the hares in 1857 around 1857 and then the lynx in red
and then the hares get another large volume in 1864, lynx in 1866.
Then hares another maximum volume at 1876 while lynx at 1876 also. The next peak seems
to be in 1886, around 1886. and 1896, 1895. And then 1906
and 1905. So what this is telling us is that. We could also look for the points
the time points at which both curves are reaching a minimum level
that this is telling is that there is a pattern in which numbers grow up to a maximum level
then they dropped the minimum level and then they raise back to maximum
level in this pattern seems to repeat it's it's a pattern that has the signature of a
cycle, and this cycle and the numbers of each species is something that
is consistent in both the population of hares and the population
of lynx. Very important point to make
here is that the fluctuations, meaning that fluctuations are oscillations in population
size, the fluctuations observed in the number of
lynx seems to be correlated to the fact that the main food supply for
lynx is hare. So,
when there is an abundance of food then the population that is feeding from that food
seems to grow, however when there is a shortage of food,
when the food supply runs short, that also seems to have an impact on those predators
that are feeding from that food supply. So this is a feature that
is well observed in this type of dataset,
and this is a very well-known data-set that is used as an example of the interactions
between species is a predator feeding from a species that is a prey.
So mathematical descriptions of these interactions between predators and prey are mathematical
tools. The date, they go way back. One of the pioneers in this type of
challenge, using mathematics and mathematical equations to come up with some
description of these interactions is Vita Volterra, you can see here is a copy of a
paper that Volterra published in 1926 in the journal Nature. The original
publication on what is considered today a prime model of predator and prey appear
and, is well, is noted here. Another person is credited with describing
given mathematical description was Alfred Lotka, who almost at the same time was studying
problems in a different context. What we'll see we revisit these equations
is that the terms of the node interactions between predator and prey are
terms that have something in common with the law of mass action
in chemistry. So here is the first page of a publication by Lotka in the 1920s.
This concludes our introduction to predator and prey dynamics.