00:01
Population models usually involve some mathematical relationships that tell us how many of a certain species exist in a region at a given time.
00:15
So we are looking at a population model in a certain forest for the number of rabbits are of t and the number of foxes, f of t.
00:29
And we have two equations.
00:32
The top equation governs the number of rabbits in the region, and the bottom one is going to govern the number of foxes.
00:41
And there are a couple of things that i would like to start off and say, look very unrealistic with this model.
00:53
So i'll just denote those from the start because this looks like an incomplete model in many ways.
01:01
I won't say incorrect, because somebody's probably starting with some modeling on numbers that they've collected in the field.
01:10
But anyway, the first problem is the r of t says that the rabbit population is just decreasing, according to the part due to the foxes.
01:42
So the negative part there at the end.
01:46
And it's not clear how the rabbits are reproducing.
02:02
And the other thing that looks a little bit unclear is what about resource limitations and disease and hunting and, you know, other things that could take out the population or cause it to change.
02:35
The second thing that looks a little bit fishy is the fox equation two.
02:45
Foxes just show exponential growth.
02:53
So whenever you have the rate of change in a population, directly proportional just to that population, that is just exponential growth.
03:04
And usually that is unrealistic.
03:06
There are usually things that take out the population.
03:17
There are resource limitations that create what's called a logistic curve.
03:24
And finally, overall, the most unrealistic thing is that the two populations of fox and rabbits do not have coupled equations.
03:47
And i would say that's probably the biggest concern is whenever you have species interacting in a certain region, especially when they depend so precariously, i'll say, on each other.
04:04
One's a food source for the foxes, and of course the foxes are a source of death for the rabbits.
04:12
You would expect a coupled equation that would determine the number in each dependent on the number in the other.
04:23
And that's something really big that's missing.
04:28
Species are interrelated is probably the way i would phrase that, and i don't see that interrelation.
04:48
I should say when they're coexisting in the same resource area.
04:55
Okay, but anyway, we are going to look at some things about these two equations.
05:01
I'll call them equation a and equation b.
05:05
The first thing to do is take equation a and try to determine the unknown parameters.
05:13
So equation a, we're going to find the a and the k that appear in that exponential.
05:26
So a is typically found from initial conditions, and k is found from another day.
05:37
Data point on the rabbit population.
05:46
Okay, so the first thing we know is that the rabbit population at times equal zero is 1600.
05:55
And so if we put that in, r of zero is 1700 minus a.
06:05
So we can solve for a.
06:07
It is 100.
06:13
Then we are given another data point five years later.
06:23
T -equal five years, what we're told is r at 5 is one quarter less, so that's 1600 minus 1600 over 4 is 400.
06:43
Oops, wait a minute, 1600 minus 400 or 1 ,200.
06:50
So there are 1 ,200 rabbits after five years.
06:53
So yeah, whatever control seems to be working, that's good.
07:04
Okay, so then we can put that in.
07:07
1 ,200 is equal to 1 ,700 minus 100 times e to the k times 5, 5k.
07:19
Let's just call 5k.
07:25
And we're going to solve for that k.
07:28
So it'll take some algebra, but let's see.
07:31
We want to pull the e to the k.
07:34
5k over by itself is 1 ,700 minus 1 ,200 over 100.
07:46
And that's just 5.
07:48
Now we take the logarithm, both sides.
07:57
Loggerithm is the opposite of the exponential function.
08:01
So the exponent function goes away, and we're left with 5k equals logarithm of 5, or k is log 5 over 5.
08:19
And i'll convert that to a decimal real quick.
08:24
It's 0 .322, i'll say.
08:32
Okay, so then we can finally write down our full rabbit as a function of time formula.
08:46
And of course, we're worried now about the rabbits going extinct.
08:52
Now, that may not be such a problem, but then the foxes would have an issue.
09:01
So, yeah, everything's interrelated.
09:06
Okay.
09:08
So we can now take equation a in its full form and find out when the rabbits go extinct, according to this equation anyway.
09:34
As i said, some interrelations will kick in probably before that happens.
09:40
Okay, so 1700...