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Now we're going to look at a very simplistic model of a food chain.
It's so simple that we're only going to consider two species of fish.
One is the predator and one is its prey.
You're going to model the change of either population, the predator and the prey
using differential equation in the form of these equations right here.
It's up to you to figure out, however, which of these corresponds to the predator
and which one to the prey.
There are several other features in this situation that must be taken into account.
The prey multiplied at a certain rate. The prey is consumed by the predator--that's kind of given.
The predators, however, only die of old age and the predators' population
also multiplies depending on the amount of prey that is present.
In addition, predators are being harvested.
Now in these equation A, B, C, and D are constants and in our program over here,
you're going to fill in their values in this dictionary called base values.
We've already provided you with the values corresponding to two keys in base value.
We've included the initial amount of prey
and the initial amount of predator population measured in tons.
Here, however, is some other important information
that you'll need to know in order to do this problem.
The rate of growth of the prey is 0.5 per year and the lifespan of the predator is 5 years.
In addition, you'll reach an equilibrium situation or a steady state situation
with the amount of prey reaches 5x10⁶ tons and the amount of predator reaches 1.0x10⁶ tons.
You're going to use this information, a little bit of logic, in these equations to calculate A,B,C, and D.
Once you've entered A,B,C, and D into base values,
we move down to look at this function called food chain.
Here you'll use the familiar forward Euler method to calculate the future populations
of the prey and the predator at each time set.
Lastly, we'll do a bit of sensitivity analysis.
We're going to add in a degree of uncertainty to this problem to make it a little bit more complex.
Let's say that A, B, C, and D as well as the initial amounts of both fish
are uncertain by plus or minus 10%.
Similar to earlier quiz, you're going to create dictionaries for the minimum and maximum values
taking this uncertainty into account and then determine which of these six parameters
A, B, C, and D and the two initial values has the largest impact on the maximum amount of prey.
Again all of these is very similar to the quiz on sensitivity analysis that you saw in the unit itself.
So if you're having any trouble, you might want to review that quiz
and of course you can always come to the forums for more help--good luck!