In Exercise 10 we modeled populations of aphids and ladybugs...

In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $ \frac {dA}{dt} = 2A(1 - 0.0001A) - 0.01AL $ $ \frac {dL}{dt} = -0.5L + 0.0001AL $ (a) In the absence of ladybugs, what does the model predict about the aphids? (b) Find the equilibrium solutions? (c) Find an expression for $ dL/dA. $ (d) Use a computer algebra system to draw a direction field for the differential equation in part (c). Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common? (e) Suppose that at time $ t = 0 $ there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. (f) Use part (e) to make rough sketches of the aphid and ladybug populations as functions of $ t. $ How are the graphs related to each other?

In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows:
$ \frac {dA}{dt} = 2A(1 - 0.0001A) - 0.01AL $
$ \frac {dL}{dt} = -0.5L + 0.0001AL $
(a) In the absence of ladybugs, what does the model predict about the aphids?
(b) Find the equilibrium solutions?
(c) Find an expression for $ dL/dA. $
(d) Use a computer algebra system to draw a direction field for the differential equation in part (c). Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common?
(e) Suppose that at time $ t = 0 $ there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change.
(f) Use part (e) to make rough sketches of the aphid and ladybug populations as functions of $ t. $ How are the graphs related to each other?

Transcript

00:01 Hey guys, so today i'm going to be talking about how we would go about analyzing a lot of terror system, which is just a very popular predator prey model system.

00:15 So let's consider two populations x and y.

00:20 Let's say we have dx over d t equal to just some function i'm just dependent on x and y and then d -y over d -t, which is another function, let's say g of xy.

00:42 And these are both going to be dependent on x and y because it's a predator prey, so the populations are going to be interacting with each other.

00:55 Okay, so for our first step, we want to find the equilibrium.

01:01 So how we do this is just setting our derivative is equal to zero.

01:06 So we have 0 equal to f of xy, equal to g of xy.

01:19 And we want these when these are simultaneously equal to 0.

01:24 So we want to solve for a point, let's say x star, y star.

01:31 And this will be our constant value, since the derivatives are equal to zero.

01:36 And what this does is tell us what the system will support.

01:40 Of the number of individuals in each population.

01:48 Okay, so for our second step, we want to try to get an expression for dy over dx.

01:57 And what this will tell us is how one population relates to the other, rather the change in one population relates to the other.

02:08 And we can apply the chain rule to find this.

02:10 So we know that d -y over d -t is equal to d -y -d -x times d -x over d -t.

02:23 And so we want to solve for this just by dividing the d -y over d -t by d -x over d -t.

02:35 And then from here, we could try to get what we call a directional field.

02:45 Direction field for this partial differential equation the d y over d x so we can use any type of computer algebra system personally i would use matlab um but you can use python or any other type of uh computer uh algebra system so um what we're trying to do is get um a vector field where we get vectors that point in the direction that they change.

03:22 So the change in x, the change in y, you'll get vector field.

03:27 So let's say for our system we have trajectories that go like this...

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