What does r and k tell us about the fishs population

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Reference no: EM132178512

In the first homework you looked at the differential equation dt .3P(1 /10000) where P was dP = 0 - P in individuals and t was in months. We're going to assume this is modeling fish in a pond.

Sketch the graph P' vs. P (flow on a line) and use that to sketch solutions for the initial conditions P(0)=100, P(0)=5000, P(0)=9000 and P(0)=13000. Label both graphs appropriately.

Now let's look at a more general version of the above differential equation: dt P(1 /K) dP = r - P sketch a graph of P' vs P (flow on a line) and sketch several solutions.

What is the limit of P(t) as t goes to infinity? How does dP/dt act for P close to zero?

What does r and K tell us about the fish's population?

The following model also includes a fishing term:

dt P(1 /K) P

dP = r - P - c

Sketch a P' vs P (flow on a line) graph of the new model. Compare1 this model with

dt P(1 /K) .

dP = r - P
Let's look at another version of this model, this time with numbers again.

dt .3P(1 /10000) 0(sin(4t) )

dP = 0 - P - 5 + 1

Using Euler's method (starting out with a ?t = 0.1 months ) construct a graph of the solution to this equation that spans two years with an initial condition of 10,000 (the pond is at a stable population).

You should be able to explain the units on the ‘50' and it's meaning as well as the meaning of the ‘4' within the sine function. It'll help if you set up your spreadsheet with variables for all the parameters:
So that your slope at any given (t, P) is calculated by:

= $F$2 * B2 *( 1 - B2 / $G$2 )- $H$2 *(sin( $I$2 * A2 )+ 1 )

(I have time in column A and population in column B). This will allow you to mess around with values for all the parameters in the model. Here are some things to consider:

1 this one word might involve a lot of work

-how does r affect the solution?

-how does c affect the solution?

-how does frequency affect the solution

-what happens for ‘coarser' values for delta t? That is, try 0.2, 0.4, etc up to 1. See if you can shed some light on what happens. The sine function is involved (to see this try using Euler's method for the plain ole logistic differential equation dt .3P(1 /10000) dP = 0 - P.

Reference no: EM132178512

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