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ver_mathstats

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## Homework Statement

On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time

*t*are given by

*s*(

*t*),

*f*(

*t*),

*h*(

*t*), and

*m*(

*t*) respectively.

The populations grow at rates given by the differential equations

*=(8/3)s - f - (1/3)h - (1/6)m*

s

s

**'**f'=(2/3)s + f - (1/3)h - (1/6)m

h'=(2/3)s + 0f + (2/3)h - (1/6)m

*m'=(46/3)s - 4*f + (1/3)h - (11/6)m

Putting the four populations into a vector

**y**(

*t*) = [

*s*(

*t*)

*f*(

*t*)

*h*(

*t*)

*m*(

*t*)]

*T*, this system can be written as

**y′**=

*A*

**y**.

Consider the initial population

**y**(0) = [14 12 11 94]

*T*. Solve for constants

*c*1 through

*c*4 in order to write y(0)=c

_{1}x1 + c

_{2}x2 + c

_{3}x3 + c

_{4}x4, where x1 to x4 were the eigenvectors.

Based on the system of differential equations from Problem 1, with the initial population from problem 2, find the function for the population of hawks,

*h*(

*t*).

## Homework Equations

## The Attempt at a Solution

I am struggling to find the function for the populations of hawks. I know the eigenvalues are -1, 2, 1/2, 1.

The values of c

_{1}, c

_{2}, c

_{3}, c

_{4}are 2, 1, 8, 3 respectively. Values of c

_{1}, c

_{2}, c

_{3}, c

_{4}are constants in the equation:

y(0)=c

_{1}x1 + c

_{2}x2 + c

_{3}x3 + c

_{4}x4, where x1 to x4 were the eigenvectors.

So I know h'=(2/3)s + 0f + (2/3)h - (1/6)m.

I know that y=c

_{1}e

^{λ1t}x

_{1}+...+c

_{n}e

^{λnt}x

_{n}.

Does that mean I have to use h'=(2/3)s + 0f + (2/3)h - (1/6)m and make it so that it fits the equation above?

I am also doing this on Matlab.

Thank you.

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