Working with differential equations to obtain a function

[ver_mathstats]

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

s'=(8/3)s - f - (1/3)h - (1/6)m
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 - 4f + (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′  =  Ay.

Consider the initial population y(0)  =  [14 12 11 94]T. Solve for constants c1 through c4 in order to write y(0)=c₁x1 + c₂x2 + c₃x3 + c₄x4, 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 c₁, c₂, c₃, c₄ are 2, 1, 8, 3 respectively. Values of c₁, c₂, c₃, c₄ are constants in the equation:

y(0)=c₁x1 + c₂x2 + c₃x3 + c₄x4, 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=c₁e^λ1tx₁+...+c_ne^λntx_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.

 

[Chestermiller]

Do you know how to solve this by finding the eigenvalues and eigenvectors of the matrix A?

 

[epenguin]

You would do better to put brackets around your fractions so that it is immediately clear (which it wasn't to me) that this is just a standard linear differential equation in four variables.

There are people here with sufficiently frequent practice in these things to be able to guess what you are not saying like what are the c's, and how does that sentence with "in order to write" end, I'm not sufficiently so to be able to guess and answer in the time I have.
But probably you have to put t = 0 into solutions you say you have which should give you something on the one hand something corresponding to the y(0) vector on the other, maybe via a linear algebraic equation.

 

[ver_mathstats]

Do you know how to solve this by finding the eigenvalues and eigenvectors of the matrix A?

I used [P,D] = eig(A) to find the eigenvectors and the eigenvalues of matrix A. But I am not entirely sure how to solve this using those values and vectors.

 

[Chestermiller]

I used [P,D] = eig(A) to find the eigenvectors and the eigenvalues of matrix A. But I am not entirely sure how to solve this using those values and vectors.

You do this using your own equation: y=c₁e^λ1tx₁+...+c_ne^λntx_n

 

[ver_mathstats]

You would do better to put brackets around your fractions so that it is immediately clear (which it wasn't to me) that this is just a standard linear differential equation in four variables.

There are people here with sufficiently frequent practice in these things to be able to guess what you are not saying like what are the c's, and how does that sentence with "in order to write" end, I'm not sufficiently so to be able to guess and answer in the time I have.
But probably you have to put t = 0 into solutions you say you have which should give you something on the one hand something corresponding to the y(0) vector on the other, maybe via a linear algebraic equation.

My apologies, I think I fixed those now. Thanks for the reply.

 

[ver_mathstats]

You do this using your own equation: y=c₁e^λ1tx₁+...+c_ne^λntx_n

Okay thank you, would I have to be using just that equation? Where would the hawk equation come in? Or does it not? y=2e^-tx1 + 1e^2tx2 + 8e^0.5tx3 + 3e^1tx4 is what I have obtained so far using y=c₁e^λ1tx₁+...+c_ne^λntx_n.

 

[curious_]

first insert your vector
A = [ fill this in]
and find the eigenvalues and eigenvectors using
[P, D] = eig(A)
if you have an initial vector insert that using
y0 = [ given vector ]'
create a new matrix, P, by ordering the column vectors based on their eigenvalues (largest to smallest) w the format
x1 = P(:, # of column w largest e.value /P(1, same number <-)
for each row of your matrix
compile that into a new matrix
P = [x1 x2 x3 x4]
here since the question is asking the largest value in the fourth row, that value in your matrix will be the answer
and to solve for the constants from your initial value, use
c = inv(P)*y0

 

[berkeman]

Welcome to the PF (cute avatar, BTW)!

first insert your vector
A = [ fill this in]
and find the eigenvalues and eigenvectors using
[P, D] = eig(A)
if you have an initial vector insert that using
y0 = [ given vector ]'
create a new matrix, P, by ordering the column vectors based on their eigenvalues (largest to smallest) w the format
x1 = P(:, # of column w largest e.value /P(1, same number <-)
for each row of your matrix
compile that into a new matrix
P = [x1 x2 x3 x4]
here since the question is asking the largest value in the fourth row, that value in your matrix will be the answer
and to solve for the constants from your initial value, use
c = inv(P)*y0

Please keep in mind that the student must do the bulk of the work on their homework problems here, so try not to post such detailed instructions for how to do it. Once the student has solved the problem one way, it is okay to post how to solve it a different way, though.

But since this thread is over a year old, the student has likely moved on anyway.

Enjoy the PF!