Question

In: Advanced Math

A vector y  =  [R(t)  F(t)]T describes the populations of some rabbits R(t) and foxes F(t). The...

A vector y  =  [R(t)  F(t)]T describes the populations of some rabbits R(t) and foxes F(t). The populations obey the system of differential equations given by y′  =  Ay where

A  = 

[−2

 15]
[−2 9 ]


The rabbit population begins at 6000. If we want the rabbit population to grow as a simple exponential of the form R(t)  =  R0e3t  with no other terms, how many foxes are needed at time t  =  0?
(Note that the eigenvalues of A are λ  = 3 and 4.)

Solutions

Expert Solution

We have

Given that the Eigen values of A are 3 and 4.

Now we have to find the Eigen Vectors

Take corresponding Eigen vector is then

We get

On solving we get

Therefore, the Eigen vector is

Take corresponding Eigen vector is then

We get

On solving we get

Therefore, the Eigen vector is

Since the Eigen vector is

It means that rabbits and foxes ratio is 3:1, but initially there are 6000 rabbits, in the same proportion there must be 2000 foxes at t=0.

Colby Messinger answered 2 months ago
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