Question

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=[−532−36​8040544​] The rabbit population begins at 24000 . If we want the rabbit population to grow as a simple exponential of the form R(t)=R0​e4t with no other terms, how many foxes are needed at time t=0 ? (Note that the eigenvalues of A are λ=4 and 8 .)

Answer 1

To have the rabbit population grow as a simple exponential of the form R(t) = R0e^4t with no other terms, we need to find the fox population at time t=0. By solving the equation A*v = 4*v, where A is the given matrix and v is the eigenvector, we find that the eigenvector corresponding to the eigenvalue of 4 is [1; -4]. Using the fact that the rabbit population at t = 0 is 24000, we can determine that the rabbit population grows as R(t) = 24000e^4t. In order for there to be no other terms in the exponential growth equation, the fox population at t = 0 should be 0.

Step-by-step explanation:

To have the rabbit population grow as a simple exponential of the form R(t) = R0e^4t with no other terms, we can use the fact that the matrix A has eigenvalues of 4 and 8. The exponential growth of the rabbit population is represented by the eigenvalue of 4, so we need to find the corresponding eigenvector. Let's solve the equation A*v = 4*v, where v is the eigenvector. We have the equation:

[-5 - 3/2; 2/3 - 36]*[v1; v2] = 4*[v1; v2]

Expanding this equation, we get a system of two equations: -5v1 - (3/2)v2 = 4v1 and (2/3)v1 - 36v2 = 4v2. Solving these equations, we find that v1 = -(1/4)v2. Therefore, the eigenvector corresponding to the eigenvalue of 4 is [1; -4].

Now, we know that the rabbit population at t = 0 is 24000 and can be represented as R(0) = R0. Using the exponential growth formula, we have 24000 = R0e^4*0 = R0. Therefore, R0 = 24000.

Since the vector y is defined as [R(t); F(t)], we know that the rabbit population at t = 0 is 24000, so R0 = 24000. Now, we only want the rabbit population to be growing exponentially at t = 0 with no other terms. Therefore, the fox population at t = 0 should be 0, because any other value would introduce additional terms in the exponential growth equation.