;line-height:107%;mso-fareast-font-family:\"Times New Roman\"; mso-bidi-font-family:\"Calibri Light\";mso-bidi-theme-font:major-latin; mso-fareast-language:EN-IN\'> Math:

A butterfly starts its life as a caterpillar, then transforms into a butterfly and then dies. Let xi(t), x2(t), and x3(t) be the number of caterpillars, live butterflies and dead butterflies, respectively, at any given time t. Suppose that () the rate at which caterpillars are born is rix2() (ii) the rate at which caterpillars are transformed into butterflies is r2xi(t). (ii) the rate at which butterflies die is r3x2(t Then....

(a) What is the dynamical description of this bio- system expressed as x-Ax?

(b) If ri/month, 2 r32/month, f 3/month, and ind the general solution for x(t)

(c) If you start with 6000 caterpillars, how many dead butterflies will you have after 10 months?

Sol91:

(a) The dynamical description of the bio-system expressed as x-Ax can be written as:

dx/dt = Ax

where x = [x1(t), x2(t), x3(t)] is the state vector representing the number of caterpillars, live butterflies, and dead butterflies at time t, and A is the 3x3 matrix given by:

A = [0 r1 0] [r2 0 r3] [0 0 0]

(b) The rate equations for the system can be written as:

dx1/dt = r1x2 dx2/dt = r2x1 - r3x2 dx3/dt = r3x2

Substituting the given values of ri, we get:

dx1/dt = 2x2 dx2/dt = 1.5x1 - 0.3x2 dx3/dt = 0.3x2

To solve this system of differential equations, we can write the state vector x as a linear combination of the eigenvectors of matrix A:

x = c1v1e?1t + c2v2e?2t + c3v3e?3t

where ci are constants to be determined, vi are the eigenvectors of A, and ?i are the corresponding eigenvalues.

The eigenvalues of A can be found by solving the characteristic equation:

det(A-?I) = 0

where I is the identity matrix. This gives:

?1 = sqrt(r1r3) ?2 = -sqrt(r1r3) ?3 = 0

The corresponding eigenvectors are:

v1 = [1, 1, 0] v2 = [-1, 1, 0] v3 = [0, 0, 1]

Using these values, we can write the general solution for x(t) as:

x(t) = c1[cos(sqrt(r1r3)t) + sin(sqrt(r1r3)t)][1, 1, 0] + c2[cos(sqrt(r1r3)t) - sin(sqrt(r1r3)t)][-1, 1, 0] + c3[1, 0, 1]

To determine the values of ci, we use the initial condition that there are 6000 caterpillars at t=0:

x(0) = [6000, 0, 0] = c1v1 + c2v2 + c3v3

Solving for ci, we get:

c1 = 3000/sqrt(r1r3) c2 = 3000/sqrt(r1r3) c3 = 6000

Therefore, the solution for x(t) is:

x(t) = 3000[cos(sqrt(r1r3)t) + sin(sqrt(r1r3)t)][1, 1, 0] - 3000[cos(sqrt(r1r3)t) - sin(sqrt(r1r3)t)][-1, 1, 0] + 6000[1, 0, 1]

(c) To find the number of dead butterflies after 10 months, we can substitute t=10 months in the solution for x(t) and find x3(t):

x3(10) = 6000 + 3000[cos(sqrt(r1r3)10) + sin(sqrt(r1r3)10)] + 3000[cos(sqrt(r1r3)10) - sin(sqrt(r1r3