Description
Problem 1a. The Fibonacci recurrence is:
Express this as a matrix-vector relation: Gxk=xk+1���=��+1 where xk=[fk,fk−1]T��=[��,��−1]� & xk+1=[fk+1,fk]T��+1=[��+1,��]�
( Where G is a 2×22×2 matrix).
Find the matrix G� for this transformation. Carry out the matrix multiplication for 8 steps. Use {\tt numpy.linalg.matrix_power} (See : Fibbonacci Numbers in nature).
The matrix is of interest. This matrix can be the starting point of carry out population distribution studies among animals in their various life-stages (more in Tuesday’s lecture)
Problem 1b Use the matrix power function from numpy.linalg.matrix_power and calculate G3,G5,G10�3,�5,�10. Do you notice anything about the elements of the matrices (you may also want to look at Assignment 1).
Problem 2 In the following problem, you are going to apply matrix-vector multiplication to study a (linear) Predator-Prey interaction.
In the forests of N. California, Dusky-footed Wood Rats provide 80% of the diet for the Northern Spotted Owl. First, try a simple linear model of the dynamical system that makes up the owls and the rats.
The populations at timestep k𝑘 are given by xk=[Owlsk,Ratsk]��=[�����,�����], where Ratsk����� are in 1000‘s1000′𝑠 of rats.
In the absence of predation (by owls), the rat pop grows by 10% per month (this is the timestep for the model). With no rats for food, the owl population falls to one-half the pop in the prev month. The predation rate of rats is p� (take p = 0.104 to start the experiment).
Since we have a linear model, there are no Owl-Rat interactions (encounters) in the model, we simulate that by an “efficiency* by which the Owls turn Rats into Owls with a conversion efficiency of 0.4 (i.e. when there are abundant Rats, the Owl pop rises by 0.4∗Rats0.4∗𝑅𝑎𝑡𝑠.
Solve this as a matrix equation by generating x1,x2,…xk𝑥1,𝑥2,…𝑥𝑘 as usual. Remember, each xi=[Owlsi,Ratsi]��=[�����,�����]. Use k=50�=50. Store the Owl numbers, in each step i�, in an array Owl[i]���[�] and the Rat numbers (in 1000’s) in another array Rats[i]����[�].
Using the arrays Owl[i]���[�], Rats[i]����[�], make (i) a plot of the Owl Pop. v/s Time and the Rat Pop. v/s Time on the same axes. Now, since this is a difference equation, the time step (of a month) is implicit in the model. The time axis is generated using the numpy function linspace with k� intervals (make endpoint=True). {\bf On the same plot} (ii) Plot Owl[i]���[�] v/s Rat[i]���[�].
Experiment with [Owl0,Rats0]=[20,40], [35,40] [45,40] [60,40] [80,40][���0,����0]=[20,40], [35,40] [45,40] [60,40] [80,40] Experiment with changing the predation rate p� (pick 4 different values to see the effects on the curves obtained).
To plot numpy arrays X,Y�,� on the same time t� axes: *plt.plot(t, X, ‘r-‘, t, Y, ‘b–‘)
To make 2 plots side-by-side; use the *matplotlib.pyplot function subplot(1,2,1) before the first plot command and subplot(1,2,2) before the second plot command)