Lab8-turnin.sagews

# Lab 8

# Name: Esha Harwalkar
# I worked on this code with:

# Please do all of your work for this week's lab in this worksheet. If
# you wish to create other worksheets for scratch work, you can, but
# this is the one that will be graded. You do not need to do anything
# to turn in your lab. It will be collected by your TA at the beginning
# of (or right before) next week’s lab.

# Be sure to clearly label which question you are answering as you go and to
# use enough comments that you and the grader can understand your code.

#1
r = 0.2
K = 100
A = 8
f(N)= 0.2*N *(1-(N/100))*((N/8)-1) #define the mathematical function
f(0)
f(A)
f(K)

0 0 0

#2:
possibleEqlist = [0, 3,5,8, 25, 50, 100] #create a list of possible equilibrium values
Eqlist = [] #create an empty list for the actual equilibrium values
for i in possibleEqlist: #for each value in the possible Eq list
    if f(i) == 0: #if f(i) equals 0 (f(N) function from #1)
        Eqlist.append(i) #add that i value to the actual equilibrium values list
b = Eqlist #assign a variable to the list of actual equilibrium values
b #call the function

[0, 8, 100]

#3
t = srange(0,200) #define what the times list is
sol = desolve_odeint(f(N),times = t, ics = [5], dvars=N) #simulate a differential equation (f(N)) with initial condition of 5.
list_plot(sol, plotjoined = True) #plot the simulation

in this graph we see that when N = 5, the system wants to return to 0, which shows that 0 is a stable point. It is going AWAY from the next equilibrium value of 8 and TOWARDS 0.

sol = desolve_odeint(f(N),times = t, ics = [50], dvars=N) #simulate a differential equation (f(N)) with initial condition of 50.
list_plot(sol, plotjoined = True) #plot the simulation

in this graph we see that when N = 50, the system wants to go towards 100. Since the arrows to the left AND right of the equilibrium at N=8 are pointing away from the Eq. point, we can confirm that the Eq. point at N = 8 is UNSTABLE.

sol = desolve_odeint(f(N),times = t, ics = [110], dvars=N) #simulate a differential equation (f(N)) with initial condition of 110.
list_plot(sol, plotjoined = True) #plot the simulation

in this graph, we see that when N = 110 ("to the right of N = 100"), the system wants to return to 100. This means that the arrows to the left and right of the Eq. point at N = 100 are pointing TOWARDS the Eq. point, so this Eq, point at N = 100 is STABLE.

#4
if (f(N) < 0 and N < b) and (f(N) > 0 and N > b): #if the change vectors point to the left of the Eq point                                                         (when N is less than the Eq point) and the change                                                             vectors point to the right of the Eq point (when N                                                         is greater than the Eq point)
    print "This is an unstable point"
elif (f(N) > 0 and N <b) and (f(N) < 0 and N > b) : #if the change vectors point to the right of the Eq                                                     point (when N is less than the Eq point) and the change                                                 vectors point to the left of the Eq point (when N is greater                                                     than the Eq point)
    print "This a stable point"
else:
    print "This is a semi-stable point"

#5
def myfunction(f, before_b, after_b): #the before and after b is the same thing as I wrote about with "N<b" or "N > b"
    if f(before_b) < 0 and f (after_e) > 0: #the same as I did in #4 for the rest of the body of this code
        print "this is an unstable equilibrium point"
    elif f(before_b) > 0 and f(after_b) < 0:
        print "this is a stable equilibrium point"
    else:
        print "this is a semi-stable point"
f(N)= 0.2*N *(1-(N/100))*((N/8)-1) #define the mathematical function
myfunction(f, 50, 110) #picked two values for N that are to the left and right of an EQ point of 100

this is a stable equilibrium point

#6
derivative(N) = diff(f, N) #takes derivative of f(N) . This is really just taking the derivative of the original differenial equation
derivative

N |--> -0.000250000000000000*(N - 8)*(N - 100) - 0.000250000000000000*(N - 8)*N - 0.000250000000000000*(N - 100)*N

#7
if derivative(b[0]) == 0: #plug in the first value of the equilibrium point list b
    print "Test failed"

the test didn't fail because the second derivative of the Eq point I inserted, was not equal to 0

 derivative(b[1]) #the value outputted will give us the result of plugging in the second value in the b list (which is 8) into the second derivative equation

0.184000000000000

 derivative(b[2])#the value outputted will give us the result of plugging in the third value in the b list (which is 100) into the second derivative equation

-2.30000000000000

 derivative(b[0])#the value outputted will give us the result of plugging in the first value in the b list (which is 0) into the second derivative equation

-0.200000000000000

b

[0, 8, 100]

#8
list = [0,8,100]
for i in list:
    if derivative(i) < 0: #plug in the first value of the equilibrium point list b
        print "stable"
    else derivative(i)> 0:
        print "unstable"

#9
new_list = [0,8,100]
def newfunction (Nprime, new_list):
    for i in list:
        if derivative(i) < 0: #plug in the first value of the equilibrium point list b
            print "stable"
        if derivative(i) > 0:
            print "unstable"
newfunction (Nprime(N),new_list)

Error in lines 8-8 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags) in namespace, locals File "", line 1, in <module> NameError: name 'Nprime' is not defined