# Notebook from AmirRazaMBA/TensorFlow-Certification
Path: 3. NLP/AZ/Text Classification/01_RNN/08_imdb_subwords.ipynb

<code>
# If the import fails, run this
# !pip install -q tensorflow-datasets
import tensorflow as tf
import tensorflow_datasets as tfds_____no_output_____imdb, info = tfds.load("imdb_reviews/subwords8k", with_info=True, as_supervised=True)WARNING:absl:TFDS datasets with text encoding are deprecated and will be removed in a future version. Instead, you should use the plain text version and tokenize the text using `tensorflow_text` (See: https://www.tensorflow.org/tutorials/tensorflow_text/intro#tfdata_example)
train_data, test_data = imdb['train'], imdb['test']_____no_output_____tokenizer = info.features['text'].encoder_____no_output_____print(tokenizer.subwords)['the_', ', ', '. ', 'a_', 'and_', 'of_', 'to_', 's_', 'is_', 'br', 'in_', 'I_', 'that_', 'this_', 'it_', ' /><', ' />', 'was_', 'The_', 'as_', 't_', 'with_', 'for_', '.<', 'on_', 'but_', 'movie_', ' (', 'are_', 'his_', 'have_', 'film_', 'not_', 'ing_', 'be_', 'ed_', 'you_', ' "', 'it', 'd_', 'an_', 'he_', 'by_', 'at_', 'one_', 'who_', 'y_', 'from_', 'e_', 'or_', 'all_', 'like_', 'they_', '" ', 'so_', 'just_', 'has_', ') ', 'her_', 'about_', 'out_', 'This_', 'some_', 'ly_', 'movie', 'film', 'very_', 'more_', 'It_', 'would_', 'what_', 'when_', 'which_', 'good_', 'if_', 'up_', 'only_', 'even_', 'their_', 'had_', 'really_', 'my_', 'can_', 'no_', 'were_', 'see_', 'she_', '? ', 'than_', '! ', 'there_', 'get_', 'been_', 'into_', ' - ', 'will_', 'much_', 'story_', 'because_', 'ing', 'time_', 'n_', 'we_', 'ed', 'me_', ': ', 'most_', 'other_', 'don', 'do_', 'm_', 'es_', 'how_', 'also_', 'make_', 'its_', 'could_', 'first_', 'any_', "' ", 'people_', 'great_', 've_', 'ly', 'er_', 'made_', 'r_', 'But_', 'think_', " '", 'i_', 'bad_', 'A_', 'And_', 'It', 'on', '; ', 'him_', 'being_', 'never_', 'way_', 'that', 'many_', 'then_', 'where_', 'two_', 'In_', 'after_', 'too_', 'little_', 'you', '), ', 'well_', 'ng_', 'your_', 'If_', 'l_', '). ', 'does_', 'ever_', 'them_', 'did_', 'watch_', 'know_', 'seen_', 'time', 'er', 'character_', 'over_', 'characters_', 'movies_', 'man_', 'There_', 'love_', 'best_', 'still_', 'off_', 'such_', 'in', 'should_', 'the', 're_', 'He_', 'plot_', 'films_', 'go_', 'these_', 'acting_', 'doesn', 'es', 'show_', 'through_', 'better_', 'al_', 'something_', 'didn', 'back_', 'those_', 'us_', 'less_', '...', 'say_', 'is', 'one', 'makes_', 'can', 'and', 'all', 'ion_', 'find_', 'scene_', 'old_', 'real_', 'few_', 'going_', 'well', 'actually_', 'watching_', 'life_', 'me', '. <', 'o_', 'man', 'there', 'scenes_', 'same_', 'he', 'end_', 'this', '... ', 'k_', 'while_', 'thing_', 'of', 'look_', 'quite_', 'out', 'lot_', 'want_', 'why_', 'seems_', 'every_', 'll_', 'pretty_', 'got_', 'able_', 'nothing_', 'good', 'As_', 'story', ' & ', 'another_', 'take_', 'to', 'years_', 'between_', 'give_', 'am_', 'work_', 'isn', 'part_', 'before_', 'actors_', 'may_', 'gets_', 'young_', 'down_', 'around_', 'ng', 'thought_', 'though_', 'end', 'without_', 'What_', 'They_', 'things_', 'life', 'always_', 'must_', 'cast_', 'almost_', 'h_', '10', 'saw_', 'own_', 'here', 'bit_', 'come_', 'both_', 'might_', 'g_', 'whole_', 'new_', 'director_', 'them', 'horror_', 'ce', 'You_', 'least_', 'bad', 'big_', 'enough_', 'him', 'feel_', 'probably_', 'up', 'here_', 'making_', 'long_', 'her', 'st_', 'kind_', '--', 'original_', 'fact_', 'rather_', 'or', 'far_', 'nt_', 'played_', 'found_', 'last_', 'movies', 'When_', 'so', '", ', 'comes_', 'action_', 'She_', 've', 'our_', 'anything_', 'funny_', 'ion', 'right_', 'way', 'trying_', 'now_', 'ous_', 'each_', 'done_', 'since_', 'ic_', 'point_', '". ', 'wasn', 'interesting_', 'c_', 'worst_', 'te_', 'le_', 'ble_', 'ty_', 'looks_', 'show', 'put_', 'looking_', 'especially_', 'believe_', 'en_', 'goes_', 'over', 'ce_', 'p_', 'films', 'hard_', 'main_', 'be', 'having_', 'ry', 'TV_', 'worth_', 'One_', 'do', 'al', 're', 'again', 'series_', 'takes_', 'guy_', 'family_', 'seem_', 'plays_', 'role_', 'away_', 'world_', 'My_', 'character', ', "', 'performance_', '2_', 'So_', 'watched_', 'John_', 'th_', 'plot', 'script_', 'For_', 'sure_', 'characters', 'set_', 'different_', 'minutes_', 'All_', 'American_', 'anyone_', 'Not_', 'music_', 'ry_', 'shows_', 'too', 'son_', 'en', 'day_', 'use_', 'someone_', 'for', 'woman_', 'yet_', '." 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'perp', 'particular', 'overwhelm', 'offered', 'nominat', 'naturally', 'locke', 'left', 'lady', 'ilt', 'iel', 'ication', 'historic', 'haunting', 'gem_', 'figures', 'figured_', 'evol', 'ery', 'eco', 'dynami', 'duct', 'doi', 'description', 'cultural', 'contrac', 'confide', 'combined', 'coin', 'cke', 'chosen_', 'amed', 'agon', 'Thomas_', 'THI', 'Nation', 'MOVIE', 'Lev', 'Jeff', 'Hoffman', 'Glen', 'Even', '1st_', ' ! ', 'yu', 'trappe', 'thir', 'tension', 'tail', 'table', 'split', 'sides', 'settle', 'schem', 'save', 'ruc', 'prime', 'posit', 'painte', 'ndi', 'marry_', 'kun', 'killing', 'isol', 'iot', 'intend', 'impres', 'horribly_', 'hing', 'heroi', 'gle_', 'fri', 'fitt', 'fighter', 'estin', 'ee_', 'drunk_', 'directly', 'dinos', 'chose_', 'changing', 'blonde_', 'benefi', 'award_', 'av', 'aki', 'ages', 'acter', 'VERY_', 'Ur', 'Tel', 'Superman_', 'Real', 'Phi', 'Palm', 'Nicol', 'Johnson', 'Jesus_', 'J_', 'Hes', 'Helen', 'Fun', 'Fle', 'Dir', 'Chap', 'vag', 'uncon', 'ues', 'types_', 'tical', 'sprin', 'sorts', 'securi', 'previ', 'porno', 'party', 'pare', 'method', 'medica', 'mber', 'landscape', 'jor', 'jail', 'imper', 'hunter', 'happening', 'gritty', 'gain_', 'flaws_', 'fak', 'extra', 'edited_', 'ecc', 'dragg', 'chie', 'cant_', 'breast', 'authorit', 'ated', 'ality', 'advise', 'advan', 'according_', 'Wors', 'Unlike', 'United_', 'Simon_', 'Riv', 'Pea', 'Michell', 'Exp', 'Child', 'Cham', 'Bourne', 'Basi', 'widow', 'walked_', 'upp', 'unforg', 'uld_', 'tting', 'till_', 'thy_', 'talents_', 'suspenseful', 'summer_', 'storm', 'screening', 'scare_', 'realizes_', 'rce', 'raw', 'qu', 'ngl', 'magic', 'lac', 'jobs', 'ister_', 'inti', 'inha', 'ill_', 'hands', 'grin', 'forward', 'examin', 'equent', 'emi', 'contact', 'concentrat', 'compu', 'competen', 'biograph', 'attach', 'amus', 'alik', 'activi', 'William', 'Myst', 'Luke_', 'Live', 'Life_', '15', 'zes', 'werewolf', 'warne', 'uring_', 'trilogy', 'swim', 'stumble', 'spite', 'spends_', 'sleep_', 'sist', 'sentence', 'rma', 'reward', 'reviewer_', 'pul', 'preten', 'performed', 'passing', 'par_', 'oph', 'livi', 'kinds_', 'journal', 'isticat', 'inva', 'idi', 'ham_', 'fte', 'few', 'featured', 'ern_', 'eag', 'dollars', 'disb', 'depth', 'cryin', 'cross_', 'content', 'contemporary_', 'colors', 'chee', 'because', 'asy', 'agent_', 'Willi', 'Warr', 'Ven', 'Vamp', 'Roch', 'ONE', 'Movie', 'Mau', 'Mass', 'MST', 'Hin', 'Hear', 'Gue', 'Gl', 'Freddy_', 'Definite', 'Captain_', 'BBC', '??? ', '80s_', '"), ', 'wol', 'weekend', 'vampires', 'underst', 'tial_', 'terrorist', 'strength_', 'starre', 'soldier_', 'snow', 'sity', 'ruin_', 'retar', 'resu', 'required', 'recommended', 'ques', 'propo', 'presents_', 'perm', 'overt', 'olds', 'occas', 'nn_', 'nen', 'nei', 'mail', 'lost', 'lion', 'libr', 'inner_', 'headed', 'happy', 'guest', 'govern', 'friendly', 'explains', 'ens_', 'effectively', 'draw_', 'downright', 'dete', 'dde', 'dare', 'cring', 'courag', 'conspi', 'comedie', 'claims_', 'cide', 'chas', 'captivat', 'bite', 'bare', 'author_', 'addition', 'Vid', 'Rh', 'Oliv', 'Nata', 'Mexican', 'Keaton_', 'Iron', 'Barb', 'ALL_', '12', '!), ', 'worthwhile', 'weake', 'ung', 'understood_', 'unbelievable', 'superf', 'stolen', 'stereotypic', 'spoiler', 'sight', 'scares', 'rut', 'remove', 'remotely_', 'releva', 'prese', 'poke', 'ndou', 'mbla', 'lucky_', 'lling_', 'legendary', 'imagery', 'humou', 'hug', 'hired', 'heck', 'guilty', 'extras', 'expected', 'everywhere', 'dry_', 'drea', 'directed', 'dimensional_', 'ddi', 'dden', 'communica', 'cham', 'buddy', 'bank_', 'azi', 'algi', 'adventures', 'accurate_', 'accompan', 'Thom', 'Still_', 'Someone', 'Serious', 'SU', 'Phill', 'Perso', 'Patrick_', 'Lei', 'Jus', 'Gho', 'Get_', 'Freeman', 'Especially_', '?).', '..."']
sample_string = 'TensorFlow, from basics to mastery'

tokenized_string = tokenizer.encode(sample_string)
print ('Tokenized string is {}'.format(tokenized_string))

original_string = tokenizer.decode(tokenized_string)
print ('The original string: {}'.format(original_string))
Tokenized string is [6307, 2327, 4043, 2120, 2, 48, 4249, 4429, 7, 2652, 8050]
The original string: TensorFlow, from basics to mastery
for ts in tokenized_string:
  print ('{} ----> {}'.format(ts, tokenizer.decode([ts])))6307 ----> Ten
2327 ----> sor
4043 ----> Fl
2120 ----> ow
2 ----> ,
48 ----> from
4249 ----> basi
4429 ----> cs
7 ----> to
2652 ----> master
8050 ----> y
BUFFER_SIZE = 10000
BATCH_SIZE = 64

train_dataset = train_data.shuffle(BUFFER_SIZE)
train_dataset = train_dataset.padded_batch(BATCH_SIZE, tf.compat.v1.data.get_output_shapes(train_dataset))
test_dataset = test_data.padded_batch(BATCH_SIZE, tf.compat.v1.data.get_output_shapes(test_data))_____no_output_____embedding_dim = 64
model = tf.keras.Sequential([
    tf.keras.layers.Embedding(tokenizer.vocab_size, embedding_dim),
    tf.keras.layers.GlobalAveragePooling1D(),
    tf.keras.layers.Dense(6, activation='relu'),
    tf.keras.layers.Dense(1, activation='sigmoid')
])

model.summary()Model: "sequential"
_________________________________________________________________
Layer (type)                 Output Shape              Param #
=================================================================
embedding (Embedding)        (None, None, 64)          523840
_________________________________________________________________
global_average_pooling1d (Gl (None, 64)                0
_________________________________________________________________
dense (Dense)                (None, 6)                 390
_________________________________________________________________
dense_1 (Dense)              (None, 1)                 7
=================================================================
Total params: 524,237
Trainable params: 524,237
Non-trainable params: 0
_________________________________________________________________
num_epochs = 10

model.compile(loss='binary_crossentropy',optimizer='adam',metrics=['accuracy'])

history = model.fit(train_dataset, epochs=num_epochs, validation_data=test_dataset)Epoch 1/10
391/391 [==============================] - 29s 74ms/step - loss: 0.6732 - accuracy: 0.5952 - val_loss: 0.6249 - val_accuracy: 0.7023
Epoch 2/10
391/391 [==============================] - 24s 62ms/step - loss: 0.5456 - accuracy: 0.8003 - val_loss: 0.4880 - val_accuracy: 0.8500
Epoch 3/10
391/391 [==============================] - 24s 62ms/step - loss: 0.4232 - accuracy: 0.8748 - val_loss: 0.4071 - val_accuracy: 0.8624
Epoch 4/10
391/391 [==============================] - 25s 63ms/step - loss: 0.3447 - accuracy: 0.9008 - val_loss: 0.3671 - val_accuracy: 0.8751
Epoch 5/10
391/391 [==============================] - 24s 62ms/step - loss: 0.2924 - accuracy: 0.9165 - val_loss: 0.3396 - val_accuracy: 0.8794
Epoch 6/10
391/391 [==============================] - 24s 62ms/step - loss: 0.2569 - accuracy: 0.9244 - val_loss: 0.3368 - val_accuracy: 0.8746
Epoch 7/10
391/391 [==============================] - 25s 63ms/step - loss: 0.2298 - accuracy: 0.9320 - val_loss: 0.3525 - val_accuracy: 0.8575
Epoch 8/10
391/391 [==============================] - 25s 63ms/step - loss: 0.2071 - accuracy: 0.9372 - val_loss: 0.3205 - val_accuracy: 0.8801
Epoch 9/10
391/391 [==============================] - 25s 63ms/step - loss: 0.1892 - accuracy: 0.9448 - val_loss: 0.3311 - val_accuracy: 0.8784
Epoch 10/10
391/391 [==============================] - 24s 63ms/step - loss: 0.1749 - accuracy: 0.9481 - val_loss: 0.3342 - val_accuracy: 0.8751
import matplotlib.pyplot as plt

def plot_graphs(history, string):
  plt.plot(history.history[string])
  plt.plot(history.history['val_'+string])
  plt.xlabel("Epochs")
  plt.ylabel(string)
  plt.legend([string, 'val_'+string])
  plt.show()

plot_graphs(history, "accuracy")
plot_graphs(history, "loss")_____no_output_____e = model.layers[0]
weights = e.get_weights()[0]
print(weights.shape) # shape: (vocab_size, embedding_dim)(8185, 64)

import io

out_v = io.open('vecs.tsv', 'w', encoding='utf-8')
out_m = io.open('meta.tsv', 'w', encoding='utf-8')
for word_num in range(1, tokenizer.vocab_size):
  word = tokenizer.decode([word_num])
  embeddings = weights[word_num]
  out_m.write(word + "\n")
  out_v.write('\t'.join([str(x) for x in embeddings]) + "\n")
out_v.close()
out_m.close()

try:
  from google.colab import files
except ImportError:
  pass
else:
  files.download('vecs.tsv')
  files.download('meta.tsv')_____no_output_____
</code>


# Notebook from gvvynplaine/scipy-cookbook
Path: ipython/LoktaVolterraTutorial.ipynb

# Matplotlib: lotka volterra tutorial

1.  1.  page was renamed from LoktaVolterraTutorial

This example describes how to integrate ODEs with the scipy.integrate
module, and how to use the matplotlib module to plot trajectories,
direction fields and other information.

You can get the source code for this tutorial here:
[tutorial_lokta-voltera_v4.py](files/attachments/LoktaVolterraTutorial/tutorial_lokta-voltera_v4.py).

Presentation of the Lotka-Volterra Model
----------------------------------------

We will have a look at the Lotka-Volterra model, also known as the
predator-prey equations, which is a pair of first order, non-linear,
differential equations frequently used to describe the dynamics of
biological systems in which two species interact, one a predator and the
other its prey. The model was proposed independently by Alfred J. Lotka
in 1925 and Vito Volterra in 1926, and can be described by_____no_output_____du/dt =  a*u -   b*u*v
dv/dt = -c*v + d*b*u*v_____no_output_____with the following notations:

*  u: number of preys (for example, rabbits)

*  v: number of predators (for example, foxes)

* a, b, c, d are constant parameters defining the behavior of the population:

  + a is the natural growing rate of rabbits, when there's no fox

  + b is the natural dying rate of rabbits, due to predation

  + c is the natural dying rate of fox, when there's no rabbit

  + d is the factor describing how many caught rabbits let create a new fox_____no_output_____We will use X=[u, v] to describe the state of both populations.

Definition of the equations:_____no_output_____

#!python
from numpy import *
import pylab as p
# Definition of parameters
a = 1.
b = 0.1
c = 1.5
d = 0.75
def dX_dt(X, t=0):
    """ Return the growth rate of fox and rabbit populations. """
    return array([ a*X[0] -   b*X[0]*X[1] ,
                  -c*X[1] + d*b*X[0]*X[1] ])_____no_output_____

### Population equilibrium

Before using !SciPy to integrate this system, we will have a closer look
at position equilibrium. Equilibrium occurs when the growth rate is
equal to 0. This gives two fixed points:_____no_output_____

#!python
X_f0 = array([     0. ,  0.])
X_f1 = array([ c/(d*b), a/b])
all(dX_dt(X_f0) == zeros(2) ) and all(dX_dt(X_f1) == zeros(2)) # => True_____no_output_____

### Stability of the fixed points

Near these two points, the system can be linearized: dX\_dt = A\_f\*X
where A is the Jacobian matrix evaluated at the corresponding point. We
have to define the Jacobian matrix:_____no_output_____

#!python
def d2X_dt2(X, t=0):
    """ Return the Jacobian matrix evaluated in X. """
    return array([[a -b*X[1],   -b*X[0]     ],
                  [b*d*X[1] ,   -c +b*d*X[0]] ])_____no_output_____

So near X\_f0, which represents the extinction of both species, we have:_____no_output_____

#! python
A_f0 = d2X_dt2(X_f0)                    # >>> array([[ 1. , -0. ],
                                        #            [ 0. , -1.5]])_____no_output_____

Near X\_f0, the number of rabbits increase and the population of foxes
decrease. The origin is therefore a [saddle
point](http://en.wikipedia.org/wiki/Saddle_point).

Near X\_f1, we have:_____no_output_____

#!python
A_f1 = d2X_dt2(X_f1)                    # >>> array([[ 0.  , -2.  ],
                                        #            [ 0.75,  0.  ]])
# whose eigenvalues are +/- sqrt(c*a).j:
lambda1, lambda2 = linalg.eigvals(A_f1) # >>> (1.22474j, -1.22474j)
# They are imaginary numbers. The fox and rabbit populations are periodic as follows from further
# analysis. Their period is given by:
T_f1 = 2*pi/abs(lambda1)                # >>> 5.130199_____no_output_____

Integrating the ODE using scipy.integrate
-----------------------------------------

Now we will use the scipy.integrate module to integrate the ODEs. This
module offers a method named odeint, which is very easy to use to
integrate ODEs:_____no_output_____

#!python
from scipy import integrate
t = linspace(0, 15,  1000)              # time
X0 = array([10, 5])                     # initials conditions: 10 rabbits and 5 foxes
X, infodict = integrate.odeint(dX_dt, X0, t, full_output=True)
infodict['message']                     # >>> 'Integration successful.'_____no_output_____

\`infodict\` is optional, and you can omit the \`full\_output\` argument
if you don't want it. Type "info(odeint)" if you want more information
about odeint inputs and outputs.

We can now use Matplotlib to plot the evolution of both populations:_____no_output_____

#!python
rabbits, foxes = X.T
f1 = p.figure()
p.plot(t, rabbits, 'r-', label='Rabbits')
p.plot(t, foxes  , 'b-', label='Foxes')
p.grid()
p.legend(loc='best')
p.xlabel('time')
p.ylabel('population')
p.title('Evolution of fox and rabbit populations')
f1.savefig('rabbits_and_foxes_1.png')_____no_output_____

![](files/attachments/LoktaVolterraTutorial/rabbits_and_foxes_1v2.png)

The populations are indeed periodic, and their period is close to the
value T\_f1 that we computed.

Plotting direction fields and trajectories in the phase plane
-------------------------------------------------------------

We will plot some trajectories in a phase plane for different starting
points between X\_f0 and X\_f1.

We will use Matplotlib's colormap to define colors for the trajectories.
These colormaps are very useful to make nice plots. Have a look at
[ShowColormaps](http://www.scipy.org/Cookbook/Matplotlib/Show_colormaps)
if you want more information._____no_output_____

values  = linspace(0.3, 0.9, 5)                          # position of X0 between X_f0 and X_f1
vcolors = p.cm.autumn_r(linspace(0.3, 1., len(values)))  # colors for each trajectory

f2 = p.figure()

#-------------------------------------------------------
# plot trajectories
for v, col in zip(values, vcolors):
    X0 = v * X_f1                               # starting point
    X = integrate.odeint( dX_dt, X0, t)         # we don't need infodict here
    p.plot( X[:,0], X[:,1], lw=3.5*v, color=col, label='X0=(%.f, %.f)' % ( X0[0], X0[1]) )

#-------------------------------------------------------
# define a grid and compute direction at each point
ymax = p.ylim(ymin=0)[1]                        # get axis limits
xmax = p.xlim(xmin=0)[1]
nb_points   = 20

x = linspace(0, xmax, nb_points)
y = linspace(0, ymax, nb_points)

X1 , Y1  = meshgrid(x, y)                       # create a grid
DX1, DY1 = dX_dt([X1, Y1])                      # compute growth rate on the gridt
M = (hypot(DX1, DY1))                           # Norm of the growth rate
M[ M == 0] = 1.                                 # Avoid zero division errors
DX1 /= M                                        # Normalize each arrows
DY1 /= M

#-------------------------------------------------------
# Drow direction fields, using matplotlib 's quiver function
# I choose to plot normalized arrows and to use colors to give information on
# the growth speed
p.title('Trajectories and direction fields')
Q = p.quiver(X1, Y1, DX1, DY1, M, pivot='mid', cmap=p.cm.jet)
p.xlabel('Number of rabbits')
p.ylabel('Number of foxes')
p.legend()
p.grid()
p.xlim(0, xmax)
p.ylim(0, ymax)
f2.savefig('rabbits_and_foxes_2.png')_____no_output_____

![](files/attachments/LoktaVolterraTutorial/rabbits_and_foxes_2v3.png)

This graph shows us that changing either the fox or the rabbit
population can have an unintuitive effect. If, in order to decrease the
number of rabbits, we introduce foxes, this can lead to an increase of
rabbits in the long run, depending on the time of intervention.

Plotting contours
-----------------

We can verify that the function IF defined below remains constant along
a trajectory:_____no_output_____

#!python
def IF(X):
    u, v = X
    return u**(c/a) * v * exp( -(b/a)*(d*u+v) )
# We will verify that IF remains constant for different trajectories
for v in values:
    X0 = v * X_f1                               # starting point
    X = integrate.odeint( dX_dt, X0, t)
    I = IF(X.T)                                 # compute IF along the trajectory
    I_mean = I.mean()
    delta = 100 * (I.max()-I.min())/I_mean
    print 'X0=(%2.f,%2.f) => I ~ %.1f |delta = %.3G %%' % (X0[0], X0[1], I_mean, delta)
# >>> X0=( 6, 3) => I ~ 20.8 |delta = 6.19E-05 %
#     X0=( 9, 4) => I ~ 39.4 |delta = 2.67E-05 %
#     X0=(12, 6) => I ~ 55.7 |delta = 1.82E-05 %
#     X0=(15, 8) => I ~ 66.8 |delta = 1.12E-05 %
#     X0=(18, 9) => I ~ 72.4 |delta = 4.68E-06 %_____no_output_____

Plotting iso-contours of IF can be a good representation of
trajectories, without having to integrate the ODE_____no_output_____

#!python
#-------------------------------------------------------
# plot iso contours
nb_points = 80                              # grid size
x = linspace(0, xmax, nb_points)
y = linspace(0, ymax, nb_points)
X2 , Y2  = meshgrid(x, y)                   # create the grid
Z2 = IF([X2, Y2])                           # compute IF on each point
f3 = p.figure()
CS = p.contourf(X2, Y2, Z2, cmap=p.cm.Purples_r, alpha=0.5)
CS2 = p.contour(X2, Y2, Z2, colors='black', linewidths=2. )
p.clabel(CS2, inline=1, fontsize=16, fmt='%.f')
p.grid()
p.xlabel('Number of rabbits')
p.ylabel('Number of foxes')
p.ylim(1, ymax)
p.xlim(1, xmax)
p.title('IF contours')
f3.savefig('rabbits_and_foxes_3.png')
p.show()_____no_output_____

![](files/attachments/LoktaVolterraTutorial/rabbits_and_foxes_3v2.png)_____no_output_____