# This example describe how to integrate ODEs with scipy.integrate module, and how
# to use the matplotlib module to plot trajectories, direction fields and other
# useful information.
#
# == Presentation of the Lokta-Volterra Model ==
#
# We will have a look at the Lokta-Volterra model, also known as the
# predator-prey equations, which are 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 one its prey. They were proposed
# independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926:
# du/dt = a*u - b*u*v
# dv/dt = -c*v + d*b*u*v
#
# 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
#
# We will use X=[u, v] to describe the state of both populations.
#
# Definition of the equations:
#
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] ])
#
# === Population equilibrium ===
#
# Before using !SciPy to integrate this system, we will have a closer look on
# position equilibrium. Equilibrium occurs when the growth rate is equal to 0.
# This gives two fixed points:
#
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
#
# === Stability of the fixed points ===
# Near theses 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:
#
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]] ])
#
# So, near X_f0, which represents the extinction of both species, we have:
# A_f0 = d2X_dt2(X_f0) # >>> array([[ 1. , -0. ],
# # [ 0. , -1.5]])
#
# Near X_f0, the number of rabbits increase and the population of foxes decrease.
# The origin is a [http://en.wikipedia.org/wiki/Saddle_point saddle point].
#
# Near X_f1, we have:
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 number, so the fox and rabbit populations are periodic and
# their period is given by:
T_f1 = 2*pi/abs(lambda1) # >>> 5.130199
#
# == Integrating the ODE using scipy.integate ==
#
# Now we will use the scipy.integrate module to integrate the ODEs.
# This module offers a method named odeint, very easy to use to integrate ODEs:
#
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.'
#
# `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:
#
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')
#
#
# The populations are indeed periodic, and their period is near to the T_f1 we calculated.
#
# == 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 [http://www.scipy.org/Cookbook/Matplotlib/Show_colormaps ShowColormaps] if you want more information.
#
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')
#
#
# We can see on this graph that an intervention on fox or rabbit populations can
# have non intuitive effects. If, in order to decrease the number of rabbits,
# we introduce foxes, this can lead to an increase of rabbits in the long run,
# if that intervention happens at a bad moment.
#
#
# == Plotting contours ==
#
# We can verify that the function IF defined below remains constant along a trajectory:
#
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 %
#
# Potting iso-contours of IF can be a good representation of trajectories,
# without having to integrate the ODE
#
#-------------------------------------------------------
# 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()
#
#
# # vim: set et sts=4 sw=4: