# [SciPy-user] Volterra system identification

Michael mnandris@btinternet....
Mon Apr 28 05:36:18 CDT 2008

```On Mon, 2008-04-28 at 11:21 +0200, Georg Holzmann wrote:
> Hallo!
>
> Are there any libraries for volterra system identification somewhere in
> the scipy/numpy world ?
> I did not find something in the web ...

>From a previous post:

import numpy as n
import pylab as p
import scipy.integrate as integrate

"""
If you look closely to the second graph, you can see that the trajectory
crosses some arrows of the direction field. I had this problem too,
before forcing matplotlib to use equal axis.
"""

alpha, delta = 1, .25
beta, gamma = .2, .05

def dr(r, f): return alpha*r - beta*r*f

def df(r, f): return gamma*r*f - delta*f

def derivs(state, t):
""" Map the state variable [rabbits, foxes] to the derivitives
[deltar, deltaf] at time t """
#print t, state
r, f = state  # rabbits and foxes
deltar = dr(r, f)  # change in rabbits
deltaf = df(r, f) # change in foxes
return deltar, deltaf

# the initial population of rabbits and foxes
r0 = 20
f0 = 10

t = n.arange(0.0, 100, 0.1)

y0 = [r0, f0]  # the initial [rabbits, foxes] state vector
y = integrate.odeint(derivs, y0, t)
r = y[:,0]  # extract the rabbits vector
f = y[:,1]  # extract the foxes vector

p.figure()
p.plot(t, r, label='rabbits')
p.plot(t, f, label='foxes')
p.xlabel('time (years)')
p.ylabel('population')
p.title('population trajectories')
p.grid()
p.legend()
#p.savefig('lotka_volterra.png', dpi=150)
#p.savefig('lotka_volterra.eps')

p.figure()
p.plot(r, f)
p.xlabel('rabbits')
p.ylabel('foxes')
p.title('phase plane')

# make a direction field plot with quiver
rmax = 1.1 * r.max()
fmax = 1.1 * f.max()
R, F = n.meshgrid(n.arange(-1, rmax), n.arange(-1, fmax))
dR = dr(R, F)
dF = df(R, F)
p.quiver(R, F, dR, dF)

R, F = n.meshgrid(n.arange(-1, rmax, .1), n.arange(-1, fmax, .1))
dR = dr(R, F)
dF = df(R, F)

p.contour(R, F, dR, levels=[0], linewidths=3, colors='black')
p.contour(R, F, dF, levels=[0], linewidths=3, colors='black')
p.ylabel('foxes')
p.title('trajectory, direction field and null clines')

#p.savefig('lotka_volterra_pplane.png', dpi=150)
#p.savefig('lotka_volterra_pplane.eps')

p.show()

> Thanks for any hint,
> LG
> Georg
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```