Overview
A simple script that recreates the min/max bifurcation diagrams from Hastings and Powell 1991.
Library Functions
Two useful functions are defined in the module bif.py.
1 import numpy
2
3 def window(data, size):
4 """A generator that returns the moving window of length
5 `size` over the `data`
6
7 """
8 for start in range(len(data) - (size - 1)):
9 yield data[start:(start + size)]
10
11
12 def min_max(data, tol=1e-14):
13 """Return a list of the local min/max found
14 in a `data` series, given the relative tolerance `tol`
15
16 """
17 maxes = []
18 mins = []
19 for first, second, third in window(data, size=3):
20 if first < second and third < second:
21 maxes.append(second)
22 elif first > second and third > second:
23 mins.append(second)
24 elif abs(first - second) < tol and abs(second - third) < tol:
25 # an equilibrium is both the maximum and minimum
26 maxes.append(second)
27 mins.append(second)
28
29 return {'max': numpy.asarray(maxes),
30 'min': numpy.asarray(mins)}
The Model
For speed the model is defined in a fortran file and compiled into a library for use from python. Using this method gives a 100 fold increase in speed. The file uses Fortran 90, which makes using f2py especially easy. The file is named hastings.f90.
module model
implicit none
real(8), save :: a1, a2, b1, b2, d1, d2
contains
subroutine fweb(y, t, yprime)
real(8), dimension(3), intent(in) :: y
real(8), intent(in) :: t
real(8), dimension(3), intent(out) :: yprime
yprime(1) = y(1)*(1.0d0 - y(1)) - a1*y(1)*y(2)/(1.0d0 + b1*y(1))
yprime(2) = a1*y(1)*y(2)/(1.0d0 + b1*y(1)) - a2*y(2)*y(3)/(1.0d0 + b2*y(2)) - d1*y(2)
yprime(3) = a2*y(2)*y(3)/(1.0d0 + b2*y(2)) - d2*y(3)
end subroutine fweb
end module modelWhich is compiled (using the gfortran compiler) with the command: f2py -c -m hastings hastings.f90 --fcompiler=gnu95
The Script
1 import numpy
2 from scipy.integrate import odeint
3 import bif
4
5 import hastings
6
7 # setup the food web parameters
8 hastings.model.a1 = 5.0
9 hastings.model.a2 = 0.1
10 hastings.model.b2 = 2.0
11 hastings.model.d1 = 0.4
12 hastings.model.d2 = 0.01
13
14 # setup the ode solver parameters
15 t = numpy.arange(10000)
16 y0 = [0.8, 0.2, 10.0]
17
18 def print_max(data, maxfile):
19 for a_max in data['max']:
20 print >> maxfile, hastings.model.b1, a_max
21
22 x_maxfile = open('x_maxfile.dat', 'w')
23 y_maxfile = open('y_maxfile.dat', 'w')
24 z_maxfile = open('z_maxfile.dat', 'w')
25 for i, hastings.model.b1 in enumerate(numpy.linspace(2.0, 6.2, 420)):
26 print i, hastings.model.b1
27 y = odeint(hastings.model.fweb, y0, t)
28
29 # use the last 'stationary' solution as an intial guess for the
30 # next run. This both speeds up the computations, as well as helps
31 # make sure that solver doesn't need to do too much work.
32 y0 = y[-1, :]
33
34 x_minmax = bif.min_max(y[5000:, 0])
35 y_minmax = bif.min_max(y[5000:, 1])
36 z_minmax = bif.min_max(y[5000:, 2])
37
38 print_max(x_minmax, x_maxfile)
39 print_max(y_minmax, y_maxfile)
40 print_max(z_minmax, z_maxfile)