# 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 model
```

Which 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)
```

Cookbook/Theoretical Ecology/Hastings and Powell (last edited 2011-03-09 06:37:28 by Warren Weckesser)