Theoretical ecology: Hastings and PowellΒΆ


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

In [ ]:
import numpy

def window(data, size):
    """A generator that returns the moving window of length
    `size` over the `data`

    for start in range(len(data) - (size - 1)):
        yield data[start:(start + size)]

def min_max(data, tol=1e-14):
    """Return a list of the local min/max found
    in a `data` series, given the relative tolerance `tol`

    maxes = []
    mins = []
    for first, second, third in window(data, size=3):
        if first < second and third < second:
        elif first > second and third > second:
        elif abs(first - second) < tol and abs(second - third) < tol:
            # an equilibrium is both the maximum and minimum

    return {'max': numpy.asarray(maxes),
            '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


    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

In [ ]:
import numpy
from scipy.integrate import odeint
import bif

import hastings

# setup the food web parameters
hastings.model.a1 = 5.0
hastings.model.a2 = 0.1
hastings.model.b2 = 2.0
hastings.model.d1 = 0.4
hastings.model.d2 = 0.01

# setup the ode solver parameters
t = numpy.arange(10000)
y0 = [0.8, 0.2, 10.0]

def print_max(data, maxfile):
    for a_max in data['max']:
        print >> maxfile, hastings.model.b1, a_max

x_maxfile = open('x_maxfile.dat', 'w')
y_maxfile = open('y_maxfile.dat', 'w')
z_maxfile = open('z_maxfile.dat', 'w')
for i, hastings.model.b1 in enumerate(numpy.linspace(2.0, 6.2, 420)):
    print i, hastings.model.b1
    y = odeint(hastings.model.fweb, y0, t)

    # use the last 'stationary' solution as an intial guess for the
    # next run. This both speeds up the computations, as well as helps
    # make sure that solver doesn't need to do too much work.
    y0 = y[-1, :]

    x_minmax = bif.min_max(y[5000:, 0])
    y_minmax = bif.min_max(y[5000:, 1])
    z_minmax = bif.min_max(y[5000:, 2])

    print_max(x_minmax, x_maxfile)
    print_max(y_minmax, y_maxfile)
    print_max(z_minmax, z_maxfile)

Section author: GabrielGellner, WarrenWeckesser