One of the courses I enjoyed most in my Ph.D. program was taught by Prof. Kirk Winemiller on population dynamics. There are various collections of models in various languages out there, and multi-model population dynamic applications. But I still think that there is some utility to rolling my own. Since 2009, I’ve gotten more into Python programming, so I thought that I would take a popular class of population dynamic models and produce a Python module to instantiate them.

A long-time standard method in population modeling is the Leslie matrix. This technique applies when one has data about the age structure of a population and produces estimates going forward by using matrix multiplication to go from the population numbers, fecundity, and survivorship numbers to get the estimate of the population in each age class at the next time step.

A similar method is the Lefkovitch approach. This is still based upon matrix operations, but the underlying data involves stages rather than age structure. This sort of model is often used to capture more complex life histories than are tracked in a Leslie matrix model.

The similarities make it straightforward to incorporate both approaches into one supporting Python class.

The following Python module defines the LMatrix class. The dependencies are the Numpy module and the interval module. I used “pip install interval” to get the interval module on my machine. If you run this module in standalone mode, it runs a test of the LMatrix model with a web-accessible example of a Leslie matrix and of a Lefkovitch matrix.

[python]

“””

popdyn.py

Trying out population dynamics in Python.

Wesley R. Elsberry

“””

class LMatrix:

“””

LMatrix

A support class for Leslie and Lefkovitch matrix use for

population dynamics.

This is a generic class that allows for an arbitrary number of age

classes or stages.

“””

def __init__(self,stAges):

import numpy as num

import numpy.matlib as M

from numpy.matlib import rand,zeros,ones,empty,eye

import interval

“””

In either Leslie age-structured or Lefkovitch stage-

structured population modeling, the central feature

is a special matrix representing both fecundity of

ages/stages and survivorship in each age/stage.

The Leslie age-structured matrix is slightly simpler,

since each iteration moves the population forward

by a time step equal to the difference between the

age classes.

The Lefkovitch stage-structured matrix,

on the other hand, may have unequal times spent in

each stage, and thus other elements of the matrix

represent the fraction of individuals that continue

to remain in the stage per time step of the model.

Those lie on the main diagonal.

The matrix in either case is an N-by-N matrix, where

N is the number of ages or stages (stAges parameter).

Because most values in the matrix are zero, we’ll

start with that.

“””

self.stAges = stAges # Keep track of how many age/stage classes there are

self.m = zeros((self.stAges,self.stAges))

self.step = 0 # We are at the beginning

self.popvec = None

self.survival = None

self.recurrence = None

self.fecundity = None

def LM_AddFecundity(self,fvector):

“””

Method to set fecundity values for an LMatrix.

This is done by setting the first row of the

matrix to the values in the vector.

A mismatch between the length of the vector and

the width of the matrix leaves both unchanged.

“””

if (fvector.shape[0] == self.stAges):

# Just replace the row

self.m[0] = fvector

# Save it in the object

self.fecundity = fvector

else:

print “Mismatch in size: %s vs. %s” % (self.stAges – 1,fvector.shape[0])

def LM_AddSurvival(self,survival):

“””

Add the values for survival that shift population members

from one age/stage to the next.

The values come in as the “survival” vector, a Numpy array.

They replace values in the m matrix in the diagonal from

[1,0] to [N-1,N-2].

“””

if (survival.shape[0] == (self.stAges – 1)):

for ii in range(1,self.stAges):

self.m[ii,ii-1] = survival[ii-1]

# Save it in the object

self.survival = survival

else:

print “Mismatch in size: %s vs. %s” % (self.stAges – 1,survival.shape[0])

def LM_AddRecurrence(self,recur):

“””

Add the values for survival of organisms remaining in the same

stage. This is for stage-structured population models only.

The input is as the vector recur, and its values replace those

in the m matrix along the main diagonal from [1,1] to [N-1,N-1].

“””

if (recur.shape[0] == (self.stAges – 1)):

for ii in range(1,self.stAges):

self.m[ii,ii] = recur[ii-1]

# Save it in the object

self.recurrence = recur

else:

print “Mismatch in size: %s vs. %s” % (self.stAges – 1,recur.shape[0])

def LM_SetOneRelation(self,fromState,toState, value):

“””

Method to set a relation that does not fall on the survival

diagonal or the recurrence diagonal. This is useful for more

complex stage-structured population modeling where organisms

from one stage may graduate to multiple other stages at defined

rates.

“””

iv = interval.Interval.between(0,self.stAges-1)

if ((fromState in iv) and (toState in iv)):

print self.m

self.m[toState,fromState] = value

print self.m

def LM_SetPopulation(self,popvector):

“””

Another central feature of these models is that the size

of the population is kept in a 1xN column vector. For the

implementation here, the actual representation is as a

Numpy array, which has no column vector as such. This will

be handled in the actual stepping method.

“””

if (popvector.shape[0] == (self.stAges)):

self.popvec = popvector

else:

print “Mismatch in size: %s vs. %s” % (self.stAges,popvector.shape[0])

def LM_StepForward(self):

“””

Do the matrix multiplication to obtain the new population

vector. Retain the previous population vector.

Handle turning population vector into a column vector for the

multiplication.

“””

# Convert the population array to a Numpy matrix and transpose it

# to get the column vector we need. Multiply the L* matrix by

# the column vector, resulting in a new column vector with the

# population at the next step.

nextpopvec = num.mat(self.m) * num.mat(self.popvec).T

# Save the old population vector

self.lastpopvec = self.popvec

# Replace the population vector with the new one, which means

# transposing it and converting to Numpy array type

self.popvec = num.array(nextpopvec.T)

# Track the number of steps taken

self.step += 1

def LM_TotalPopulation(self):

“””

Return the total population size. Sums the “popvec” vector.

“””

if (None != self.popvec):

# Population vector as array multiplied by column vector of 1s is a sum

t = num.mat(self.popvec) * ones(self.stAges).T

return t[0,0]

else:

return 0.0

if __name__ == “__main__”:

“””

Generic initialization suggested at

http://www.scipy.org/NumPy_for_Matlab_Users

“””

# Make all numpy available via shorter ‘num’ prefix

import numpy as num

# Make all matlib functions accessible at the top level via M.func()

import numpy.matlib as M

# Make some matlib functions accessible directly at the top level via, e.g. rand(3,3)

from numpy.matlib import rand,zeros,ones,empty,eye

# Define a Hermitian function

def hermitian(A, **kwargs):

return num.transpose(A,**kwargs).conj()

# Make some shorcuts for transpose,hermitian:

# num.transpose(A) –> T(A)

# hermitian(A) –> H(A)

T = num.transpose

H = hermitian

import interval

# Check it against an existing example data set

# http://www.cnr.uidaho.edu/wlf448/Leslie1.htm

ex1 = LMatrix(4)

fex1 = num.array([0.5, 2.4, 1.0, 0.0])

ex1.LM_AddFecundity(fex1)

sex1 = num.array([0.5, 0.8, 0.5])

ex1.LM_AddSurvival(sex1)

pex1 = num.array([20, 10, 40, 30])

ex1.LM_SetPopulation(pex1)

print pex1

print ex1.m

ex1.LM_StepForward()

print ex1.popvec

# It checks out!

# Another example, this time of a stage-structured population

# http://www.afrc.uamont.edu/whited/Population%20projection%20models.pdf

ex2 = LMatrix(3)

fex2 = num.array([0.0, 52, 279.5])

ex2.LM_AddFecundity(fex2)

sex2 = num.array([0.024, 0.08])

ex2.LM_AddSurvival(sex2)

rex2 = num.array([0.25, 0.43])

ex2.LM_AddRecurrence(rex2)

pex2 = num.array([70.0,20.0,10.0])

ex2.LM_SetPopulation(pex2)

print pex2

print ex2.m

ex2.LM_StepForward()

print ex2.popvec

print ex2.LM_TotalPopulation()

ex2.LM_StepForward()

print ex2.LM_TotalPopulation()

ex2.LM_StepForward()

print ex2.LM_TotalPopulation()

for ii in range(22):

ex2.LM_StepForward()

print ex2.popvec

# Tests OK!

[/python]

Output from the standalone run:

[python]

[20 10 40 30]

[[ 0.5 2.4 1. 0. ]

[ 0.5 0. 0. 0. ]

[ 0. 0.8 0. 0. ]

[ 0. 0. 0.5 0. ]]

[[ 74. 10. 8. 20.]]

[ 70. 20. 10.]

[[ 0.00000000e+00 5.20000000e+01 2.79500000e+02]

[ 2.40000000e-02 2.50000000e-01 0.00000000e+00]

[ 0.00000000e+00 8.00000000e-02 4.30000000e-01]]

[[ 3835. 6.68 5.9 ]]

3847.58

2093.1914

5811.535142

[[ 19837904.89838918 393232.36554185 30519.85368983]]

[/python]