Eigenvector Decomposition and Northern Spotted Owl Conservation
Joshua Track
MATH 280
03/23/2015
The northern spotted owl is a bird that resides around the Pacific Northwest, with its habitat
spanning approximately from northern California to southern Canada, and occupying a substantial
portion of Washington and Oregon. A good part of the economy in this region had been rooted in
logging and timber for over a hundred years, so the natural habitat of the northern spotted owl
subsequently diminished. As a result of the ever-shrinking natural habitat, the population of the
northern spotted owl had dropped to a point where the bird was listed as a threatened species under
the Endangered Species Act in 1990. (Conservation Northwest)
After being listed as a threatened species, several organizations such as the Northwest
Ecosystem Alliance, and the United States Fish and Wildlife Service studied the populations of
the northern spotted owl, and sought a way to further conservatory efforts. In 1991, Charles Biles,
a mathematician at Humboldt State University, and Barry R. Noon, a conservationist with the US
Forest Service published an article in the UMAP Journal outlining the mathematics used for the
purpose of spotted owl conservation. In The Spotted Owl, Biles demonstrates how systems of
equations involving the spotted owl populations at different stages in life, and their tendency to
survive and reproduce can yield the percentage of either growth or decline in northern spotted owl
population using an eigenvector decomposition (which will be explained further in this paper).
The method of eigenvector decomposition has since become standard practice in the study and
modeling of northern spotted owl conservation efforts.
Throughout this paper, there will be a variety of terminology used for the purpose of
explaining the processes involved in northern spotted owl conservation. It seems appropriate to
provide definitions for some of the less common terminology associated with the mathematical
processes outlined herein, as well as some terminology used in explaining conservatory efforts.
Two mathematical terms that will be seen regularly in this paper are eigenvalue and eigenvector.
An eigenvalue for an arbitrary matrix A is a scalar λ such that the matrix equation Ax = λx has a
solution for some non-zero vector x. An eigenvector for an arbitrary matrix A is a non-zero vector
x such that Ax = λx for some scalar λ. (Lay, A9) How these operations work specifically will be
shown later in a basic walkthrough. It should be noted that this paper is written with the assumption
that the reader minimally has a fundamental understanding of basic linear algebraic concepts and
a several other mathematical operations which are essential in understanding the applications of
this particular study.
A non-mathematical term that seems like a good idea to define is fecundity, as this is a
term that is frequently used in the development of systems of equations for northern spotted owl
conservation study, and will be used several times throughout the course of this paper. Fecundity,
as it pertains to the northern spotted owl, refers to the ability of owls in a particular age bracket to
produce offspring. This is important because fecundity is tantamount in determining the rate of
growth or decline of this particular population.
The next part of this paper will demonstrate the basic process of an eigenvector
decomposition. The first part will consist of a detailed outline using an arbitrary 3 x 3 matrix, the
scalar λ, and the identity matrix, denoted I. The process of eigenvector decomposition is described
by the following characteristic equation:
det (A- λI) = 0
For those unfamiliar with the terminology, this means that the eigenvalues can be found by
finding the product of λ and the identity matrix, I, subtracting that product from the matrix A,
finding the determinant of the resultant matrix thereafter, and factoring to find the values of λ. In
most cases, the determinant will yield a polynomial that must be factored to find the values for λ,
which are the eigenvalues. Consider the first problem, with an arbitrary 3 x 3 matrix. Underneath
each of the units in this matrix, there is a symbol to describe what each matrix is corresponding to
the in explanations listed earlier.
1 2
det [4 5
7 8
determinant
3
1 0 0
6] – λ [0 1 0] = 0
9
0 0 1
-λ
(A
I)
The first order of business is to multiply the scalar λ by the identity matrix, I. This yields:
1 2
det [4 5
7 8
3
λ
6] - [0
9
0
0 0
λ 0] = 0
0 λ
Next is a basic matrix subtraction problem, which will consolidate these two matrices into one
matrix yielding:
1−λ
2
det [ 4
5−λ
7
8
3
6 ]=0
9−λ
After the matrix is consolidated, we must then find the determinant as follows:
5−λ
1-λ |
8
4
6
6
4 5−λ
|–2|
|+3|
|
7 9−λ
9−λ
7
8
1-λ (λ2 -14 λ-3) – 2 (-λ-6) + 3 (7 λ-3)
After completing the process of finding the determinant, combining like terms, and setting the
result equal to zero, the resulting polynomial equation for λ is as follows:
−λ3 + 15 𝜆2 + 18λ = 0
λ2 - 15 λ -18 = 0
To find the eigenvalues of this equation, we must solve the equation for λ. The values of λ are the
eigenvalues. This can be accomplished by factoring, or utilizing the quadratic formula. For the
sake of making this paper look more mathematically adept, the next step will outline the use of the
quadratic formula. It must be noted, however, that later problems demonstrated will show the
eigenvalues after having been calculated using the assistance of Maple software.
λ=
λ=
15 ± √225+72
2
15
2
3
± √33
2
In this case, the result is two real eigenvalues. In further examples, the results may not be as nice
looking, and approximations to several significant figures may be used. It must also be reiterated
that the previous case was completely arbitrary and meant only as a primer to the problems which
will be demonstrated later in this writing, and as a basis for how to obtain eigenvalues via
eigenvector decomposition.
Now, consider the second problem, which is listed in a supplement to the Lay text available
online. The problem in review is question number one from Case Study: Dynamical Systems and
Spotted Owls. Let’s first review the information provided in the case study, an explanation will
then be provided as to how to construct the dynamical system needed to perform the subsequent
eigenvector decomposition. “The most recent spotted owl data available gives the following for
the matrix A: Juvenile Survival .33, Sub-adult Survival .85, Adult Survival .85, Sub-adult
Fecundity .125, Adult Fecundity .26.” (Lay, pp. 3) After inputting the data in a stage-matrix, an
explanation for the transcription will be afforded. The transcribed matrix yields:
0 . 125
[. 33
0
0
. 85
. 26
0 ]
. 85
The first column of the matrix describes the behavior and tendencies of juvenile spotted owls.
Since juveniles do not tend to reproduce, their fecundity is described in the matrix as zero. The
middle column describes the survival rate and fecundity of the sub-adult owls. The top row
describes the fecundity, and the bottom two rows describe the survival rate. The same holds for
the third column as it applies to adult owl population. According to Lay in the spotted owl
supplement, the model for this matrix is called the stage-matrix model for a population. (Lay, 1)
Next, the previous matrix will be applied to the eigenvector decomposition formula, recall the
formula is det (A-λI), yielding:
0
[. 33
0
1 0 0
. 26
0 ] – λ [0 1 0 ]
0 0 1
. 85
. 125
0
. 85
The process outlined here is the same as the process outlined in the previous example with the
arbitrary matrix. Therefore, the next step is to multiply the λ vector by the identity matrix I.
0
[. 33
0
. 125
0
. 85
. 26
λ 0
]
[
0
0 λ
. 85
0 0
0
0]
λ
After subtracting the λ identity matrix from the fecundity and survival matrix, the remaining matrix
yields:
−𝜆
[. 33
0
. 125
−𝜆
. 85
. 26
0 ]
. 85 − 𝜆
Now, the determinant of the matrix must be found.
-𝜆|
. 33
0
−𝜆
0
. 33 −𝜆
| - .125 |
| + .26 |
|
0 . 85 − 𝜆
. 85 . 85 − 𝜆
0 . 85
.85 𝜆2 - 𝜆3 + .0378675 + .04125 𝜆
With the assistance of Maple software, the eigenvalues from the polynomial shown above are
listed as:
0.9371354439
0.04356772197 - 0.1962385450i
0.04356772197 - 0.1962385450i
Note, the bottom two eigenvalues are complex, which are beyond the scope of this paper, though
they are listed here to show the complete result of the decomposition. The eigenvalue of import is
the one listed at the top. According to the Spotted Owl Case Study, “If λ is a real number greater
than 1 and all the other eigenvalues are less than 1 in magnitude, then the population is increasing
exponentially. In this case the eigenvector gives the stable distribution of the population between
classes, and yields the percentages found in each class if scaled so that its entries sum to 1.” (Lay,
pp. 2)
According to Lay, the important number in this decomposition is 0.9371354439. This
number corresponds to the increase or decrease in spotted owl population. If we were to decompose
a data set resulting with a real eigenvalue greater than one, the conclusion would be that the
population is increasing, or has increased. In the case of this particular problem, the resulting
eigenvalue is less than 1, which means that the data set for this particular case indicates a decrease
in spotted owl population.
The last example will review the most recent available set of data, whereupon the current
state of affairs of northern spotted owl population will be determined. In an article entitled Status
and Trends of Northern Spotted Owl Populations, there is a plethora of data which will be utilized
to find the current survival rate of northern spotted owls. The following images are tables that are
included in the aforementioned articles and will be used as the primary reference for constructing
the matrix which will be the basis for impending eigenvector decomposition. The first image is a
table of average survival rates for 1, 2, and 3+ year old northern spotted owls:
(Davis, Dugger, Mohoric, Evers, Aney, pp. 11)
It should be noted that the survival rates listed are for several different regions, and vary
from region to region. Because of this variance, the survival rates for the three groups will be
averaged to obtain an overall population growth or decline model in a basic stage-matrix. The next
table is taken from the same journal, and outlines fecundity:
(Davis, Dugger, Mohoric, Evers, Aney, pp. 12).
By utilizing the method of finding an arithmetic mean with the data provided by the
aforementioned authors, we can conclude that the average survival rate of 1 year old spotted owls
is .70422, for 2 years old: .80144, and 3+ years old: .84567. The corresponding fecundity averages
are as follows: 1 year old: .06467, 2 years old: .25911, and for 3+ years old: .34378. The
eigenvector decomposition model from the previous examples with the survival and fecundity
averages taken from recent data yields the following stage-matrix model:
. 06467
[. 70422
0
. 25911
0
. 80144
. 34378
0 ]
. 84567
With this matrix, the formula for eigenvector decomposition will be applied, and the necessary
mathematical operations will be applied to find the eigenvalues, and thereafter determine the
overall population growth and decline.
. 06467
[. 70422
0
. 06467
[. 70422
0
. 25911
0
. 80144
. 25911
0
. 80144
. 06467 − 𝜆
[ . 70422
0
.06467- 𝜆 |
−𝜆
. 80144
1 0 0
. 34378
]
–
λ
[
0 1 0]
0
0 0 1
. 84567
. 34378 𝜆 0
0 ]- [0 𝜆
. 84567 0 0
. 25911
−𝜆
. 80144
. 70422
0
| - .25911|
0
. 84567 − 𝜆
0
0]
𝜆
. 34378
]
0
. 84567 − 𝜆
0
. 70422
| + .34378|
. 84567 − 𝜆
0
−𝜆
|
. 80144
-𝜆3 + .91034𝜆2 + .12778𝜆 + .03972
-𝜆3 + .91034𝜆2 + .12778𝜆 + .03972 = 0
After inputting the above matrix in Maple, the software results yielded the following eigenvalues,
shortened to five significant figures:
1.06528
-.07747 + .01768i
-.07747 + .01768i
After reviewing the data, and the resulting eigenvalues after decomposition, it can be seen
that for the most recent data set that could be acquired the 𝜆 value for the real eigenvalue is
1.06528, and the subsequent eigenvalues are less than 1, indicating a notable growth in northern
spotted owl population. This is good. Thanks to the wit of mathematicians, and the diligent efforts
of conservationists in the Pacific Northwest, the northern spotted owl population can hopefully
live to thrive and grow another year. More information about northern spotted owl conservation
efforts can be found on the references page of this paper.
References
Biles, C., & Noon, B. (1990). The Spotted Owl. The UMAP Journal, 11.2, 99-109.
Conservation
Northwest
(2015).
The
Northern
Spotted
Owl.
Retrieved
from:
http://www.conservationnw.org/what-we-do/wildlife-habitat/northern-spotted-owl
Davis, R., Dugger, K., Mohoric, S., Evers, L., & Aney, W. (2011) Status and Trends of Northern
Spotted Owl Populations and Habitats. United States Department of Agriculture, United
States Forest Service, Pacific Northwest Research Station.
Lay, D. (2012). Eigenvalues and Eigenvectors. Linear Algebra and its applications, (pp. 265-324).
United States: Addison-Wesley.
Lay, D. (2003). Case Study: Dynamical Systems and Spotted Owls. United States: AddisonWesley. Retrieved from:
http://media.pearsoncmg.com/aw/aw_lay_linearalg_updated_cw_3/cs_apps/lay03_05_c
s.pdf