From: "Richard A. Hinrichsen"
Subject: Population Waves
Date: Wed, 29 Sep 1999 15:54:46 +0000
Newsgroups: sci.math
Keywords: population dynamics, Leslie matrices
I have a question about population waves.
Leslie matrices are being used by the
National Marine Fisheries Service to
model the dynamics of salmon populations
that are nearly extinct. The goal is
to find strategies that increase the Perron
eigenvalues. It occurred to me that population waves
will also be important in determining whether a
population will persist.
The difference equation governing the dynamics
of the age-structured population is:
v(t+1) = L*v(t),
where L is a Leslie matrix, and v(t) is the age-specific
population vector for year t (the intial age-specific
vector is v(0)).
As usual, the first row of L
contains the age-specific fecundities, and the
diagonal below the main diagonal contains the
age-specific survival probabilities (See Leslie
1945 for further information).
The entries of the nxn matrix L are nonnegative
as are the entries of the nx1 vectors v(0),v(1),... .
I wonder what form of the initial age population
vector gives rise to the greatest oscillations? This
could be used to get an upper bound on the
importance of fluctuations to extinction
risk. The matrices we use are primitive, the
eigenvalues are distinct and come in conjugate
pairs (except for the Perron eigenvalue). My intuition
is that the initial vector will have a single nonzero
entry.
An example of a Leslie matrix we are using is
L =
0 0 0 9.023 48.779
0.0582 0 0 0 0
0 0.8 0 0 0
0 0 0.8 0 0
0 0 0 0.652 0
Which is "close" to an imprimitive matrix
since the birth rate in the last age group
is much greater than the others. This means
that population waves (oscillations) will be important.
We can show that the population
trajectory is
v(t) = w1*c1*lambda1^t
+(r2)^t*{cos(theta2*t)(c2*w2+c3*w3)+isin(theta2*t)*(c2*w2-c3*w3)}
+(r4)^t*{cos(theta4*t)(c4*w4+c5*w5)+isin(theta4*t)*(c4*w4-c5*w5)}
where v(0) = c1*w1 + c2*w2 + c3*w3 + c4*w4+c5*w5
(such a linear combination of the eigenvectors with weights
c1, c2, ..., c5 is possible because the e.vectors are linearly
independent).
In the above, lambda1 is the Perron eigenvalue, r2 is the modulus
of the second and third eigenvalues, and r4 is the moduls of the
fourth and fifth eigenvalues. w1,w2,w3,w4,w5 are eigenvectors
corresponding to the five distinct eigenvalues, such that
w2=Conj(w3), w4=Conj(w5), and the weights are such that c2=Conj(c3),
c4=Conj(c5), and c1 is positive and real. (w1 is real and can be
chosen to have positive entries).
theta2 is the the angle that the 2nd eigenvalue makes with the positive
x axis in the complex plane, and theta4 is the angle of the 4th
eigenvalue. The
eigenvalues are such that lambda2=Conj(lambda3) and
lambda4=Conj(lambda5).
The problem is to choose v(0) to excite the greatest population waves in
the salmon population (waves of maximum amplitude), given that
||v(0)||=1, where ||.|| is the l2 (Euclidean) norm. The waves are
associated
with the terms in the {} in the above equation.(Note that v(0), the
initial
vector, must be nonnegative). Are there general rules for choosing
v(0) based on the entries of L?
For discussion of population waves, see Bernadelli (1941).
Thanks in advance.
Reference:
Leslie, P.H. 1945. On the use of matrices in certain population
mathematics.
Biometrika 33:183-212.
Bernadelli, H. 1941. Population waves. Journal of Burma Research Society
31: 1-18.
--
Richard A. Hinrichsen
Lakeview Medical Dental Building
3216 NE 45TH PLE STE 303W
Seattle, WA 98105-4028