Iakovos Matsikis
Stochastic Population Dynamics
Work done in collaboration with :
Stuart Townley and
Dave Hodgson
Overview
 MODELING STOCHASTIC POPULATION ENVIRONMENTS
 Life Cycle Graph
 Population Projection Matrices
 Noisy Vital-Rates
 SECOND MEAN ANALYSIS
 Mean and Second Mean Growth Bounds
 Lyapunov Matrix Equations
 GROWTH or EXTINCTION ?
 Transient and Asymptotic Behavior
 Examples
 FURTHER RESEARCH
Modeling the environment
A population can be described by age-stages.
An individual grows through the different age-stages.
The vital rates characterizing the populations evolution could
be :
1. Transition probabilities Gij from one stage to another.
2. Transition probabilities Pii from a stage to itself.
3. Fecundity rates Fj1, leading to new individuals.
F51
F41
Life Cycle graph
F31
F21
1
P11
G12
2
P22
G23
3
P33
G34
4
P44
G45
5
P55
The life cycle graph is isomorphic to the following linear,
discrete time system:
x(t  1)  Ax(t )
(1)
where A in (1) is the Population Projection Matrix and is equal to
 P11
G
 12
A 0

 0
 0
F21
F31
F41
P22
0
0
G23
P33
0
0
G34
P44
0
0
G45
F51 
0 
0 .

0 
P55 
Here Fj1, Gij and Pii are the fecundities and transition probabilities
from the life cycle graph.
We now assume that the fecundities change randomly in time.
This means that the fecundities become
F j1  F j1   t f j1 ,
j  1,..., n -1
where σt are a family of independent, identically distributed random
variables and fj1 weights, measuring the effect of noise into the system.
The above assumption transforms equation (1) into the following
linear, noisy discrete time system
x(t  1)  Ax(t )   t Bx(t )
(2).
The matrix B appearing in (2) has the following form
0
0

B  0

0
0
f 21
f31
f 41
0
0
0
0
0
0
0
0
0
0
0
0
f51 
0 
0 .

0
0 
Mean Analysis
Our objective is to predict the evolution of the population described
by the previously derived, noisy, discrete time, linear system
x(t  1)  ( A   t B) x(t ).
A natural candidate is the mean exponent of a trajectory which is
defined as
1
1  lim
log x(t ) 1 .
t  t
Since we are interested in noisy systems we can take the average
over all mean exponents λ1, to obtain the ensemble average of the
system (Lyapunov exponent)
s  E ( lim
t 
1
log x(t ) 1 ).
t
Here E denotes expectation over the growth rates of every trajectory
of the noisy system.
We take another road!
Second Mean Analysis
To measure the population dynamics as they unfold in time we
define a norm
x
P

xT Px ,
where x is the population vector, P is a positive definite matrix and
for illustrative purposes we assume in what follows that σt is N(0,σ).
Assuming an initial condition x(0), then (x(t))TPx(t) is a random
variable for every t, independent of x(0). Its first step expectation is


E ( x(1)T Px(1))  x(0)T AT PA   2 BT PB x(0) 
 x(0)T Px
1 (0).
Proceeding inductively we obtain
E ( x(t  1)T Px(t  1))  x(0)T Pt x(0).
The previous arguments imply that the second mean dynamics of
the system will be determined by the matrices Pt, given by the
Lyapunov iteration
Pt 1  AT Pt A   2 BT Pt B, where P0  P.
If P is symmetric, we can identify it with a vector π and pass to an
eigenvalue-eigenvector equation by rewriting the above equation
as
 (t  1)   ( A, B,  ) (t ).
Τhe second-mean growth bound λ2 is defined to be the minimum ρ
such that
E ( x(t ) P )  M  x(0) P , for all x(0)
2
t
2
or if we pass to the eigenvalue-eigenvector characterization
described by the previous equation, the maximum eigenvalue of
Ψ(A,B,σ).
We can compute the second mean exponent via the next power
method

Yk  A Pk A   B Pk B 

S k  Yk
 k  0,1, 2,...

Yk

Pk 1 
Sk

T
2
T
(3)
For any initial P0, Sk will converge to λ2.
We can use this scheme to even analyze several noise terms
k
x(t  1)  Ax(t )    ti Bi x(t )
(4)
i 1
where the different σi are differently distributed random
variables.
Growth or Extinction?
The arguments from the previous section can be used to predict the
transient and asymptotic behavior of a noisy system of the form (4).
Example 1
To show this relation we consider the following example for the
invasive shrub Ardisia elliptica that evolves according to the PPM
0
0
0
0.5135 7.0345 33.8955 184.5300 
 0
0.0985 0.4320 0.0190 0.0135

0
0
0
0


 0
0.3680 0.7180 0.0410 0.0115 0.0115
0
0 


0
0
0.1980
0.6075
0.0340
0
0.0145
0

A
 0
0
0
0.2300 0.7005 0.0730 0.0280
0 


0
0
0.0525 0.1815 0.6700
0
0 
 0
 0
0
0
0
0.0160 0.2455 0.8805 0.0170 


0
0
0
0
0
0.0570 0.9780 
 0
We assume that the environment is subject to disturbances σt
following either a normal distribution N(0,0.5) or a Poisson distribution
with parameter λ=0.04 and acting on the PPM according to
0
0
0
0.5135 4.2575 11.3525 14.5300 
 0
0.0005 0.0680 0.0190 0.0135

0
0
0
0


 0
0.0270 0.0410 0.0410 0.0345 0.0115
0
0 


0
0
0.1080
0.0775
0.0180
0
0.0145
0

B
 0
0
0
0.1340 0.0065 0.0410 0.0280
0 


0
0
0
0.0385
0.0605
0.0570
0
0


 0
0
0
0
0.0160 0.1095 0.0055
0 


0
0
0
0
0
0.0290
0 
 0
We use the Lyapunov iteration (4) to compute the second moment
growth bound and compare with different realizations of the
stochastic process x(t).
Transient and Asymptotic Dynamics
Example 2
We consider another example, the desert tortoise Gompherus
agassizii. Here we assume that noise follows either a Ν(0,0.5) or a
Poisson distribution with parameter λ=0.04. The PPM is the following
0
0
0
0
1.131 1.724 2.224 
 0
0.716 0.567

0
0
0
0
0
0


 0
0.149 0.567
0
0
0
0
0 


0
0
0.149
0.604
0
0
0
0

A
 0
0
0
0.235 0.560
0
0
0  and B is of the form


0
0
0
0.225 0.678
0
0 
 0
 0
0
0
0
0
0.249 0.851
0 


0
0
0
0
0
0
0.016
0.86


0
0

0

0
B
0

0
0

0
0 0 0 0 1.131 1.724 2.224 
0 0 0 0
0
0
0 
0 0 0 0
0
0
0 

0 0 0 0
0
0
0 
.
0 0 0 0
0
0
0 

0 0 0 0
0
0
0 
0 0 0 0
0
0
0 

0 0 0 0
0
0
0 
Perturbation Analysis
Pseudospectra
We can moreover derive envelopes that bound the systems
variance given by
2
vart ( x(t ))  E ( x(t )  x (t ) )  E ( x(0) Pt x(0))  A x (0) .
2
T
t
Variances
Further Research
1.
Study the ensemble effects of a family of differently distributed
noise variables in the transient and asymptotic behavior of a
population.
2.
Derive covariance matrices and the associated pseudospectra
that will allow us to judge if any initial correlation between
antagonistic species persists or not.
3.
Apply Lyapunov theory techniques to density-dependent, noisy
population environments.
Thank you for your attention