Probabilistic Uncertainty in Population Dynamics
a*
b
b
c
OSEI, B. M. , HOFFMANN, J. P. , and ELLINGWOOD C. D. , BENTIL, D. E.
a
Departments of Mathematics and Computer Science, Eastern Connecticut State University,
83 Windham Street, Willimantic, CT 06226
and
b
c
Botany and Agricultural Biochemistry, and Mathematics and Statistics, University of Vermont,
Burlington, VT 05405
UNITED STATES OF AMERICA
Abstract: Analysis and characterization of either environmental or demographic uncertainty in population
dynamics models are critical to species conservation and invasion. In particular, a probabilistic
uncertainty methodology can be employed to analyze the fluctuations in environmental variables for
model parameters of a prototype, generalized growth model in population dynamics. This generalized
model encapsulates a myriad of sub-models. Estimates of the extinction time as well as the boundary
classifications are determined.
Key-Words:- Probabilistic uncertainty, population dynamics, environmental noise, generalized growth models.
[8]). In this paper, we focus on probabilistic
uncertainty.
1 Introduction
The phenomenon of uncertainty in ecological
modeling plays a very important role in spatial data
and processes. Data on population dynamics often
contain fluctuations, which are either due to
environmental or demographic noise. In particular,
environmental stochasticity can be measured due to
fluctuations in the parameters of the model. For
example, Croxal et al. [1] report that the Wandering
Albatross exhibits fluctuations that cause an increase
in their population to above their carrying capacities
which in turn causes a corresponding negative
decrease in their intrinsic growth rates. Fluctuations in
model parameters can greatly impact on the
conclusions made out of model predictions.
There are two ways to incorporate uncertainty or
noise in population dynamic models: stochastic and
deterministic approaches. For the stochastic approach,
noise is considered to be probabilistic ([2], [3], [4],
[5]). In addition, depending on the modeling scenario,
noise can be added linearly by some well know
probability distribution (for example, Gaussian,
Cauchy and exponential). On the other hand, for
deterministic uncertainty, noise is considered as a
small perturbation of the model parameters ([6], [7],
2 Suite of Growth Models
We present a probabilistic uncertainty analysis for a
suite of models, which are obtained from a
generalized growth model of the form:
dN a 1 b b
 N K  Nb 
dt b
(1)
where K is the carrying capacity, b is a constant which
describes the rate of growth and density dependence
and a is a scaled growth rate [9]. This generalized
model is an extension to the one originally described
by [10] for two independent growth mechanisms. This
generalization describes a suite of approximately 11
known sub-models for parameter ranges given in table
1 below.
Based on equation 1 we will derive results for
fluctuations due to probabilistic noise, in particular
environmental noise.
1
where m( N ) and v( N ) are the mean and variance of
the change in N per unit time t. Equation (5) is called
the Diffusion equation (other names are FokkerPlanck equation in Physics and ChapmanKolmogorov equation in Mathematics). Based on this
equation we can calculate the equilibrium distribution,
bearing in mind that at equilibrium the flux across the
surface is zero, that is, pˆ ( N , t ) must satisfy the
ordinary differential equation
1 d
m( N ) pˆ ( N , t ) 
 v( N ) pˆ ( N , t )   0 (6)
2 dN
Equation (6) simplifies to the equilibrium
probability distribution
3 Probabilistic Uncertainty
Elderstein-Kershet [11] showed that the flow of
particles across a surface undergoing growth is given
by
N ( x, t )
J ( x, t )

 f ( x, t )
(2)
t
x
where J ( x, t ) is the flux across the surface and
f ( x, t ) is the growth function for the species. Let
p  N , t  represent the probability density function for
a population of size N ( x, t ) at time t. The flux
equation (equation 2) reduces to
p ( N , t )
J ( N , t )

(3)
t
N
noting that f ( x, t )  0 for probabilities since
there is no growth.
pˆ ( N ) 
Model type
Exponential
Momomolecular
Logistic
Gompertz
Richards
Generalized von
Bertalanfy: Putter
No. 1
a > 0, b = 1/3, K>0
Generalized von
Bertalanfy: Putter
No. 2
a > 0, b > 0, K>0
Generalized von
Bertalanfy
a = 0, b = 1, K>0
Linear
a = 0, b = ½, K>0
Quadratic
a = 0, b = 0, K>0
rth power
Table 1: A list of parameter values and their model
types

m( s ) 
w( N )  exp  2 
ds  ,
 N v( s ) 
2 N
 v( N ) p( N , t )  .
p( N ) 
N
 m( s) 
1
exp  2 
ds 
v( N )
 N v( s ) 
(9)
is the speed density and differs from the equilibrium
distribution by the normalization constant. The
integrals,
y ( N )  p( N )  w( s)ds
(10)
z ( N )  w( N )  p( s)ds ,
(11)
N
N
are functions that help to determine boundary
classifications.
Given any population dynamic model it is
possible to relate the quantities in equation (5) to
biological phenomena. For example, [112] relates this
model to genetic drift. In this paper, we relate this
methodology to population growth.
Consider a general stochastic growth equation
such as
dN
 F (N )
dt
(4)
Incorporating this in the general particle flow,
equation (equation 3) gives
p  N , t 

1 2

 m( N ) p( N , t ) 
v( N ) p( N , t ) (5)
dt
(8)
which is the scale density function whereas
The flux, J ( N , t ) , is now made up of two terms
which are the diffusion term and the term that
accounts for movement due to some external force. In
short, J ( N , t ) is given by
1 
(7)
provided the integral exists and where c is a
normalization constant. Other integrals that can be
calculated to provide more information and answers to
biological questions (see [12]) are as follows
Values of a and b, K
a < 0, b = 1, K=0
a < 0, b = 1, K>0
a > 0, b = -1, K>0
a > 0, b = 0, K>0
a > 0, b < 0, K>0
a > 0, b = 1, K>0
J ( N , t )  m( N ) p ( N , t ) 
 m( s) 
c
exp  2 
ds 
v( N )
v
(
s
)
 N

(12)
Suppose there is a biological parameter, say r that
undergoes a fluctuation with variance  , then
equation (12) can be simplified in general as
2 N 2
2
dN
 f ( N )   g ( N ) z (t )
dt
On the other hand, if this probability is zero (0) then
that path is repelling.
(13)
f ( N )  rNf1 ( N )
and g ( N )  Nf1 ( N ) .
discrete equivalent we have
For
ot
a
w
e
bl
ra
g
te
le
rab
eg
int
N)
w(
where z(t) is a random variable in continuous time and
so
that
F ( N )  (r   z (t )) Nf1 ( N ) ,
in
)n
(N

p(n
) no
t in
) int
egra
ble
p(N
y(N) integrable
tegrable
p(N) in
t 
rable

Naturally
attracting
Exit
Not Attainable
Fig 1: Boundary
uncertainty
Regular
Attainable
classifications
for
probabilistic
(16)
4 Generalized Model with Fluctuating a
(see for example, [12]).
Biological conclusions based on the quantities
m(N), v(N), w(N), y(N), z(N) and p(N) will be made
from the boundary classification. Figure one is a
diagram of the summary of the boundary classification
[12]. Note that the boundaries are N  0 , N  K and
N   . The boundary classifications can biologically
determine if a population will go extinct in a finite
time or as t   . For example if N  0 is attracting
then the population is approaching extinction as
t   (population is on the decline), on other hand
N  0 is attainable if the population becomes extinct
in a finite time. In general, a boundary is attracting if
the probability of the limit of a sample path is positive
as t   , mathematically this can be represented by
P lim N (t )  a  0 .
z(N) not integrable

m( N )  f ( N )
2
p(N) is not integrable
Naturally
Entrance Repelling
for some M  0 , then under some smoothness
conditions for m and  , X(t) satisfies the diffusion
equation (5). Thus by the Ito’s lemma we have
v( N )   2  g  N  
ble
tegra
2
not in

m( x, t )   ( x, t )  M 1  x
2
y(N)
(b)
2
integ
z(N)
(14)
Here f(N) is the growth component whereas the
second term on the right hand side of both equations
(13) and (14) respectively are the probabilistic
(stochastic: continuous and discrete) components. It
must be noted that z t is white noise and z(t), in the
limit of small autocorrelations, is also white noise.
Based on Ito’s lemma, both f(N) and g(N) can be
related to m(N) and v(N) via the diffusion equation
(equation 5).
Given the Ito’s stochastic differential equation
dX (t )  m( X (t ), t )dt   ( X (t ), t )dW (t ) , (15)
suppose X(t) is a solution of (15) and satisfies the
following conditions
(a) m( x, t )  m( y, t )   ( x, t )   ( y, t )  M x  y
teg
rab
le
Nt 1  f ( Nt )   g ( Nt ) zt
Consider the generalized growth equation (1) and let a
fluctuate such that a  a0   z (t ) then the differential
equation corresponding to this fluctuation is given by
dN a0 1b b
 N  K  N b    N 1b  K b  N b  z (t ) . (18)
dt
b

Thus g(N) is given by g ( N )  N 1b K b  N b

and
hence we can compute m(N) and v(N) by using
equations (16) as
m( N ) 
a0 1b b
N K  Nb 
b
v( N )   N
2
2(1 b )
K
b
N

b 2
Calculating w(N), p(N), y(N) and z(N) we have
(17)
3
(19)

 a0 b  s1b  K b  sb  

w( N )  exp 2 
ds 
2 21b 
b
b 2
K

s
N  s






 2a
 exp   02
 b

p( N ) 
K
b
 K
N
 Nb 
2 a0

sb1
ds    K b  N b  b2 2
b
b
 s  
2 a0
5 Generalized Model with Fluctuating
K
The differential equation with fluctuating carrying
capacity K, is given by
(20)
dN a 1b b
 a b 1b
 N  K0  N b  
K 0 N z (t ) .
(23)
dt b
b
By using equation (16) we can find m(N ) and v(N )
b  
2
2
 2 N 21b   K b  N b 
 b 2 N 2(1b )  K b  N b 
2

(21)
as
 
2 a0 b 2 2 1
m( N ) 
We can show from equations (8) to (11), (20) and
(21) that at the N = 0 boundary
p( N )  N

2
2
2 a0
b 2

2
(22)
y ( N )  N 1
 2  2a0
z ( N )  N 1
 2  2a0
Considering the case when b < 0, we note that
w(N) is not integrable when a0   2 2 and integrable
when a0  
2
(24)
  a b 1b 
v( N )  
K0 N 
 b

It can be shown by using equations (8) that

a b s1b K 0b  s b 
w( N )  exp  2 
ds 
as  b 2
 N

(25)
  2 N b  b N b 
 b  0
 exp  2 2b  K 0 
2 
 a K 0 
2 a0
w( N )  N b
a 1b b
N K  Nb 
b

and w( N )  C when b  0 , where C is a constant.
Calculating p (N ) by using equation (9) we
2 . On the other hand p(N) is integrable
obtain
when a0   2 and not integrable when a0   2 2 .
In both cases y(N) and z(N) are not integrable. Thus
the population is naturally repelling when a0   2 2
2
 2N b
p  N 0  exp  2 2 2b
 a  K0
p( N ) 
N 21b 
and attainable when a0   2 2 (see figure 1). The
population will therefore never go extinct when
a0   2 2 (approaching the carrying capacity) but
will be moving towards extinction due to the
stochasticity when a0   2 2 .
The roles are reversed when b > 0, that is the
population will therefore never go extinct when
a0   2 2 but will be moving towards extinction due
 b Nb 
 K0  2  


(26)
2
 b 
where p  N 0   
 and b  0 . For b  0 ,
 a 
K
p( N )  2 , where K is a constant.
N
By using a similar analysis as in section 3 above,
it can be shown that there will be no extinction when
the carrying capacity fluctuates for b < 0. Equation
(23) illustrates this view clearly. As N  0 , the
second term of equation (23) tends to zero (0) faster
than the first term. The stochastic term vanishes faster
than the deterministic term thus pushing the
population away from the boundary and hence a
repelling boundary. Conclusions can be made at only
one equilibrium point, N = 0 since the other
equilibrium point N = K is fluctuating.
When b  0 then
to the stochasticity when a0   2 2 . A similar
analysis will show that the population will never go
extinct in the neighborhood of the carrying capacity.
4
w( N )  constant
p( N )  N 2(1b )
y ( N )  N b 1
conservation biology problems related to invasive
species spread and endangered species modeling.
Given the importance of the definition of the
generalized models in evolutionary algorithms, we
have discussed the concept of probabilistic
uncertainty. We have adopted the methodology by
[10] for analyzing probabilistic uncertainty and its
usefulness in examining parameter variability. In
particular, Ito’s lemma has been adopted in this case.
Analyses of this generalized model shows that for a
fluctuating growth rate, and for models where b  0 ,
at the N  0 boundary, the populations will be
moving away from this boundary when the growth
rate is greater half of the square of the variance in the
fluctuations (increasing towards the carrying
capacity). On the other hand if the growth rate is less
than half of the square of the variance of the
fluctuation then the population will go extinct in finite
time. The roles are reversed when b  0 . At the
carrying capacity boundary fluctuations in the growth
rate does not cause any extinctions.
When the carrying capacity is fluctuating the
N  0 boundary is attainable when b  0.5 and thus
the population will go extinct in finite time.
Otherwise, the population will increase exponentially
to the carrying capacity when b  1 .
Allee Effect occurs when a population goes below
a threshold level, forcing it to go extinct. This occurs
when the intrinsic growth rate decreases as density or
abundance reduces to low levels. There is no denying
that a fluctuating growth rate will influence this
dynamic. We are now considering the relationship
between the variability in the model parameters and
Allee Effect [16].
From our results, it is clear that the type and
strength of the fluctuations also influences the
dynamics of the phenomenon under consideration.
(27)
z ( N )  N b 1
Hence w( N ) is always integrable and p ( N ) is
integrable and therefore the boundary is attainable.
When 0  b  0.5 , the N = 0 boundary is still
attainable and hence the population will be going
extinct in finite time. On the other hand, when
0.5  b  1 then the population will increase
unboundedly.
6 Discussion and Conclusion
Both environmental and demographic noise in
fluctuating environments have been partially
responsible for the decline of many threatened and
endangered species as well as invasive species and for
the overexploitation and collapse of numerous living
resources, including commercial fisheries [3]. We
have shown in this paper that the type of model used
and the population dynamic that is fluctuating can
lead to varied conclusions. How can one therefore be
sure whether the right model has been used especially
in the presence of noise? For example, models based
on differential equations, integro-difference equations
neural nets, metapopulations, cellular automata and
discrete simulations techniques have been use to
predict spatial spread such as biological invasions (see
for example [11], [12] and [13]). Thus, the choice of
the right model is as important as the results of the
simulation.
To do model selection, one needs to derive
appropriate “generalized growth dynamic models”
based upon which the best candidate model for
describing the phenomenon under consideration can
be found. One such model selection technique is
Evolutionary Algorithms. Evolutionary Computations
(EC) have been inspired by analogies to evolutionary
concepts
(Darwinian
evolution)
such
as
recombination, mutation and selection. EC methods
are exceptionally robust and pliable tools for search
and optimization ([14], [15]). Here, a combined EC
and information theoretic approach is used to
orchestrate a competition among a community of
candidate models that successfully evolves prototype
mathematical models for a myriad of ecological and
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Acknowledgements
This work was supported in part by grants from
Eastern Connecticut State University, the NSF (DMS
9973208) to DEB, USDA Hatch and DOE
Computational Biology grants to the University of
Vermont.
6