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S
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•I
I
Advances and Applications in Statistics
Volume 25, Number 1,2011, Pages 31-45
Avai:able online at http://pphmj.comljournals/adas.htm
Published by Pushpa Publishing House, Allahabad, INDIA
ON ULTIMATE EXTINCTION
MEAN BEHAVIOUR
PROBABILITIES
AND
OF SPATIAL PATTERNS
I. C. Kipchirchir
School of Mathematics
University of Nairobi
P. O. Box 30197-00100
Nairobi, Kenya
e-mail: [email protected]
Abstract
-.,,~
The
three
basic
spatial
patterns
of
organisms
are
clustering
op
(overdispersion),
randorrmess and uniformity (underdispersion).
paper, deterministic
"
In this
and stochastic models are employed to describe
the dynamics of the number of individuals in a habitable site called
a unit.
Stochastic
model
discriminates
deterministic model does not, however.xhe
spatial
patterns
is equivalent to the deterministic model. The probabilities
extinction
and ultimate
whereas
mean of stochastic model
mean number of individuals
of ultimate
in a unit are
determined using the stochastic model. The analysis demonstrated that
if spatial pattern is uniform (underdispersed),
certain;
explosion
if spatial
pattern
is clustered
ultimate extinction is
(overdispersed),
ultimate
is certain and if spatial pattern is random, ultimately
it
stabilizes.
© 2011 Pushpa Publishing House
2010 Mathematics Subject Classification: 60G99.
Keywords and phrases: dispersion, spatial pattern, deterministic model, stochastic model,
ultimate extinction probability, ultimate mean.
Received September 6, 2011
1. C.
32
Kipchirchir
1. Introduction
1.1. Dispersion
Dispersion is the description of the pattern of distribution of organisms in
space (Southwood [7]) and is often referred to as spatial distribution. Spatial
distribution is a visual description and not a probability distribution.
Probability distribution models are used to quantify and classify the
dispersion of organisms. They have been widely used in entomological
research to describe the dispersion of insects.
In order to describe dispersion, it is assumed that organisms are confined
to discrete habitable sites called units (sampling units). We further suppose
that a random variable X represents the number of individuals that a unit may
contain and let m
=
E(X)
and v
= var(X).
If the individuals are distributed randomly, then the spatial distribution is
said to be random and the population pattern is also said to be random. If the
individuals are distributed uniformly, then the spatial distribution is said to
be regular and the population pattern is said to be uniform or
underdispersed. If the individuals are distributed in clusters, then the spatial
distribution is said to be contagious and the population pattern is said to be
clustered or overdispersed
or aggregated or clumped or patchy. The
population pattern is often referred to as spatial pattern.
Random spatial pattern is described by Poisson distribution (v
=
m),
uniform spatial pattern is described by the binomial distribution (v < m) and
clustered spatial pattern is described by the negative binomial distribution
(v > m).
In particular, the negative bin~a1
is defined as
_ _ (k
P(X-x)-
+ xx -
having mean m
= kp
distribution with parameters k and p
1J( 1+pP )X( 1+p1)k
,x-O,1,2,
and variance v
+ p).
= m(1
_
.
... ,k>O,p>0(1)
On Ultimate Extinction Probabilities and Mean Behaviour...
33
Anscombe [1] gave statistical analysis of insect counts based on the
negative binomial distribution. The negative binomial parameter k is
considered as a dispersion parameter. Small values of k (k ~ 0) are
associated with overdispersion whereas large values of k (k ~
(0)
are
associated with randomness.
Young and Young [8] reviewed measures of aggregation, namely,
variance to mean ratio, index of clumping, index of mean crowding and
index of patchiness with respect to Poisson and negative binomial
distributions. The four measures of aggregation revealed that decreasing
values of k are associated with increasing measures of aggregation (departure
from randomness).
Kipchirchir [3] demonstrated analytically that the negative binomial
parameter k is a measure of dispersion by analysing equicorrelation matrix in
relation to coefficient of determination, partial correlation and principal
components with respect to k. The analysis demonstrated that small values of
k are associated with overdispersion whereas large values are associated with
randomness.
For fixed m
p ~ 0 or
or
p~
p~
=
kp and reparameterizing
p = -1
P
+p
, then k ~
00
0 implies a random spatial pattern while k ~ 0 or p ~
or
00
1 implies a clustered (an overdispersed) spatial pattern.
1.2. Deterministic model
The deterministic model assumes that each organism reproduces and dies
on a completely predictable basis at c~~ta.:o.trate. Let
Ylt
be the number of
individuals in a unit at time t and suppose that they reproduce at rate A and
die at rate
fl.
The deterministic equation of a tinear birth-death process is
dn,
(.
)
dt = A - fl nt
(2)
with solution
(3)
1. C. Kipchirchir
34
which is the exponential growth model. If 'A
'A > 11,
nt ~
=
11, nt
= no
(constant); if
(explosion) and if 'A < 11, nt ~ 0 (extinction). The
00
deterministic model does not discriminate the various spatial patterns.
Once
shortage
of resources
(environmental
resistance)
preludes
exponential growth, we can deduce the logistic growth by introducing the
limit imposed by environmental resistance represented by K into (2).
Commonly such limit is imposed by exhaustion of either food supplies or
space. The deterministic logistic equation is
(4)
with solution
nt
=
1
+ ye
K
-(A.-J-l)t'
y
= (K -
no)/no·
The limit K is often referred to as the carrying capacity and (K - nt)/ K
(5)
IS
a
proportionate measure of the total resources unutilised (McLeone and
Andrews [4]). If environmental resistance is due to exhaustion of resources,
then we can interprete (K - nt)/ K as a proportionate measure of the total
resources the environment cannot provide.
If
'A = 11 (nt = no),
there is an unstable equilibrium. However, if
nt = K, there is a stable equilibrium around which the number of individuals
in a unit fluctuates; .any perturbations are nullified by opposite forces
proportional to K - nt bringing the....••.•....•
number of individuals in a unit back
-
,'"
towards the equilibrium leveL If 'A 3-~Il, nt ~ K, that is, nt will stabilize at
K (explosion is avoided) whilst if 'A"<"Il,
nt ~
0 (extinction). Thus,
extinction is still inevitable although it is delayed since ('A - 11) (1 - nt / K)
> ('A - 11). In particular, if 'A > 11 and nt < K, then the growth of the
number of individuals in a unit follows a sigmoid curve (logistic curve).
,
On Ultimate Extinction Probabilities
The
growth
microorganisms
the growth
of many
species
and Mean Behaviour...
populations
of animals,
35
plants
and
follows the sigmoid curve, but it must not be assumed that
of these populations
is entirely
equation, for numerous mathematical
represented
by the logistic
equations can produce a sigmoid curve
(McLeone and Andrews [4]).
Introducing
immigration
into (2) allows us to avoid extinction.
immigrants arrive in a unit randomly at rate v. The deterministic
a linear birth-death process with immigration
Suppose
equation of
is
(6)
with solution
(7)
(A> 11), linearly (A
which changes exponentially
=
11) with time towards
(A < 11)with time towards v/(Il- A).
infinity and exponentially
In practice, the number of individuals
in a unit may fluctuate around
nt
or K or
v/(Il- A), so there is need to consider a probability distribution of
number
of individuals
possible realizations,
in a unit so as to capture
the behaviour
moreover, knowledge of the probability
over all
distribution
can
provide insight into the behaviour of various spatial patterns.
Provided
deterministic
that population
numbers
never
model may' enable sufficient
become
biological
approaches simultaneously
deterministic
analysis
is vi-!~~(Renshaw
then
understanding
gained about the system and if at any time population
small, then a stochastic
too small,
numbers
to be
do become
[6]). So pursuing
both
ensures that we do not become trapped either by
simplicity or detailed stochastic analysis.
1.3. Stochastic model
Let
a
X(t) be the number of individuals in a unit at time t and let Pn(t)
I. C. Kipchirchir
36
P(X(t)
=
Pn(t)
=
n), n
= 0, 1, 2, ... be the probability
of t and L;=oPn(t)
is a function
extinction
probability
probability.
the number of individuals
a particular realization
distributional
of X(t).
Then
Po(t)
is the
= 1. In particular,
t and limt--+oo Po (t) is the ultimate extinction
at time
The probability
distribution
{Pn (t) }~=o tells us the likely range of
structure
in a unit at time t. It does not explicitly tell us what
of the process
{X(t)}
looks like, but gives instead the
properties of a large ensemble of such realizations.
in a unit is m(t) = E(X(t))
The mean number of individuals
ultimate mean number of individuals
noting that 'mean behaviour'
in a unit is limt--+oom(t).
by m(t),
represented
and the
It is worth
may be very different
from the behaviour of individual realizations.
The differential-difference
equations of a stochastic
birth-death
process
are
+ IlIPl(t),
Po(t)
=
-Aopo(t)
p~(t)
=
An-lPn-l(t)
n
= 0,
- (An + Iln)Pn(t)
+ Iln+lPn+l(t),
n
where An and Iln are the birth and death rates, respectively
[2]).
They
equations
are
so-called
for the birth-death
individuals
forward
... , (8)
(Medhi [5]; Bhat
differential-difference
If 1,.0 = 0, and if the number
process.
reaches zero at any time, then it remains at zero thereafter
hence zero is an absorbing
and Iln = nil,
process
Kolmogorov's
= 1,2,3,
n
> 1,
(Feller-Arley
state. In the special case when An = nA, n
we have a linear birth-death
process)
n ;;::1, we have a linear pure
and when
mitIi process
"">.,~
An
of
and
>
°
process or linear growth
= nA, n e
°
and
Iln
=
0,
(Yule-Furry process).
If a biological process has beer; developing long enough to ensure that it
has stabilized, then steady-st-ate or stable probabilities
derivation assumes that both population
during the observed time span.
may be obtained. Their
explosion and extinction are unlikely
On Ultimate Extinction Probabilities and Mean Behaviour...
37
Equations (8) represent the transient behaviour of the individuals in a
unit. Setting p~(t)
= 0 and letting Pn = limt~<XlPn(t) (assuming it exists)
denote stable or steady-state probability, (8) reduces to the steady-state
equations for the birth-death process. The steady-state behaviour provides us
with an approximation to that of transient behaviour, for large t, that is,
asymptotic behaviour.
If the initial number of individuals in a unit is no, that is, X(O)
=
no,
then the initial conditions are
n
n
=
=1=
no,
no.
(9)
Let
<Xl
g(s, t)
=
LPn(t)sn
n=O
(10)
be the probability generating function of the sequence {Pn(t)};=o. Then the
extinction probability at time t, poet) is g(O, t). The partial derivatives of
g(s, t) with respect to sand t are, respectively,
ages, t) _ ~
as
- ~npn
n=O
() n-l
t s ,
ages, t) _ ~
'() n
at
- ~Pn t s .
n=O
2. Linear Birth-death
(11)
with 1liti'nigration (Kendall) Process
..•.~
This is a linear growth process with immigration (Medhi [5]; Bhat [2]).
nA + v, n
>
°
where v > 0 is the immigration rate. Since 1..0
=1=
0,
The rates of this process are 'An
=
state. The differential-difference equations (8) become
and Iln
°
= nil,
n
> 1,
is not an absorbing
I.
38
c. Kipchirchir
po(t)
=
-vpo(t) + ).1.Pl(t), n
p~(t)
=
((n -l)A + v)Pn_l(t) - (n(A +).1.) + v)Pn(t)
+ (n + l)).1.Pn+l(t),
= 0,
n
=
(12)
1, 2, 3, ....
Now, from (10) and (11), we obtain the partial differential equation
8g~, t) + ().1.- AS)(S -1) 8g~; t)
=
v(s -l)g(s,
t)
(13)
having auxiliary equations
dt _
T-
().1.-
ds
_
dg(s, t)
AS)(S -1) - v(s -l)g(s, t)·
(14)
On solving (14) using initial conditions (9), we obtain
1- Nt) )v/A.(a.(t) - s(a.(t) + ~(t) _l))no
l-s~(t)
l-sNt)
,
{(
g ()s t =
,
( 1- ~(t)
1 - s~(t)
)v/A.(~(t) + s(l- 2~(t)))no
1- s~(t)
,
(15)
A
= ).1.,
where
(16)
and
(17)
Extinction probability
at time t is
.• ~
(18)
On Ultimate Extinction Probabilities
and Mean Behaviour...
39
and the mean number of individuals at time t is
+ _v_(e(A.-Il)t
n e(A.-Il)t
m(t)
o
=
{
-1),
A. - ~
(19)
no,
vt +
(A. > u}, linearly (A. =~) with time towards
which changes exponentially
(A. < u) with time towards v/(~ - A.).
infinity and exponentially
From (16), (17) and (18), ultimate extinction probability is
(20)
and from (19), ultimate mean number of individuals is
m(t)
lim
= {~ ~ A.'
t~oo
which are independent
(21)
00,
no.
of
2.1. Dispersion and the Kendall process
In particular, if
no
= 0, then (15) reduces to
_ ( 1- P(t)
g(s,t)where
l-sP(t)
)v/A.
(22)
,
P(t) is given by (17) and on reparameterizing
Nt)
.
_
P (t ) -
A.
~
{
(
~
1- e
.>.~
(A.-Il)t)
,A.
!I.
A.t,
:
A.
7=
u,
••
~
=
p(t~),
1+ P t
then
(23)
= ~
and (22) becomes
g(s, t)
=
(1 + p(t) - sp(t)fv/A.
(24)
1. c. Kipchirchir
40
which
IS
the probability generating function of the negative binomial
distribution with parameters k
=
viA and p(t). Extinction probability at
=
(1 - Nt)r/t..,
time tis
po(t)
(25)
where ~(t) is given by (17) and the mean number of individuals at time tis
_V_(eCt..-Il)t
fl
-1)
m(t) = k p(t) = A {
(26)
'
vt,
which changes exponentially (A > u), linearly (A
= fl)
with time towards
infinity and exponentially (A < fl) with time towards v/(fl - A).
The ultimate extinction probability and the ultimate mean number of
individuals are given by (20) and (21), respectively, since they are
independent of
Since k
(k ~
00
=
no.
*'
then A ~ 0 or v ~
or ~(t) ~ 0) and A ~
00,
implies a random spatial pattern
or v ~ 0, implies a clustered (an
00
overdispersed) spatial pattern (k ~ 0 or ~(t) ~ 1).
From (17),
lim ~(t)
A < u,
= {~'
t~OCJ
(27)
1
,
and hence for A ;:::u, ultimate spatial pattern is overdispersed whereas for
A<
fl
and A ~ 0,
lim
t~<p, t..~o
~(t.) = 0
(28)
and hence ultimate spatial pattern is random. In this case, from (20), the
ultimate extinction probability is
On Ultimate Extinction Probabilities
lim
t~CO,A.~O
and Mean Behaviour
)V/A. =e-v/~
Po(t) = lim ( 1--A)V/A. = lim (vi
1---.!::
A.~O
Il
A.~O
viA
...
41
(29)
and from (21), the ultimate mean number of individuals is
m(t)=~.
lim
t~co, A.~O
°
We observe that A ~
(30)
Il
is reminiscent
of a linear death with immigration
process which we discuss in the sequel.
3. Linear Death with Immigration
Letting
linear
A=
death
°
in linear birth-death
with immigration
process
Process
with immigration
and since
process, we obtain
AO"* 0,
°
is not an
absorbing state. Letting A = 0, (14) reduces to
dt _
T-
Il(S
ds
_
dg(s, t)
-1) - v(s -l)g(s, t)
(31)
which on solving using initial conditions (9), we obtain
which is the product
and pOisson(~(l-
of probability
generating
functions
of Bin( no, e-~t)
e-~t)}
Extinction probability
po(t)
=
at time tis
(1.;
e-~~:~~~{-~(1- e-~t)}
(33)
...;.~
and the mean number of individuals is
(34)
which is the sum of the means of Bin( no, e-~t) and pOisson( ~ (1 - e-~t)).
,.
1. C. Kipchirchir
42
The ultimate extinction probability is
lim
Po(t)
= e-v/Il
(35)
t~oo
and the ultimate mean number of individuals is
lim
m(t)
=
v/IJ-
(36)
t~oo
which are independent
3.1. Dispersion
no.
of
and linear death with immigration
In particular, if
no
process
= 0, then (32) reduces to
(37)
which is the probability
generating function of pOisson(~(l-
e-Ilt))
which
describes a random spatial pattern.
at time tis
Extinction probability
(38)
and the mean number of individuals at time t is
(39)
which is a curve with negative curvature and converges to v/IJ-, indicative of
stability. From (34),
lim
t~oo
which is the probability
g(s, t) ~ exp{~ (s - I)}
:
generating
ultimate spatial pattern is random.
IJ-
function of Poissonlv/u)
(40)
and hence the
On Ultimate Extinction Probabilities
The ultimate
individuals
extinction
probability
aad Mean Behaviour...
and the ultimate
43
mean number
are given by (35) and (36) since they are independent
of
of no,
moreover, they confirm (29) and (30), respectively.
Furthermore,
from (32),
lim g(s, t) = (1- e-J.lt + se-J.ltto
v-'}o
which is the probability
generating
the spatial pattern is uniform.
(41)
function of Bin( no, e -J.lt) and hence
In this case, from (35), ultimate
extinction
probability is
lim
po(t) = lim e-v/J.l = 1
t-'}oo, V-'}O
(42)
V-'}O
which implies ultimate extinction is certain and from (36), the ultimate mean
number of individuals is
lim
t-'}oo, V-'}O
m(t) = lim vi/-! = O.
(43)
V-'}O
We observe that v ~ 0 is reminiscent
of a linear death process which
we discuss in the sequel.
4. Dispersion and Linear Death Process
Letting v
=
0 in the linear death with immigration
linear death process
and since
"-0
process, we obtain a
= 0, 0 is an absorbing
state. Letting
v = 0, (31) reduces to
dt
T=
ds
/-!(s -1)
=
dg(s, t)
0
(44)
which on solving using initial conditions (9), we obtain
(45)
which is the probability
generating'function
hence spatial pattern is uniform.
Extinction probability
of Bin( no, e -J.lt), no > 0 and
•. ;:
at time t is
(46)
1. C. K.ipch~chir
44
which implies extinction at time t is not certain and the mean number of
individuals is
(47)
which changes exponentially with time towards zero, moreover,
° is an
absorbing state.
From (46), ultimate extinction probability is
lim Po(t)
=
(48)
1
t~oo
which implies ultimate extinction is certain and from (47), ultimate mean
number of individuals is
lim m(t)
=
t~oo
°
no, moreover, they
which are independent of
(49)
confirm
(42) and (43),
respectively.
5. Conclusions
The mean (19) of the stochastic model (12) is equivalent to the
deterministic model (7). For clustered and random spatial patterns,
absorbing state whereas for a uniform spatial pattern,
°
°
is not an
is an absorbing state.
From stochastic analysis, the ultimate extinction probability is
lim Po(t)
t-oco
clustered (overdispersed),
e-v/~,
random,
1,
uniform (underdispersed)
=
O'
.
{
(50)
and the ultimate mean number of.IDtIividualsis
-;,~
CO,
lim m(t)
t~oo
=
~;
{
clustgred (overdispersed),
random,
).l.
0,
uniform (underdispersed).
(51)
On Ultimate Extinction Probabilities and Mean Behaviour. ..
45
Thus, if spatial pattern is uniform, then ultimate extinction is certain and
if spatial pattern is overdispersed, then ultimate explosion is certain. If spatial
pattern is random, then ultimate extinction is not certain and ultimate mean
stabilizes.
Intuitively, overdispersion overly promotes survival leading to explosion
whereas underdispersion overly inhibits survival leading to extinction and
randomness is a compromise between these two extreme situations.
Acknowledgement
The author acknowledges EAUMP-ISP for their partial support.
References
[1]
F. J. Anscombe, The statistical analysis of insect counts based on the negative
binomial distribution, Biometrics 5 (1949), 165-174.
[2]
B. R. Bhat, Stochastic Models: Analysis and Applications, New Age International
(P) Ltd. Publishers, 2000.
[3]
1. C. Kipchirchir, The negative binomial parameter k as a measure of dispersion,
ICASTOR Journal of Mathematical Sciences 4(2) (2010), 197-207.
[4]
R. R. McLeone and J. G. Andrews, Mathematical Modelling, Butterworth
and Co.
Publishers Ltd., 1976.
[5]
[6]
J. Medhi, Stochastic Processes, New Age International (P) Ltd. Publishers,
E. Renshaw, Modelling Biological Populations
1982.
in Space and Time, Cambridge
University Press, 1991.
[7]
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[8]
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Proceedings
Conference on Applied Statistics ~.Apiculture,
.•..
-.:",~
.. ..
of the Kansas State University
1990, pp. 13-20 .
ADVANCES AND APPLICATIONS IN STATISTICS
Editorial Board
M.I. Ageel
Tae-Sung Kim
King Khalid University
Saudi Arabia
Won Kwang University
South Korea
Olcay Akman
Kuldeep Kumar
illinois State University
USA
Bond University
Australia
Raghunath Arnab
Dejian La~
University of Botswana
Botswana
University of Texas at Houston
USA
Shaul K. Bar-Lev
Tamas Lengyel
University of Haifa
Israel
Occidental College
USA
K. S. Chen
Ming- Ying Leung
National Chin-Yi University of Technology
Taiwan
University of Texas at San Antonio
USA
Antonio Di Crescenzo
Jae-HakLim
Universita di Salerno
Italy
Hanbat National University
South Korea
YixinFang
Lianfen Qian
Georgia State University
USA
Florida Atlantic University
USA
Wai Cheung Ip
Yuan Chung Sheu
The Hong Kong Polytechnic University
Hong Kong
National Chiao Tung University
Taiwan
M. D. Jimenez-Gamero
Isao Shoji
University de Sevilla
Spain
University of Tsukuba
Japan
John Sterling Kao
Karan P. Singh
University of San Francisco
USA
University of North Texas Health Science Center
USA
Jong-Min Kim
Igor Zurbenko
University of Minnesota'
USA
University at Albany
USA
-~~
.'"
l~llmagingEditor
K. K.Azad
Universliy bf Allahabad
India