Ecological Modelling 136 (2001) 237– 254
www.elsevier.com/locate/ecolmodel
Structural uncertainty in stochastic population models:
delayed development in the eastern barred bandicoot,
Perameles gunnii
Charles R. Todd a,*, Pablo Inchausti b, Simone Jenkins c, Mark A. Burgman d,
Meei Pyng Ng e
a
Freshwater Ecology, Arthur Rylah Institute for En6ironmental Research, Department of Natural Resources and En6ironment,
PO Box 137, Heidelberg, Vic. 3084, Australia
b
NERC Centre for Population Biology Imperial College, Silwood Park, Ascot, Berkshire SL5 7PY, UK
c
Department of Biological Sciences, Monash Uni6ersity, Clayton, Vic. 3168, Australia
d
School of Botany, The Uni6ersity of Melbourne, Park6ille, Vic. 3052, Australia
e
Department of Mathematics and Statistics, The Uni6ersity of Melbourne, Park6ille, Vic. 3052, Australia
Received 2 March 2000; received in revised form 26 September 2000; accepted 13 October 2000
Abstract
Uncertainty about which model structure best describes the life history of a species may be a problem for the
development of some population viability analysis (PVA). This paper describes the development and exploration of
two structurally different stochastic population models when there is uncertainty about the life history of a species.
Delayed reproduction was observed in a protected population of the small marsupial Perameles gunnii (eastern barred
bandicoot) at Woodlands Historic Park, Victoria, Australia. This previously undocumented feature of P. gunnii may
be considered to be either a component of the seasonal breeding cycle or it may be delayed development to sexual
maturity. A delayed development model is compared to a standard development model where the parameter estimates
of each model were obtained from a long-term mark-recapture study at Woodlands Historic Park. While the growth
rate of the delayed development model is less than that of the standard model, the predicted risks of extinction/
quasiextinction were higher for the standard model. This discrepancy is the result of different interpretations placed
upon the available data underpinning the two models, the most important of which is the difference between the
estimated variance in survivorship of the sub-adult stage. The results highlight the need for conservation assessments
based on stochastic modelling to explore the degree to which the predicted extinction risk is affected by the
incomplete knowledge of the species’ basic biology and parameter values. © 2001 Elsevier Science B.V. All rights
reserved.
Keywords: Model uncertainty; Stochastic population models; Eastern barred bandicoot; Demographic models; PVA
* Corresponding author. Tel.: + 61-3-94508742; fax: + 61-3-94508730.
E-mail address: [email protected] (C.R. Todd).
0304-3800/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S0304-3800(00)00427-0
238
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
1. Introduction
Small, isolated populations face an uncertain
future. Chance events such as fluctuations in the
vital rates, uneven sex ratios, and inbreeding depression may impact upon these populations and
drive them to extinction (Gilpin and Soulé, 1986).
Assessing the viability of small populations is an
essential prerequisite for determining conservation
priorities, for assigning effort and resources in the
form of conservation actions, and for projecting
and evaluating the results of recovery plans
(Boyce, 1992; Possingham et al., 1993; Hamilton
and Moller, 1995). Population viability analysis
(PVA; reviews in Boyce, 1992; Ralls et al., 1992;
Caughley, 1996; Beissinger and Westphal, 1998) is
a process for assessing the viability of a population using, in part, the projections from a stochastic model that summarises the demographic
information as well as other relevant ecological
information (e.g. territoriality, density dependence, trends in habitat change) about the population in question. Most PVA models incorporate
a deterministic processes (e.g. density dependence)
and explicitly include (one or more of) three main
sources of chance variation in demographic rates:
demographic
stochasticity,
environmental
stochasticity, and spatial variability. In considering these random, natural sources of variation, a
model can provide results that are relevant to
conservation purposes such as the risk of extinction, the chance of population persistence, expected persistence time, and the projected range
of population abundance (Boyce, 1992; Beissinger
and Westphal, 1998).
Models of endangered species are often formulated when there is incomplete field data to estimate the demographic parameters, or insufficient
knowledge of the basic ecology of the species.
Especially in the case of rare and/or endangered
species, the deficiency in the basic demographic
ecological information generally stems from the
rarity or threatened condition that motivates the
viability assessment in the first place. This lack of
information translates into uncertainty about
model structure, its functional forms and/or
parameter estimates and introduces an added
component of uncertainty to model results
(Akçakaya et al., 1997). Pascual and Hilborn
(1995) stressed the need to actively search for and
evaluate several alternative models. However,
only a handful of the 58 PVA studies reviewed by
Groom and Pascual (1998) actually considered or
evaluated alternative model structures.
1.1. Eastern barred bandicoots
Perameles gunnii (eastern barred bandicoot) is a
small marsupial species endemic to south-eastern
Australia. P. gunnii previously occurred in southeastern South Australia, the western basalt plains
of Victoria and most of Tasmania (Seebeck et al.,
1990). However, P. gunnii is now extinct in South
Australia, functionally extinct in the wild but
extant in a number captive and semi-protected
populations in Victoria, and declining in Tasmania (Clark et al., 1995a; Watson and Halley,
1999). Adult P. gunnii are small to medium sized,
omnivorous, ground-dwelling marsupials that
grow to 270 –350mm in length, generally weigh
500 –900g and prefer grassland habitat (Seebeck, 1995). P. gunnii is considered to be critically
endangered in Victoria (Department of Natural
Resources and Environment, 1999) and has been
considered at least endangered for the past 10–15
years (based upon the International Union for
Conservation of Nature (IUCN, 1994, 1996 threat
categories).
The stochastic population models developed for
P. gunnii by Lacy and Clark (1990), Clark et al.
(1995b) both contributed to management decisions about the future direction of the last wild
population of P. gunnii at Hamilton, in southwestern Victoria. Both these models are regularly
cited in management plans for P. gunnii (Backhouse et al., 1994; Watson and Halley 1999). Lacy
and Clark (1990) essentially considered two different models, one with and one without catastrophes. From the results of their simulations, Lacy
and Clark (1990) concluded that the P. gunnii
population at Hamilton faced continued decline
at a rate commensurate to imminent extinction
(extinction likely by the year 2000). Clark et al.
(1995b) also considered two models, one with and
the other without removals. In their model, animals were removed at an average rate per 3
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
months equivalent to the removal of 114 individuals over 2 years to provide stock for the captive
breeding program (see Seebeck, 1990; Backhouse
et al., 1994 for reviews of management plans,
translocations and captive breeding programs relevant to this period). These removals from the
Hamilton population took place in full acknowledgement that they could accelerate the decline of
this population and induce its eventual extinction
(Maguire et al., 1990). The conclusions of Lacy
and Clark (1990), Clark et al. (1995b) about the
predicted viability of P. gunnii should be viewed
with caution since their models used estimates of
demographic rates acquired from short-term
mark-recapture studies. Consequently, these estimates may have reduced statistical reliability and
are not representative of the demographic variability of P. gunnii throughout its geographic
range. In addition, the assessments of Clark et al.
(1995b) did not include density-dependence, a feature that has been shown to affect strongly the
PVA estimates of extinction risk (Ginzburg et al.,
1982; Burgman et al., 1993; Beissinger and Westphal, 1998).
A long term mark-recapture study of the protected P. gunnii population at Woodlands Historic
Park, Victoria (Woodlands) indicated the need to
consider another aspect in the modelling of the
species. Most of the females captured at Woodlands had not been sexually active in the first
6 – 10 months after post-emergent juvenile status
( 9 –11 months of age). This observation of a
significantly longer time taken to the first reproduction event at Woodlands can be described in
at least two ways. Firstly, sexually inactive females may be considered to be sub-adults exhibiting a delay in maturation, or development,
approximately twice as long as previously thought
(Heinsohn, 1966; Brown, 1989; Dufty, 1995; Seebeck, 1995). Secondly, sexually inactive females
may be considered to be adults from 6 months of
age, but in such poor condition that they are
unable to reproduce. The latter may be an example of depressed reproduction that may occur in
late summer, and in times of drought, when
breeding may cease altogether (Backhouse et al.,
1994). Neither Lacy and Clark (1990) nor Clark et
al. (1995b) considered the impact of either a delay
239
in development or a period of depressed reproduction on the viability of P. gunnii at Hamilton.
The viability analysis of the Woodlands’ population requires exploring the impact that delayed
sexual maturation or breeding depression may
have on the estimated risk of decline.
In this paper, we explore the role that uncertainty about model structure and the related
parameter estimates can have on predicting extinction risks for P. gunnii. We introduce a number of structural modifications to the models of
Lacy and Clark (1990), Clark et al. (1995b) to
reflect the observed delay in reproduction of P.
gunnii. We start by developing a standard model
to incorporate the occurrence of density-dependent interactions affecting sub-adult and adult
survival rates and then time varying fecundity
reflecting seasonal change. We then incorporate
delayed development and subsequently include
density-dependence and variable fecundity under
assumptions of delayed development. Demographic parameters for all the models considered
in this paper were estimated using the long-term
mark-recapture study of P. gunnii at Woodlands.
We analyse the influence that relative changes in
the average values of demographic rates have on
the geometric growth rate and the sensitivity of
the quasiextinction risk to the uncertainties in the
initial population size and the initial age distribution for both the standard and delayed development models. This exploration indicates that
uncertainty in model structure and model parameters may yield unreliable or erroneous predictions
of the risk of extinction.
2. Field methods
All field work was conducted in the nature
reserve of Woodlands Historic Park (Woodlands)
(37°39% S, 144°51% E). Approximately one half of
the nature reserve is considered to be optimal
bandicoot habitat (Dufty, 1994a), primarily consisting of open grassy-woodland, with a belt of
open forest dominated by Grey Box (Eucalyptus
microcarpa) in the south-eastern corner. This belt
was chosen for the location for a mark-recapture
study as it seems to be the preferred habitat for P.
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C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
gunnii in the park (Department of Natural Resources and Environment, 1999).
The 2 year mark-recapture live trapping program commenced in September 1994 towards the
end of a drought, which had abated by March
1995 (Australian Bureau of Meteorology,
1998a,b). The average monthly rainfall during the
trapping period (51.7 mm) was higher than the
long-term average monthly rainfall (46.8 mm;
Australian Bureau of Meteorology, 1998a,b). Previous mark-recapture studies of P. gunnii (Brown,
1989; Minta et al., 1990; Dufty, 1991, 1994b)
collected data over shorter periods of time and
consequently their parameter estimates may have
been seriously affected by the prevailing weather
conditions at the time of sampling. The potential
for fluctuations in the weather to distort the data
collected in this study was minimised because
sampling covered the full range of weather conditions (dry and wet periods) during two cycles of
seasonal change.
Bandicoots were live-trapped every 2 months
from September 1994 until July 1996 in two separate grids of 6.25ha. Each grid incorporated 36
trapping stations (6× 6) spaced 50m apart. Trapping sessions were conducted over six consecutive
nights, three nights on each of the two grids using
two small mammal, gravity-fall, cage traps at each
station. Trapped bandicoots were individually
identified from ear tattoos and their sex, body
mass and location of capture amongst other attributes were recorded. The number of lactating/
enlarged nipples and/or pouched young were also
recorded. Records of each individual were kept
over the whole trapping period from which life
history patterns were observed. Individuals were
classified as one of three stage classes: juvenile,
sub-adult or adult.
Two alternative life histories for P. gunnii are
considered in this paper. The standard model for
P. gunnii (Heinsohn, 1966; Brown, 1989; Dufty,
1995; Seebeck, 1995) involving direct development, a three stage model (juvenile, sub-adult and
adult), and a 3-month time step transition between the stages. In the alternative model involving delayed development, the definition of stages
could not be strictly based upon age as in the
standard model. The juvenile stage is treated
equivalently to the standard model covering the
first 3 months of life (see Lacy and Clark, 1990;
Backhouse et al., 1994; Dufty, 1995; Seebeck,
1995). Sub-adults are defined as young that are
independent of their mother but not yet sexually
active. Under this hypothesis, it may take between
6 and 10 months for a sub-adult to make the
transition to adulthood, i.e. becomes sexually active. Adults are defined as independent and sexually mature individuals which was verified by the
examination of an individual’s pouch.
3. Population modelling
Stochastic population models have been developed for a number of small populations over the
past decade (see Beissinger and Westphal, 1998;
Groom and Pascual, 1998 for a recent review).
Several of these models include some form of
density dependence (Price and Kelly, 1994; Kenny
et al., 1995; Maguire et al., 1995; Song, 1996;
Brook and Kikkawa, 1998; Drechsler et al., 1998;
Root, 1998; Carter et al., 1999). The models of
Lacy and Clark (1990), Clark et al. (1995b) were
used as the starting point for the stochastic models of P. gunnii at Woodlands. We consider a
female only model as P. gunnii are considered
polygynous (Coulson, 1990; Dufty, 1994a) and
males are not thought to limit the number of
breeding females (McCarthy et al., 1994). Both
Lacy and Clark (1990), Clark et al. (1995b) chose
a 3 month period as the time step for their
models. The basic pre-breeding census model for
the standard model of P. gunnii is shown in Table
1. Environmental stochasticity in survival was
modelled using the method of Todd and Ng,
(submitted) that allows selecting a random deviate
at each time step from a probability distribution
with defined mean and standard deviation (S.D.)
such that the deviate always lies in the unit interval. Environmental stochasticity in fecundity was
modelled by a distribution bounded at zero with
mean f and S.D. |f. Demographic stochasticity
was incorporated using a binomial distribution to
model the number of individuals surviving between consecutive time steps, and a Poisson distribution to model the number of offspring
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
produced per individual (Akçakaya, 1991; Burgman et al., 1993; McCarthy et al., 1994).
241
1966; Dufty, 1994a) and to obtain shelter. These
density-dependent interactions reflecting contest
competition amongst females may be included in
the standard model by multiplying the adult survival by the density-dependence factor da (the
ratio of total female carrying capacity to the
current number of adult females; Fig. 1, e.g.
McCarthy, 1995; Drechsler et al., 1998). The subadult resource share is likely to be affected by
both the number of female adults occupying
prime territories as well as the number of female
sub-adults. Consequently, sub-adult survival is
further reduced by the density-dependence factor
for sub-adults, ds (the ratio of the total carrying
3.1. The standard model including
density-dependence
Adult bandicoots appear to be territorial (Coulson, 1990; Dufty, 1994a). It was observed that the
female adult generally claimed better quality habitat than the male counterpart and remained relatively sedentary. Furthermore, resources are not
shared evenly as adults often claim the prime
resource sites and sub-adults are forced to the
periphery of quality habitat to forage (Heinsohn,
Table 1
Model equationsa
Transition equations
Characteristic polynomial
Standard model (direct development)
Deterministic
At+1 = s2×At+s1×SAt
SAt+1= s0×f×At
Stochastic including density dependence At+1 = Bin[At, da×s2]+Bin[SAt, ds×s1]
SAt+1 = Poi[s0×f×At ]
Density-dependence parameters
1
if AtBka,
da=
ka/At if At]ka,
ds=
!
!
"
1
ks/(At+SAt )
if (At+SAt )Bks,
if (At+SAt )]ks.
Delay model (delayed development)
Deterministic
"
At+1 = s2×At+k×s1×Sbt
Sbt+1 = (1−k)×s1×Sbt+k×s1×Sat
Sat+1 = (1−k)×s1×Sat+s0×fd×At
Stochastic including density dependence At+1 = Bin[At, dda×s2]+Bin[Sbt, dds×k×s1]
Sbt+1 = Bin[Sbt, dds×(1−k)×s1]+Bin[Sat,
dds×k×s1]
Sat+1 = Bin[Sat,
dds×(1−k)×s1]+Poi[s0×fd×At]
Density-dependence parameters
1
if AtBkad,
dda=
kad/At if At]kad,
dds=
u 2−s2u−s1s0f =0
!
!
1
ksd/(At+Sbt+Sat )
u 3−(s2+2s1(1−k))u 2
+s1(1−k)(2s2+s1(1−k))u
−s 21(s0fdk 2+s2(1−k)2) =0
if (At+Sbt+Sat )Bksd,
if (At+Sbt+Sat )]ksd.
a
Summary of the demographic models built for P. gunnii. At, SAt, Sat and Sbt are the number of female adults, sub-adults,
sub-adults in the first and second pseudo-stages at time t; s0, s1 and s2 are the survival rates of juvenile, sub-adult and adults females;
k is the probability of passage to the next stage (or pseudo-stage); f is the number of juvenile females offspring produced per female
adult.; fd is the mean number of female offspring produced per female adult, accounting for delayed development. The factors da,
ds, dda and dds are density-dependence multipliers that may reduce the proportion of animals surviving to the next time step (see
Fig. 1 and main text for further details).
242
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
Fig. 1. The effects of the density-dependence in use in the
models constructed. In the absence of any stochastic effects the
model predicts population growth provided that the growth
rate is greater than 1. If the population is less than the
carrying capacity then the population will approach the carrying capacity following the trajectory plotted. The trajectory
plotted is for the delay model. The effects of density-dependence on the standard model, while not exactly the same as
shown, are too similar to present on the same plot for the
same growth rate.
capacity to the current number of adult and subadult females; Fig. 1). The standard model with
density dependence (Table 1) assumes the existence of a maximum or ceiling abundance such
that population abundance either increases
asymptotically to the carrying capacity whenever
its abundance is smaller than TCC, or instantly
decreases to TCC whenever the abundance is
greater than TCC (Fig. 1). When environmental
and demographic stochasticity are included into
the standard model, this formulation of densitydependence may allow the population abundance
to rise transitorily above TCC, reflecting exceptionally favourable circumstances of high resource
abundance.
3.2. The delay model including density-dependence
One approach to model delayed development of
P. gunnii would be to create an extra sub-adult
stage. However, adding an extra sub-adult stage
would imply a fixed passage of time or constant
duration of 6 months of the sub-adult stage. The
fixed passage of time does not match the observed
pattern of development of P. gunnii at Woodlands, where the sub-adult transition times were
of at least 6 months under the alternative life
history proposed. In addition, a fixed duration
approach would assume the existence of a stable
age distribution within the sub-adult stage
(Caswell, 1989) that is very unlikely to occur
given the observed temporal variability of demographic rates at Woodlands. To avoid the assumption of a stable age distribution, Caswell
(1989) suggested using the negative binomial distribution to construct two unobservable pseudostages within the sub-adult stage to allow variable
transition times for individuals in the sub-adult
stage (Table 1). Density-dependence in the delay
model was included in the same manner as in the
standard model. Demographic stochasticity and
environmental stochasticity in fecundity and survival rates were modelled in the same manner as
described for the standard model. The pseudostages (Sat and Sbt ; see Table 1) were assumed to
be perfectly and positively correlated because
these pseudo-stages are a model construct introduced to reflect delayed sexual maturation and are
not identifiable in reality.
3.3. Variable fecundity rates-cyclic fecundity
The starting date and the duration of the P.
gunnii breeding period are variable throughout its
geographic range. For instance, while P. gunnii is
considered a winter and spring breeder in Tasmania (Heinsohn, 1966; Reimer and Hindell, 1996),
reproduction occurs all year round but is depressed in late summer and ceases altogether in
times of drought (Backhouse et al., 1994). There
was no evidence of breeding between January and
March 1995 at Woodlands. We found a negative
correlation between the average bimonthly diurnal temperature and the estimated number of
pouched young recorded per female adult (r= −
0.94, Fig. 2) which suggests that breeding
depression occurred in mid to late summer at the
time. We investigated a variant of the standard
and delay models by considering two breeding
periods of high and low fecundity occurring during the cooler autumn/winter (W) months and
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
the warmer spring/summer (S) months, respectively. In this variant model (called cyclic fecundity), the duration of each breeding period was set
equal to 6 months. Since the time step for the
standard and the delay models was of 3 months,
fecundity deviates need to be sampled twice from
the low fecund state before switching to the high
fecund state. Consequently, the initial conditions
of the cyclic fecundity model required specifying
the fecundity state (low or high) from which
fecundity deviates were randomly drawn at the
start of the simulation.
3.4. Parameter estimation
Mark-recapture is widely considered to be the
best method for the estimation of demographic
rates (e.g. parameters required by PVA models,
White and Burnham, 1999). A mark-recapture
study in Woodlands provided 321 capture histories of marked bandicoots of which 221 (101
females and 110 males) were adults, 79 sub-adults
(19 females and 60 males) and 31 juveniles (13
females and 18 males). The means for the fecundity and survival rates for each stage and their
Fig. 2. Fecundity rates for both standard and delayed development plotted with temperature over the capture period. The
short dashed line with solid square is the mean diurnal bimonthly temperature; the solid line with open circles is the
bi-monthly fecundity rate under standard interpretation of the
P. gunnii life history; and the long dashed line with closed
triangles is the bi-monthly fecundity rate under delayed development.
243
associated variances were estimated from the
mark-recapture data for each model (i.e. standard
and delay).
3.4.1. Sur6i6al rates
The capture histories of marked bandicoots
from Woodlands were analysed using MARK
(White, 1999) for the estimation of the survival
rates for both the direct development (standard
model) and the delayed development (delay
model) life histories. The mean and S.D. of the
bimonthly for survival rates for female bandicoots
at Woodlands were obtained from the mark-recapture analysis (Table 2). Since the time step of
both the standard and delay models has been set
to 3 months, the bimonthly estimate of the mean
and S.D. had to be converted to a 3 month
description. This assumes that the probability of
survival remained constant over 3 months.
Temporal variation of survival rates is usually
modelled in most PVA approaches by randomly
selecting deviates from probability distributions
whose means and variances are estimated from
field data (Boyce, 1992, Burgman et al., 1993;
Beissinger and Westphal, 1998). By definition, the
probability distribution describing the environmental variation of survival rates must produce
variates whose value is to be restricted to the
[0–1] interval. Some approaches for restricting
survival rates to the unit interval have the undesirable effect of producing a bias in the original
estimates of the mean and S.D. of the survival
rates. This can lead to an underestimation of the
extinction risk (Todd and Ng, (submitted)). The
method developed by Todd and Ng is implemented in this paper to obtain survival variates
restricted to the unit interval without bias to the
original estimates of the means and S.D. of the
survival rates.
3.4.2. Fecundity
The pouches of captured P. gunnii were examined for evidence of breeding. During lactation,
the nipples of P. gunnii increase in length from
their normal size of 5–10 to 15–30mm. (Heinsohn, 1966). As it takes 3 weeks for a suckled
nipple to revert to its normal size (Heinsohn,
1966), the presence of enlarged nipples was used
as an indicator of recent breeding even when the
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
244
Table 2
Estimated trimonthly mean survival and standard deviation
Standard
Bimonthly estimatea
Trimonthly estimate
Average over time, (v)c
|ˆ =
var(S. i )−var(S. i Si )d
var(Z) =9/4×v×|ˆ 2e
var (Z)
Delay
s0
s1
s2
s0
s1
s2
0.6585b
0.5344
0.6168
0.2697
0.1009
0.3177
0.8190
0.7412
0.7589
0.2908
0.1444
0.3800
0.8367
0.7654
0.8460
0.0500
0.0048
0.0690
0.6543
0.5292
0.6018
0.2673
0.0967
0.3110
0.8665
0.8066
0.8621
0.0445
0.0038
0.0620
0.8309
0.7574
0.8402
0.0547
0.0057
0.0753
a
The mean bimonthly estimate of the survival rate over all the data.
0.65853/2 = 0.5344 =s0.
c
The average of the estimate of the survival rate for each 2-month interval (each capture event).
d
MARK (White, 1999) estimates variance components allowing sampling error to be removed providing an estimate for temporal
variation, see Burnham et al. (1987), Gould and Nichols (1998) for details.
e
Converting the standard deviation (S.D.) to a 3 month estimate using the relation var(Z) = (g%(vy ))2var(Y) (Rao, 1965; Bulmer,
1967; Lawless, 1982), where g(Y)= Y 3/2 (as in point b above), g%(Y)= (3Y 1/2)/2 and (g%(Y))2 =9Y/4.
b
Table 3
Relative cumulative frequency distributions for litter size
Pouched young
0
1
2
3
Meand
S.D.
Direct development (standard model)a
Delayed development (delay model)b
fc
fW
fS
fd
fdW
fdS
0.3918
0.4705
0.9195
1.0
0.6091
0.6426
0.0476
0.1465
0.8571
1.0
0.9744
0.3283
0.7203
0.7797
0.9790
1.0
0.2605
0.4407
0.3487
0.4330
0.9138
1.0
0.6523
0.5205
0.0335
0.1338
0.8550
1.0
0.9888
0.3083
0.6838
0.7510
0.9763
1.0
0.2945
0.4578
a
The number of adult females captured under direct development was 559 over 2 years, 273 in autumn/winter, and 286 in
spring/summer.
b
The number of adult females captured under delayed development was 522 over 2 years, 269 in autumn/winter, and 253 in
spring/summer. The discrepancy between the numbers recorded under direct development and delayed development is attributed to
reassigning some individuals under delayed development as sub-adults and therefore removed from the breeding pool.
c
Under direct development, fecundity is denoted by: f for overall fecundity; fW for autumn/winter fecundity; fS for spring/summer
fecundity. Under delayed development, fecundity is denoted by: fd for overall fecundity; fdW for autumn/winter fecundity; fdS for
spring/summer fecundity.
d
The means and standard deviations (S.D.) were calculated for females producing only female young based upon an even sex
ratio of pouched young.
young are not found inside the pouch. The fecundity for P. gunnii at Woodlands was estimated as
the ratio of the total number of female young to
the total number of female adults, assuming an
even sex ratio of young in the pouch (McCracken,
1990). The estimation of fecundity for the delay
model required accounting for those individuals
still considered to be sub-adults which had the
effect of increasing the mean fecundity rate and
lowering its variation compared to the estimates
of the standard model (Table 3).
The frequency distribution of bimonthly fecundity estimates did not resemble any common
probability distribution (i.e. normal, Poisson, uniform), therefore the temporal variability of fecundity was modelled as follows. Using a similar
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
method to Lacy and Clark (1990), we calculated a
cumulative frequency distribution over the range
of recorded numbers of offspring per adult female
for each model structure considered. The fecundity deviates were then randomly sampled every
time step from these cumulative frequency distributions. The fecundity deviate could only take the
values of the numbers of pouched young observed, i.e. 0, 1, 2, or 3. The fecundity values for
a female only model (i.e. number of female young
produced per female at each time step) were obtained by halving (assuming an even sex ratio) the
fecundity deviates, and their resulting means and
S.D. are shown in Table 2.
3.4.3. Density-dependence parameters
The strength of the density-dependence was
determined by the values of the parameters ka
and ks for the standard model and kad and ksd
for the delay model (Table 1). The parameter
estimates were obtained by solving the deterministic transition equations (Table 1) when the population is at equilibrium and at a stable age
distribution. For example, At + 1 =At (equilibrium) and TCC = A +SA (stable age distribution) the total carrying capacity. Furthermore, if
A + SA ]ks then the deterministic transition
equations including density-dependence reduce to
A = s2A+s1s0 fAks/TCC
which can be rearranged in to the form
ks=
(1− s2)TCC
s0s1 f.
and if A ]ka, then TCC = ka +SA by definition
and SA =ka s0f which implies
ka =
s1TCC
(1+s1 −s2)
Density-dependence in the delay model is modelled similarly to the standard model. The
parameters kad and ksd are the density-dependence scaling factors for delayed development and
their values were found in a similar manner to the
scaling factors for the standard model where
kad=
s1kTCC
,
s1k+ (1− s2)(1+ m)
ksd=
245
TCC
,
s1(1−k +km)
and
m =
s0 fd /(1− s2)
Based upon the size of Woodlands and the habitat available to the bandicoots, it was assumed
that the park could support 600 animals (Seebeck,
personal communication). Further, assuming an
even sex ratio, the total carrying capacity for a
female only model at Woodlands was TCC= 300
where the estimates of the density-dependence
parameters becomes ka= 227.9, ks=291.7,
kad=177.2 and ksd= 314.2.
3.5. Simulations
Quasiextinction risk curves, defined as the
probability that the population falls below a prespecified abundance at least once during the projection (Ginzburg et al., 1982), is a useful
summary of the predicted extreme behaviour of
endangered populations (Ferson et al. 1989; Burgman et al., 1993; McCarthy et al., 1994). The
predicted quasiextinction risks from both the
standard and delay models were compared under
three scenarios: (1) no density-dependence; (2)
TCC = 300; and (3) TCC= 300 and seasonal fluctuations in fecundity (cycling fecundity). The first
scenario represents a reference for comparing the
two alternative life history viewpoints in the absence of any density-dependent effects. The second scenario introduces the best estimate of the
carrying capacity reflecting the limited resources
at Woodlands and the third combines this best
estimate with seasonal variation in fecundity to
simulate the observed changes in fecundity.
A forecast horizon of 10 years (equivalent to 40
3-month transitions) was used and 2500 replications were generated for each simulation. In simulations that included the cycling fecundity model,
fecundity was initially set to the high fecundity
state but switched to low fecundity after the first
time step. Survival rates, within each time step,
were assumed to be perfectly correlated in all
simulations. Correlation between survival rates
could not be estimated and while it was assumed
that the correlations would likely be both large
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
246
and positive this remains unknown. However,
Burgman et al. (1993) state that, for linear dependencies, perfectly correlated survival rates form an
upper bound on the extinction risk. Fecundity
was assumed to be independent of adult survival.
Seebeck and Bowley (1994) estimated the population size at Woodlands to be 500 individuals
in November 1993, this figure was thought to be a
good approximation to the population in mid
1994 (Seebeck, personal communication). Assuming an even sex ratio in the total population, the
initial population size for all models was 250
females, with the population at the associated
stable age distribution (adults – sub-adults, 189 –
61: standard; 150 –100: delay).
4. Results
Elasticity analysis allows comparing the relative
effect that a small change in the model parameters
has on the geometric population growth rate (or
dominant eigenvalue of the transition matrix u;
Caswell, 1989). The (geometric) growth rate for
the standard and delay models were 1.0054 and
0.9845, respectively. The difference is due to the
different model structures adopted, embodied in
the characteristic polynomials for each model
(Table 1). We considered a 9 10% change in the
mean estimates of the fecundity and survival rates
for the standard and delay models (Table 4).
Changes in adult survival produced the single
largest change to the growth rate for both models.
Adult survival, being the most sensitive parameter, is consistent with a number of other studies
(Doak et al., 1994; Escos and Alados, 1994; Heppell et al., 1994; Beissinger, 1995; Wiegand et al.,
1998; Fisher et al., 2000). The growth rate of the
delay model was more sensitive to changes in the
sub-adult survival rate and less sensitive to
changes in the adult survival rate compared to the
growth rate from the standard model.
Table 4
Simple sensitivity analysis of changes to the mean survival rates and fecundity for the deterministic models and associated annual
growth rate (u 4)
Standard model
None
Ds0
+Ds0
−Ds1
+Ds1
−Ds2
+Ds2
−Df
+Df
s0
0.5344
0.4810
0.5879
–
–
–
–
–
s1
0.7412
–
–
0.6671
0.8153
–
–
–
–
s2
0.7654
–
–
–
0.6888
0.8419
–
–
f
0.6091
–
–
–
–
–
0.5482
0.6700
us
1.0054
0.9857
1.0245
0.9857
1.0245
0.9443
1.0679
0.9857
1.0245
%D
0.00
−1.96
1.90
−1.96
1.90
−6.07
6.22
−1.96
1.90
u 4s
1.0216
0.9440
1.1014
0.9440
1.1014
0.7953
1.3004
0.9440
1.1014
s0
s1
s2
fd
ld
%D
l4d
0.5292
0.4763
0.5821
–
–
–
–
–
–
0.8066
–
–
0.7260
0.8873
–
–
–
–
0.7574
–
–
–
–
0.6816
0.8331
–
–
0.6523
–
–
–
–
–
–
0.5871
0.7175
0.9845
0.9688
0.9995
0.9527
1.0165
0.9347
1.0370
0.9688
0.9995
0.00
−1.60
1.52
−3.23
3.25
−5.06
5.33
−1.60
1.52
0.9395
0.8808
0.9980
0.8239
1.0678
0.7633
1.1564
0.8808
0.9980
Delay model
None
−Das0
+Ds0
−Ds1
+Ds1
−Ds2
+Ds2
−Df
+Df
a
Changes in parameters are denoted by −D for −10% change and +D for +10% change.
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
247
Fig. 3. Quasiextinction risk curves generated from the standard and delay models under the three scenarios modelled. The open
symbols represent predicted outcomes from the standard model and closed symbols the delay model. Scenario (1) indicated by
triangles; scenario (2) indicated by squares; and scenario (3) indicated by circles.
The introduction of density-dependence increased the predicted risks of quasiextinction for
both the direct and delayed development life histories (Fig. 3). For example, the likelihood that
the population falls below 40 females at least once
over the 10 year forecast period for the three
scenarios was 58.6, 68.8 and 64.0% for the delay
model, and 71.9, 86.3 and 84.2% for the standard
model. However, the inclusion of density-dependence did not impact on the predicted risks of
extinction (Fig. 3; quasiextinction curves at zero
abundance) to as a great an extent: 3.9, 4.0 and
2.9% for the delay model; and 11.0, 14.4 and
12.8% for the standard model. The inclusion of
cycling fecundity only marginally lowered the influence of including density-dependence on the
quasiextinction predictions in the standard model
(Fig. 3). However, the impact on the delay model
was more marked with cyclic fecundity reducing
the predicted risk of extinction and quasiextinc-
tion in the delay model at low population levels
( B20). For each scenario considered, modelling the occurrence of delayed sexual maturation
lowered the predicted extinction and quasiextinction risks compared to the standard model (Fig.
3), with the decrease being quite distinct for scenarios (2–3). Interestingly, the standard model
has the largest geometric growth rate, and the
growth rate is greater than unity, yet the standard
model also predicts the highest risk of quasiextinction for each scenario considered.
We carried out two sensitivity analyses to assess
the extent to which the extinction risk predicted
by the standard and delay models were affected
by the initial population size and by the initial age
structure. Whilst the initial population size of P.
gunnii at Woodlands was estimated using field
observations, there remains some uncertainty
about the accuracy of this estimate. We considered several values reflecting this uncertainty in
248
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
the estimate of the initial female population size
ranging from 100 to 350 individuals, and with the
initial population at the corresponding stable age
distribution (McCarthy et al., 1994). As expected,
the risk of extinction increases with decreasing
initial population size (Fig. 4a). The predicted
Fig. 4. Sensitivity to different initial population structures.
The open symbols represent predicted outcomes from the
standard model and closed symbols the delay model. The
three scenarios modelled were: scenario (1) indicated by triangles (no density-dependence); scenario (2) indicated by squares
(TCC= 300); and scenario (3) indicated by circles (TCC = 300
and cycling fecundity). (a) The risk of extinction within 10
years plotted against varying the initial total female population size. (b) The risk of extinction within 10 years plotted
against varying the initial female adult population size, with a
total initial population size of 250.
risks of extinction for the three scenarios diverged
with increasing initial population size for the standard model and converged for the delay model.
This suggests that the standard model is more
sensitive to larger initial population estimates
where the delay model was more sensitive to
smaller initial population estimates under the
three scenarios. However, there is some complication to this pattern when the initial population
size compounds with density-dependence (TCC=
300). For example, the standard model with density-dependence exhibits a slight increase in the
risk of extinction once the initial population rises
above the carry capacity. At initial population
levels equal to or below the carrying capacity,
cycling fecundity appears to minimise the predicted risk of extinction relative to the other two
scenarios in the delay model. Compared to the
standard model, the delay model appeared to be
relatively insensitive to the effects of density-dependence when the initial population levels were
below the carrying capacity.
While the total initial population size could be
reasonably estimated, estimates of the initial ageclass distribution for the Woodlands P. gunnii
population in 1994 could not be obtained and
may have consequences for the interpretation of
predictions (Burgman et al., 1994). Analysis of the
mark-recapture data at each capture event indicated a ratio of at least 3:1 adults to sub-adults in
the first half of the study and an increase in this
ratio thereafter. A range of initial adult: sub-adult
population distributions totalling 250 individuals
were considered to test the sensitivity of the models to the initial age-class distribution. As the
initial adult population size increased, the predicted risk of extinction decreased in the standard
model (Fig. 4b). However, the delay model was
relatively insensitive to uncertainties in the initial
age-class distributions as the predicted risks of
extinction were similar for all age-class distribution. Again, the cycling fecundity scenario generated predictions lower than the other two
scenarios for the delay model. Around the stable
age distributionfor both models (189 adults, standard; 150 adults, delay) the predicted risks of
extinction behave consistently.
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
Fig. 5. The trajectories of the mean population size over 10 years
generated from both the standard and delay models under the
three scenarios modelled. The open symbols represent predicted
outcomes from the standard model and closed symbols the delay
model. Scenario (1) indicated by triangles; scenario (2) indicated
by squares; and scenario (3) indicated by circles.
5. Discussion
We found that both model structure and the
actual estimates of model parameters had a large
influence in the outcomes of the analysis undertaken
of P. gunnii Woodlands. The delay model has a
lower growth rate than the standard model and yet
when stochasticity is included the delay model
generally predicts lower risks of quasiextinction.
Both models are dependent upon, and sensitive to,
the accurate estimation of the initial size of the
population and of the initial age-class distribution.
However, the sensitivity analysis indicated that
assuming the initial population abundance to be 250
individuals did not significantly effect the predicted
risks of extinction risk (Fig. 4a). Furthermore, the
assumption that this initial population abundance
was equal to the corresponding stable age distribution for both models seemed reasonable as this
assumption also did not significantly affect the
predicted extinction risk (Fig. 4b).
Density-dependence affected the quasiextinction
risk predicted by the standard and delay models in
a similar manner, however the inclusion of cyclic
fecundity moderated the effects of limited resources
and hence attenuated the impact of density-depen-
249
dence on the extinction risk. In general, the delay
model was less sensitive to the inclusion of densitydependence than the standard model.
The inclusion of variation and random processes
highlight the importance of modelling naturally
occurring variation in the demographic rates (compare the predicted risk from the delay and standard
models with no density-dependence in Fig. 3) even
when the mean population size increases (Fig. 5).
The inclusion of limited resources (density-dependence) placed further restrictions on the standard
model so that not only do parameters vary and risks
increase but also population growth was bounded
to the extent that the mean projected population size
also declined over the forecast period (Fig. 5).
The relative variability (as measured by the
coefficient of variation) for most other demographic
rates were similar for both the standard and delay
models, with the exception of sub-adult survival
whose variability was markedly higher for the
standard model (Table 5). The latter could explain
the difference in predicted risk of quasiextinction
between the two models, even in the absence of
density-dependence (Fig. 3). The data analysed
under assumptions of a delayed development life
history produced an increase in the estimate of the
mean and a decrease in the estimate of the variance
in sub-adult survival. These estimate of the mean
and variance were based on three successive capture
events, rather than only one capture event as in the
direct development analysis.
The most noticeable differences amongst previous estimates of P. gunnii demographic rates and
the estimates developed in this paper for both life
histories were for mean sub-adult survival, variation in juvenile survival and both mean and variation in fecundity (Table 5). The estimates of mean
sub-adult survival obtained by Lacy and Clark
(1990) and Clark et al. (1995b) were much lower
than the estimates developed in this paper (Table
5). The parameter estimates obtained by Clark et
al. (1995b) are open to interpretation. Clark et al.
(1995b) based their estimation of survival rates on
the study of Lacy and Clark (1990) study, except
their sub-adult survival rate (0.56) differed from
that of Lacy and Clark (1990; 0.315). Lacy and
Clark (1990) collapsed two sub-adult stages together and multiplied the associated survival rates
to obtain their estimate of average sub-adult sur-
0.3110
0.0620
0.0753
0.5205
0.3083
0.4578
–
–
58.76
7.69
9.94
79.79
31.18
155.45
–
–
Mean
S.D.
0.0408
0.0174
0.1050
0.3808b
–
–
–
–
0.5
0.315
0.75
1.1a
–
–
–
60, 150, 300
59.45
51.27
9.02
105.50
33.69
169.17
–
–
C.V. (%)
0.5292
0.8066
0.7574
0.6523
0.9888
0.2945
0.9520
300
0.3177
0.3800
0.0690
0.6426
0.3283
0.4407
–
–
S.D.
0.5344
0.7412
0.7654
0.6091
0.9744
0.2605
–
300
Mean
Lacy and Clark (1990)
C.V. (%)
Delay
S.D.
Standard
Mean
8
5.5
14
35
–
–
–
–
C.V. (%)
S.D.
0.5
0.56
0.75
1.1
–
–
–
–
0.0750
0.0840
0.1125
0.475
–
–
–
–
Clark et al. (1995)
Mean
15
15
15
43
–
–
–
–
C.V. (%)
b
Lacy and Clark (1990) stated the fecundity to be 2.2 young per female which equates to 1.1 female young per female adult, assuming an even sex ratio.
The S.D. for fecundity was calculated using the distribution for litter size from Lacy and Clark (1990): VAR = {sum {rel. freq. (i )*[(X(i )−mean)2]}. The coefficient
of variation (C.V.) was calculated then multiplied by 1.1 to produce the S.D. for female young per female adult. The S.D. for Clark et al. (1995b) was calculated
similarly.
a
s0
s1
s2
f
fW
fS
k
TCC
Parameters
Table 5
Mean vital rates and standard deviations (S.D.) for past and present models
250
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
vival. Clark et al. (1995b) appeared to have calculated the sub-adult survival by taking the arithmetic
average of the survival rates of the two sub-adult
stages initially defined by Lacy and Clark (1990).
We think that Lacy and Clark’s (1990) estimate of
mean sub-adult survival clearly reflects the belief
that the sub-adult population suffered a greater
mortality under the conditions that were found in
Hamilton at the time (Seebeck et al., 1990 for a
description). However, a more suitable study over
a longer period of time may have revealed a different
estimate. This illustrates a difficulty that confounds
the management of rare and/or endangered species:
the need to act often conflicts with the need to gain
greater comprehension of the species life history.
The estimates of mean adult survival in this paper
were very similar to the Lacy and Clark (1990)
estimates, but the temporal variability was one half
of that estimated by Lacy and Clark (1990); 14%)
and Clark et al. (1995b; 15%). Estimates of mean
juvenile survival rates for both possible life histories
were similar but not quite as low as the Lacy and
Clark (1990) estimate of 0.5 based on best guess.
The large coefficient of variation of juvenile survival
(Table 5) may reflect the paucity of juvenile capture
histories since only 13 records of first capture being
a female juvenile. It is unclear whether the disparity
between the estimates of mean and variance of the
demographic rates between the Hamilton and
Woodlands studies reflect a real difference between
the environments at each location or whether it
could be attributed to the difference in the length
of the mark-recapture studies made at each location.
It is likely that Lacy and Clark (1990) and Clark
et al. (1995b) underestimated the magnitude of the
temporal variability in Hamilton due to data collected over too short a period which Lacy and Clark
(1990) acknowledge as a possibility.
Data collected over the 24 months of the mark-recapture study reveal that fecundity changes
throughout the year (Fig. 2) and that the overall
mean female fecundity rate is 0.61 or 0.65 (direct
or delayed development) with coefficients of variation 80 and 105%, respectively (Table 5). However,
Lacy and Clark (1990), Clark et al. (1995b) estimate
a female fecundity rate to be 1.1, with a coefficient
of variation of 35 and 43%, respectively (Table 5).
The outcomes of the model are not highly sensitive
251
to fecundity, but differences of this magnitude have
the potential to change qualitatively the assessment
of the relative values of management alternatives.
The comparisons between the previous models
developed for P. gunnii and the models developed
for this paper reveal that if the previous estimates
of the survival rates were appropriate then the
models may well underestimate the risk of extinction
based upon the estimate of fecundity alone. The data
collected for this paper came from animals monitored in a protected park and the estimates of mean
survival of all stages are expected to reflect this.
However, fecundity is not expected to change greatly
from population to population, and is expected to
be independent of survival.
Breeding depression (Backhouse et al., 1994) has
been documented in at least one other congeneric
bandicoot species (Perameles nasuta) that has highly
seasonal reproduction (Scott et al., 1999). Reimer
and Hindell (1996) pointed out that seasonal stress
as indicated by low body condition could occur in
summer when the ground becomes harder and more
difficult to dig for food. Reimer and Hindell (1996)
suggested that P. gunnii must increase the energy
intake to meet the increased energy demand during
lactation. If so, P. gunnii sub-adults born in late
winter or early spring may not be established enough
in late spring as to exact the necessary food sources
that enable the young adults to develop the required
body condition to lactate in early summer, particularly during periods of drought as in late 1994 and
early 1995. Most of the individuals considered to
be sub-adults exhibiting delayed reproduction were
caught at times of drought in late 1994 and early
1995. These animals may well have been adults of
poor body condition unable to sustain the energy
requirements for lactation. If this were the case then
it needs to be acknowledged that breeding depression may occur at any time of the year depending
on the current environmental conditions. The bimonthly fecundity is lower in drought affected
spring 1994 compared to the bimonthly fecundity
measured (Fig. 2) when above average rains occurred in spring 1995 following by above average
autumnal rains (Australian Bureau of Meteorology,
1998a,b). This would suggest that the season normally associated with breeding and individual
growth can actually be highly variable and may also
be subjected to the effects of breeding depression.
252
C.R. Todd et al. / Ecological Modelling 136 (2001) 237–254
The models developed in this paper predict the
P. gunnii population at Woodlands to be in decline
and at an appreciable risk of extinction. Usually
structural uncertainty is dealt with through a
choice of models. The standard model with cycling
fecundity, adequately describing breeding depression as well as delayed reproduction in adults
breeding for the first time (given the argument
above), suitably models the P. gunnii population
at Woodlands. However, the delay model with
cycling fecundity, adequately describing breeding
depression and delayed sexual maturation, also
suitably models the Woodlands population. The
choice comes down to whether delayed development is more likely to represent actual population
dynamics than is the standard or direct development life history. It has generally been accepted
that P. gunnii exhibits the developmental traits of
the standard model in all other documented circumstances (Heinsohn, 1966; Brown, 1989; Dufty,
1995; Seebeck, 1995). However, the observation of
delayed reproduction in animals considered to be
old enough to reproduce and the reason why this
occurs makes the choice of model difficult. Further research is required to establish the cause for
delayed reproduction, whether it is developmental
or whether it is environmental (an expression of
breeding depression). Until there is clarification on
which life history should be adopted to describe
the P. gunnii population at Woodlands both the
standard model with cycling fecundity and the
delay model with cycling fecundity should be used
to undertake any further population viability analysis on this population. One possible approach
would be to take the conservative position. That
is, given a series of biologically plausible model
structures that predict similar or comparable
quasiextinction risks, then one should err on the
side of caution by selecting the model that predicts
the highest risk of extinction/quasiextinction but
always refer to the estimated quasiextinction risks
obtained by other models considered.
ing of management options for rare and/or endangered species particularly in circumstances of
incomplete data or lack of ecological knowledge
(Boyce, 1992; Burgman et al., 1993; Hamilton and
Moller, 1995; Beissinger and Westphal 1998).
However, uncertainty about model structure and
model parameters may yield unreliable or erroneous estimates of the risk of extinction (McCarthy et al., 1994; Beissinger and Westphal,
1998). The assumptions about the structure and
the input parameters of these models should be
carefully explored prior to the use of their results
as a basis for decision making. The increasingly
frequent use of stochastic population models to
assess the viability of natural populations should
necessitate the assessment of the sensitivity of the
outcomes of population models for both their
uncertain structure and parameter values.
6. Conclusions
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Acknowledgements
The authors gratefully acknowledge John Seebeck and Alan Lill for their contributions to this
manuscript. The authors wish to further acknowledge the comments from or the assistance of Reşit
Akçakaya, the Australian Bureau of Meteorology
(Melbourne), Andrew Bearlin, Hal Caswell, Martin Drechsler, Scott Ferson, Ben Kefford, Mick
McCarthy, Simon Nicol, Alan Robley, and Gary
White. A substantial body of this manuscript was
completed whilst C.R. Todd was undertaking a
Ph.D. in the School of Forestry and Resource
Conservation, at The University of Melbourne.
This work was supported by a Melbourne Research Scholarship and a contribution by the Department of Natural Resources and Environment
to C.R. Todd. C.R. Todd developed part of this
manuscript whilst visiting the Department of Ecology and Evolution, State University of New York,
Stony Brook, New York and is grateful to Professor Jim Rohlf for providing him with this opportunity.
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