Ecological Modelling 248 (2013) 71–79
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Ecological Modelling
journal homepage: www.elsevier.com/locate/ecolmodel
Demographic analysis of sperm whales using matrix population models
Ross A. Chiquet a,∗ , Baoling Ma b , Azmy S. Ackleh a , Nabendu Pal a , Natalia Sidorovskaia c
a
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USA
c
Department of Physics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA
b
a r t i c l e
i n f o
Article history:
Received 19 February 2012
Received in revised form 30 July 2012
Accepted 30 September 2012
Available online 10 November 2012
Keywords:
Sperm whale
Matrix models
Sensitivity and elasticity analysis
Asymptotic growth rate
Extinction probability
a b s t r a c t
The focus of this study is to investigate the demographic and sensitivity/elasticity analysis of the endangered sperm whale population. First, a matrix population model corresponding to a general sperm whale
life cycle is presented. The values of the parameters in the model are then estimated. The population’s
asymptotic growth rate , life expectancy and net reproduction number are calculated. Extinction time
probability distribution is also studied. The results show that the sperm whale population grows slowly
and is potentially very fragile. The asymptotic growth rate is most sensitive to the survivorship rates,
especially to survivorship rate of mature females, and less so to maturity rates. Our results also indicate
that these survivorship rates are very delicate, and a slight decrease could result in an asymptotic growth
rate below one, i.e., a declining population, leading to extinction.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
The sperm whale, Physeter macrocephalus, is the largest odontocete, or toothed whale (Rice, 1989), and can be found throughout
the world’s oceans, gulfs, and seas (Rice, 1989; Whitehead, 2002).
The studies of sperm whale populations in the literature are dominated by the studies of the geographic structure of sperm whale
populations, which in recent years have mainly been through
genetic techniques and assessment of regional and global population estimates of sperm whales. The genetic studies, including
(Lyrholm et al., 1999; Lyrholm and Gyllensten, 1998; Richard et al.,
1996) and others, have used mitochondrial DNA, which is DNA
inherited from the mother, and also nuclear microsatellite. Samples for these genetic studies can come from commercial catches,
bycatches, historical artifacts, and strandings, just to name a few
(Whitehead, 2003). The genetic analyses of Lyrholm and Gyllensten
(1998) suggest that present-day sperm whale populations have
very low mitochondrial DNA diversity and show little geographical
differentiation over their global range.
In the assessment of regional and global population estimations,
scientist use different techniques, such as catch-per-unit-effort
analyses, length-specific techniques, mark-recapture techniques,
acoustic data (Ackleh et al., 2012), and ship or aerial surveys
(Whitehead, 2003). These techniques were used to study populations of sperm whales in the Gulf of Mexico (Fulling et al., 2003;
∗ Corresponding author. Tel.: +1 337 482 5290.
E-mail address: [email protected] (R.A. Chiquet).
0304-3800/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ecolmodel.2012.09.023
Waring et al., 2009a,b), in the southern Australian waters (Evans
and Hindell, 2004), in the Eastern Caribbean Sea (Gero et al.,
2007), in the Northeastern (Barlow and Taylor, 2005) and Southern (Whitehead and Rendell, 2004) Pacific, in the Northern Atlantic
(Waring et al., 2009a,b), in the Mediterranean Sea (Gannier et al.,
2002) and in other waters around the world. These techniques,
along with others, were also used by Whitehead (2002) to get estimates of the global population size and historical trajectory for
sperm whales.
Even with all this research dedicated to sperm whales, very little
is known about the sperm whale’s population dynamics. There is a
definite lack of research dedicated to the study of stage-structured
population models applied to sperm whales. This is probably due
to the lack of reliable estimates for the vital rates of sperm whales.
Evans and Hindell (2004) suggest that techniques used to study
smaller cetaceans, such as the bottlenose dolphins and orcas, and
large baleen whales, such as the humpback and bowhead whales,
are harder to apply to sperm whales. There are very few articles in
the literature in which vital parameters are given for sperm whales,
much less age-specific vital parameters needed to develop stagestructured models used to study these populations.
In this paper, we develop a stage-structured matrix population model for sperm whales. The parameters for our model were
obtained from the limited information in the literature or by
construction using other vital rates as in Doak et al. (2006) and techniques from Caswell (2001). We present the best and worst cases for
these vital rates in terms of survivorship rates, along with the values that were used in the analysis of the population dynamics. We
use our model and these parameters to construct the asymptotic
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R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
Table 1
Vital rates values.
Fig. 1. Life cycle graph for female sperm whales with (right) and without (left)
the consideration of death stage. Numbers represent different stages. 1: calf; 2:
immature (juvenile); 3: mature; 4: mother; 5: post breeding; 6: mortality.
growth rate for the population. We also calculate other deterministic values for the population such as life expectancy, lifetime
reproduction number, and the inherent net reproduction number
R0 . We then perform sensitivity and elasticity analysis to each of the
vital rates to see which ones affect these values the most. Also, we
calculate the extinction time and distribution of sperm whales from
the matrix population model, treating transitions and reproduction
as independent events.
2. Stage-structured model and parameter estimations
In Section 2.1, we develop a stage-structured matrix model to
describe the dynamics of the female sperm whale. The female population is divided into stages similar to those used by Fujiwara and
Caswell (2001) for the North Atlantic right whale. In Section 2.2,
we use techniques from Caswell (2001) and Doak et al. (2006) to
construct vital rates for the female sperm whale. We will use these
parameters in our model to study the dynamics of the population.
2.1. Stage-structured population matrix model
The life cycle of the female sperm whale is typical of most large
whales. After calves are born, they are suckled by their mother
for about 2 years (Best et al., 1984) and reach maturity at the age
of 9 (Doak et al., 2006). The interbirth interval for sperm whales
differs from 3–5 years (Boyd et al., 1999; Doak et al., 2006) to
4–6 years (Best et al., 1984; Rice, 1989; Whitehead, 2003) which
includes a gestation period of 14–16 months (Evans and Hindell,
2004). Thus, we divide the female sperm whale population into
five stages: calves (stage 1), juveniles/immature (stage 2), mature
females (stage 3), mothers (stage 4) and post breeding females
(stage 5). The life cycle for the female sperm whales is given in
Fig. 1.
A mature female transits from stage 3 to stage 4 each time she
produces a calf, and takes care of the calf throughout stage 4, thus
the time taken for a female on stage 4 is the same as that for a calf
to grow into an immature, which again is about 2 years for sperm
whale (Best et al., 1984). After this 2 year period, the female sperm
whale then goes into stage 5 where after the interbirth interval,
the female can return to stage 3 to give birth to another calf. If the
female is no longer able to reproduce due to age or other natural causes, the female will remain in stage 3. For our model and
throughout the rest of the paper, we will take our time unit to be
one year. Let Pi = i (1 − i ) and Gi = i i , where i is survivor probability of stage i and i is the probability of an individual in stage i to
move onto stage i + 1 for i = 1, · · · , 4. The transition probability from
stage 5 to stage 3 is given by 5 . Therefore, Pi gives the probability
of surviving and staying in stage i, while Gi gives the probability
of surviving and moving to stage i + 1 for i = 1, · · · , 4. G5 gives the
Vital rates
Worst case
Estimated values
Best case
1
2
3
4
5
1
2
3
4
5
0.8841
0.8841
0.9390
0.9390
0.9390
0.4783
0.1085
0.2198
0.4934
0.4934
0.9070
0.9424
0.9777
0.9777
0.9777
0.4732
0.1151
0.2586
0.4920
0.4920
0.9850
0.9850
0.9800
0.9800
0.9800
0.4888
0.1244
0.3436
0.4875
0.4875
probability of surviving and moving to stage 3 from stage 5. From
Caswell (2001) and Doak et al. (2006), we define
i =
(i /)Ti − (i /)Ti −1
(i /)Ti − 1
,
(1)
where Ti is the duration of stage i and is the population’s asymptotic growth rate, i.e., the growth rate of the population at the stable
stage distribution. Since i is actually used in the calculation of ,
we first set to one. Then, as described in Caswell (2001), we use
an iterative process to get better estimations of i . With = 1, we
calculate the projection matrix for our model. The eigenvalues of
this projection matrix will yield a second estimation for , which
we use to estimate i again. We then repeat this process until we
get a projection matrix whose entries are compatible with its own
eigenvalues. This is how we obtain the estimated values for i in
Table 1. Now, we define the fertility number as
√
b1 = 0.53 3 4 ,
(2)
which depends on the mature female survivor probability, the
probability of giving birth after survival, and the survivor probability of the mother caring for the calf (Caswell, 2009). The 0.5
comes from the assumption that the sex ratio is about equal at
birth (Whitehead, 2003). Thus, we obtain the following model corresponding to the life cycle shown in Fig. 1(left):
n(t + 1) = An(t),
(3)
where n(t) is a vector representing the population of female sperm
whales at each stage. The projection matrix A is given by
⎛P
1
0
b1
0
0
⎞
⎜G P 0 0 0 ⎟
⎜ 1 2
⎟
⎜
⎟
⎟
A=⎜
P
0
G
0
G
5 ⎟.
2
3
⎜
⎜
⎟
⎝ 0 0 G3 P4 0 ⎠
0
0
0
G4
(4)
P5
2.2. Estimating model parameters
Despite the small number of references for vital rates of sperm
whales in the literature, we estimate the birth, mortality, and transition rates for model (3) and (4). Sperm whale mortality is the least
known aspect of a sperm whale’s life history (Whitehead, 2003).
In 1982, International Whaling Commission’s Scientific Committee
estimated the annual mortalities of 0.055 for female sperm whales
and 0.093 for infants. However, according to the experts, these estimates, for the infants especially, were based on “extremely shaky
evidence” (Whitehead, 2003). Whitehead (2001) provides an estimate of 0.021 for the annual mortality rate of female and immature
sperm whales in the eastern tropical Pacific. Doak et al. (2006) give
a minimum of 0.02 and a maximum of 0.061 for the annual adult
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
mortality of sperm whales in general. Evans and Hindell (2004)
provide estimated annual mortality rates for female sperm whales
in the southern Australian waters. The average annual mortality
rate for those sperm whales is about 0.0223.
For the analysis of model (3) and (4), we consider the range of the
annual female sperm whale mortality to be 0.02–0.061, and thus we
have the range of the survivor rates to be 0.939 ≤ 3 ≤ 0.98. We use
the value of 0.0223 we calculated from Evans and Hindell (2004) in
our initial analysis for the mortality rate or 0.977 for the survivor
rate. We assume that the mortality rate of the post breeding whales
and of the mothers are the same as the adult mortality rate, i.e.,
3 = 4 = 5 . The mortality of juveniles and immatures is assumed
to be 0.5–2 times the mortality rate of adults. Thus 0.8841 ≤ 1 ,
2 ≤ 0.9850. For our initial calculations, we use the value of 0.907
from the International Whaling Commission.
Recall from Section 2.1, the interbirth interval for sperm whales
differs from 3–5 years (Boyd et al., 1999; Doak et al., 2006) to 4–6
years (Best et al., 1984; Rice, 1989; Whitehead, 2003). In our analysis, we use 4 years as the interbirth interval. As in Doak et al. (2006),
we take the annual fecundity rate to be one half the reciprocal of
the interbirth interval, where the one half is to account for only the
female sperm whales. Thus using (2), the value of b1 in our model
will be 0.125. For the transition rates i for i = 1, · · · , 5, we use Eqs.
(1) and (2) with the assumptions from Section 2.1 that a sperm
whale remains a juvenile until age 2 (Best et al., 1984) and reaches
maturity at the age of 9 (Doak et al., 2006). Therefore, the ranges of
the vital rates for model (3) and (4) are given in Table 1. The best
case and worst case parameters are in terms of the survivorship
rates.
Based on the estimated values in Table 1, the projection matrix
(4) is estimated to be
⎛
0.4778
0
0.1250
0
0
⎞
⎜ 0.4292 0.8339
⎟
0
0
0
⎜
⎟
⎜
A=⎜
0
0.1085 0.7249
0
0.4810 ⎟
⎟.
⎝ 0
⎠
0
0.2528 0.4967
0
0
0
0
0.4810
(5)
0.4967
The dominant eigenvalue of A gives the population’s asymptotic
growth rate . The corresponding right eigenvector provides the
stable stage distribution (i.e., the proportion of individuals of each
stage within the population). Once the stable stage distribution has
been reached, the population undergoes exponential growth at rate
. Throughout this paper, we analyze the population matrix model
(3) with the projection matrix A as given in (5).
3. Model analysis
In this section, we first calculate the fundamental matrix for
the sperm whale in order to perform demographic analysis on
the model (3) and (4) with the estimated parameters presented in
Table 1. We calculate the life expectancy, the asymptotic growth
rate , and inherent reproductive number R0 for the population. Lineage extinction time of sperm whales is then calculated
by transforming the projection matrix A, as given in (5), into a
multi-type branching process (Caswell, 2001). We also obtain the
graph of the probability distribution of extinction time assuming
demographic stochasticity. We perform sensitivity and/or elasticity
analysis using techniques similar to Caswell (2009) to help identify
the life-history stage that contributes the most to the deterministic
values found for the population.
3.1. Fundamental matrix and lifetime reproduction number
The fundamental matrix is one of the primary tools used
in the analysis of population models. The entries of the
73
fundamental matrix give the mean number of visits by an individual
to any transient state. The first step in calculating the fundamental
matrix for sperm whales is to decompose the projection matrix A
as given in (5), into A = T + F, where
⎛
0.4778
0
0
0
⎞
0
⎜ 0.4292 0.8339
⎟
0
0
0
⎜
⎟
⎜
T =⎜
0
0.1085 0.7249
0
0.4810 ⎟
⎟,
⎝ 0
⎠
0
0.2528 0.4967
0
0
0
0
0 0
0.1250
0
0
0
0
0
0
0
0
0
0.4810
(6)
0.4967
and
⎛
⎜0 0
⎜
F = ⎜0 0
⎜
⎝0 0
0 0
0
⎞
0⎟
⎟
0⎟
⎟.
(7)
0⎠
0
Here, T in (6), describes the individual transition probability and F
in (7), the individual fertility number. The fundamental matrix is
then defined by N = (I − T)−1 , with I being the identity matrix. Thus,
we get
⎛
1.9149
0
⎜ 4.9482
⎜
6.0203
⎝ 6.0980
5.8279
0
0
0
0
22.6203
20.6603
7.4193
11.3631
12.3653
7.0906
10.8596
11.8175
N=⎜
⎜ 12.1393 14.7694
0
0
⎞
⎟
⎟
21.6181 ⎟
⎟,
10.8596 ⎠
(8)
12.3653
where N(i, j) gives the expected number of visits over a life time to
stage i from an individual starting at stage j.
The first column represents the female calf. On average, a female
calf spends about 1.9 years as a calf, about 12.14 years as a mature
female, 6.10 years as a mother, which means a calf is expected
to give birth 6.10 times on average. The value 6.10 is also known
as the expected lifetime reproduction number from the calf stage.
The third column corresponds to mature females, so N(4, 3) ≈ 11.36
indicates that a mature female is expected to give birth 11.36 times,
on average. This number is higher than 6.10 because of the mortality
likelihood in transiting from calf to mature female.
Fig. 2(left) indicates how the expected lifetime reproduction
number N(4, 1) from the calf stage varies in response to changes
in the vital rates (Caswell, 2009). It shows that the number of
reproduction events is most elastic to the mature female survivor
probability ( 3 ) and very elastic to survivor probabilities of the
mothers ( 4 ), post-breeding adult females ( 5 ), and the juveniles
( 2 ). Specifically, a 1% increase in 3 ( 4 or 5 ) results in approximately a 23% (12% or 10%, respectively) increase in the breeding
number. Surprisingly, the probability of giving birth by an adult
female 3 has no major influence on the expected reproduction
number. The probability of transiting from mother to post-breeding
female 4 has a negative effect on the reproduction event. This can
be explained since the larger the value of 4 is, the shorter time
the calf is taken care by the mother, thus the less healthy or not
fully developed the calf is, resulting in a smaller lifetime reproduction number for the calf. The number of expected reproductions
from the calf stage is a random variable, and from the fundamental matrix, we only get the mean value N(4, 1). To explore more of
the individual stochasticity, we calculate the elasticity of the variance in the expected lifetime reproduction number (Caswell, 2009).
Fig. 2(right) indicates that the vital rates having a large influence
on the reproduction events also have a large effect on the variance.
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R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
Expected lifetime reproduction number
Variance in expected lifetime reproduction number
25
45
40
20
35
30
25
Elasticity
Elasticity
15
10
20
15
5
10
5
0
0
−5
−5
s1
s2
s3
s4
s5
g1 g2 g3 g4 g5
s1
s2
s3
s4
s5
g1 g2 g3 g4 g5
Fig. 2. (left) The elasticity of the expected lifetime reproduction number to each vital rates; (right) the elasticity of variance of the expected lifetime reproduction number
to each vital rates. si = i are survival probabilities, and gi = i are probabilities of maturation.
3.2. Life expectancy
Life expectancy, the average longevity, is one of the most important demographic characteristics of a population (Carey, 2003).
Using techniques similar to those in Caswell (2009), the expectancy
vector for the sperm whale is
E = (30.9282 35.2996 44.8430 44.8430 44.8430).
(9)
E is calculated by summing the columns of the fundamental matrix
N in (8). The entries of E represent the life expectancies of each of the
five stages of sperm whales, starting with that stage. For instance,
the first entry of E implies that the life expectancy for a calf is about
31 years. The third entry tells us that a mature adult will live, on
average, an additional 45 years. We see that the difference in the life
expectancies of calves and mature sperm whales is quite significant.
Also, with the assumption that the mortality rates of stages 3–5 are
the same, the life expectancies are the same for these stages.
Scientist are most interested in the life expectancy at birth, so
we calculate the elasticity of life expectancy of a female calf with
respect to the vital rates (Caswell, 2009). Fig. 3(left) shows this
elasticity. Interestingly, the elasticity of the life expectancy for the
calves is relatively large in terms of the survivorship probabilities
and very small in terms of the maturation probabilities. In particular, the larger the amount of time spent as a mature female, the
longer the female lives. Specifically, 1% increase in the value of 3
would result in about 17% increase in the life expectancy.
3.3. Asymptotic growth rate
To investigate the dynamics of the population of sperm whales,
one key value to study is the asymptotic growth rate . Under
the assumption that the vital rates are invariant of time and environment, if > 1, the population will grow, while for < 1, the
population will decrease. Based on the estimation of vital rates
Population growth rate λ
Life expectancy (female calf)
18
0.45
16
0.4
14
0.35
12
0.3
Sensitivity
Elasticity
10
8
6
0.25
0.2
0.15
4
0.1
2
0.05
0
−2
0
s1
s2
s3
s4
s5
g1
g2
g3
g4
g5
s1
s2
s3
s4
s5
g1
g2
g3
g4
g5
Fig. 3. (left) The elasticity of life expectancy for a female calf to each of the vital rates; (right) the sensitivity, to each vital rate, of the population growth rate .
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
800
800
700
700
600
600
500
500
400
400
300
300
200
200
100
100
0
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
0
0.96
0.97
λ: μ=1.001; 95% CI [0.97743,1.0236]
0.98
0.99
75
1
1.01
1.02
1.03
1.04
1.05
λ: μ=1.0011; 95% CI [0.98582,1.016]
Fig. 4. The histogram of growth rate assuming the parameters follow uniform distributions (left) or normal distribution (right).
in Table 1, the asymptotic growth rate for model (3) and (4) is
≈ 1.0096, which indicates that the population is growing at a
rate of 0.96% per year. Because the calculation of involves uncertainties, interval estimates of can be obtained as follows. From
the discussion in Section 2.1, we assume that the female adult
survival rate 3 follows a uniform distribution on [0.939, 0.98],
3 = 4 = 5 , 1 follows a uniform distribution on [0.75 3 , 1.9 3 ],
2 = (1/2)( 1 + 3 ) and the interbirth interval for sperm whales I
follows a uniform distribution on [3, 5]. Bootstrap resampling is
used here to estimate the mean and confidence intervals of using
100,000 bootstrap samples. The histogram for the bootstrap values
of is shown in Fig. 4(left). The average asymptotic growth rate is
approximately 1.001. The percentile confidence interval method is
applied to estimate the 95% confidence interval of , where the endpoints of the 95% confidence interval are given by the 2500th and
97500th sorted bootstrap values of (Efron and Tibshirani, 1993).
For these data, that interval is [0.9774, 1.0236]. Even for the best
case parameters given in Table 1, we get ≈ 1.0302, which is still
only a growth rate of about 3% per year. For the worst case parameters, ≈ 0.9641. This means that given the worst case parameters,
the population of sperm whales would eventually go to extinction. If instead of uniform distributions normal distributions are
assumed, the results for are shown in Fig. 4(right). In this case,
the mean asymptotic growth rate is 1.0011, while the 95% confidence interval is [0.9857, 1.0161]. Similar simulations are used for
the inherent growth rate R0 (Fig. 5(left) and (right)).
There are various ways to increase the asymptotic growth rate
. Efforts could be focused on increasing the survivorship of calves
and juveniles, or decreasing the mortality rate of post breeding
females, etc. We conduct sensitivity analysis on each of the model’s
parameters to identify the life-history stage that contributes most
to the population growth . Sensitivity analysis reveals how very
small changes in each vital rate parameters will affect when the
other elements in the matrix A are held constant. Sensitivities thus
1500
1500
1000
1000
500
500
0
0
0.5
1
1.5
2
2.5
R0: μ=1.1913; 95% CI [0.53252,2.4551]
3
0.5
1
1.5
2
2.5
3
R0: μ=1.084; 95% CI [0.67098,1.7658]
Fig. 5. The histogram of the lifetime reproductive number R0 assuming the parameters follow uniform distributions (left) or normal distribution (right).
76
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
compare the absolute effects of the same absolute changes in the
vital rates on . The sensitivity of the asymptotic growth rate to
changes in the elements aij of A are given by Caswell (2009)
vi wj
∂
=
,
w,
v
∂aij
(10)
where w and v are the right and left eigenvectors of the projection
matrix A corresponding to . The values we used for w and v are
3.4. Inherent net reproductive number
The generation growth matrix for sperm whale is
⎛
⎜
⎜
FN = ⎜
⎜
⎝
w = (0.0850 0.2077 0.3617 0.1783 0.1672)T
and
v = (1.000 1.2389 2.0067 1.7651 1.8820)T .
The sensitivity matrix of to elements of A is
⎛
0.0501
0
0.2131
0
0
0
0
0.2456
0.4275
0
⎜ 0.0620 0.1516
⎜
SA = ⎜
⎜
⎝
0
0
0
0
0.3761 0.1854
0
0.1977
0
⎞
0
⎟
⎟
0
⎠
0.1977 ⎟
⎟.
(11)
0.1854
Elasticity analysis estimates the effect of a proportional change in
the vital rates on population growth. The elasticity of an element
aij in matrix A, eij , is given by Caswell (2009)
eij =
aij ∂
∂ log =
.
∂aij
∂ log aij
(12)
In essence, elasticities are proportional sensitivities, scaled so
that they are dimensionless. Elasticities thus compare the relative effects on with the same relative changes in the values of
the demographic parameters. This allows a direct comparison of
the effect of demographic parameters with different units on the
asymptotic growth rate . The elasticity matrix of to elements of
A is
⎛
0.0237
0
0.0264
0
0
⎞
⎜ 0.0264 0.1252
⎟
0
0
0
⎜
⎟
⎟.
EA = ⎜
0
0.0264
0.3070
0
0.0942
⎜
⎟
⎝ 0
⎠
0
0.0942 0.0912
0
0
0
0
0.0942
(13)
0.0912
The elasticity matrix shows that is most elastic to P3 , the probability that an adult female survives and stays as an adult, and is
second most elastic to the probability that an immature female survives and stays as an immature (P2 ). Specifically, 1% increase in P3
results in about 30.7% increase in . Fig. 3(right) demonstrates the
sensitivity of to each of the vital rates. It can be seen that the sensitivity of is affected strongly by the survivor probabilities and
much less by maturation. The asymptotic growth rate is most
sensitive to 3 and less so to 4 , 5 , and 2 . Among all the maturation rates, is most sensitive to the transition probability from
immature to mature, and from mature to mother.
The approximated stable stage distribution of the population is
T
(0.0850 0.2077 0.3617 0.1783 0.1672) .
(14)
So when the stage distribution approaches stability, approximately
36% of the population are mature adults, while 21% are immature
juveniles. The percentage of female calves (8.5%) is approximately
one half of the percentage of mothers (17.83%), since we assume
the sex ratio at birth is 1:1 and that the female will take care of the
one newborn calf for about 2 years before giving birth again. The
expected reproductive value for each stage of the population is
(1.0000 1.2389 2.0067 1.7651 1.8820)T .
(15)
The mature adults contribute most to reproduction, closely followed by post-breeding adults.
1.5174 1.8462
2.8275 2.5825
2.7023
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
⎞
⎟
⎟
⎟
⎟
⎠
(16)
The inherent net reproductive number R0 for a newborn female
calf is the dominant eigenvalue of FN. For (16), R0 ≈ 1.5174. Note
that only female offsprings are included in R0 , while all offsprings,
regardless of sex, are counted in the fundamental matrix N. Comparing R0 ≈ 1.52 with N(4, 1) ≈ 6.10, we notice that R0 is less than
a half of N(4, 1) . This is mainly due to the mortality likelihood of
calves transiting from one stage to another.
Fig. 6 shows the elasticity (left) and sensitivity (right) of R0 to
each of the vital rates for a newborn calf. It can be seen that the sensitivity of R0 is roughly proportional to that of . This indicates that
the vital rates having large effects on will also have large effects
on R0 . As with , R0 is affected strongly by survivor probability and
much less by maturation. R0 is most sensitive to 3 , less so to 4 ,
5 , and 2 .
3.5. Extinction time
We now investigate the extinction time of the population. The
transition matrix T in (6) does not include death as a state, so we
create the matrix T̃ given by
⎛
0.4778
0
0
0
0
⎞
⎜ 0.4292 0.8339
⎟
0
0
0
⎜
⎟
⎜ 0
0.1085 0.7249
0
0.4810 ⎟
⎟,
T̃ = ⎜
⎜ 0
⎟
0
0.2528 0.4967
0
⎜
⎟
⎝ 0
0
0
0.4810 0.4967 ⎠
0.0930
0.0576
0.0223
0.0223
(17)
0.0223
which includes death. The last row of T̃ gives the death probability
at each stage. We first consider the lineage extinction probability for
the sperm whale on each stage. As mentioned in Caswell (2001) the
extinction of the lineage is founded by an individual. The probability
q(t) = (q1 (t), · · · , q5 (t))T of extinction probability of each of the five
stages is defined as
qi (t) = P[n(t) = 0|n(0) = e(i) ],
(18)
for i = 1, · · · , 5. By transforming the projection matrix A into a
multi-type branching process, we obtain the probability generating function of each stage for total offspring production X (Caswell,
2001):
1 (s) = 0.4778s + 0.4292s + 0.0930;
GX
1
2
2 (s) = 0.8339s + 0.1085s + 0.0576;
GX
2
3
3 (s) = (0.7249s + 0.2528s + 0.0223)(0.1250s + 0.8750); (19)
GX
3
4
1
4 (s) = 0.4967s + 0.4810s + 0.0223;
GX
5
4
5 (s) = 0.4810s + 0.4967s + 0.0223,
GX
5
3
where s = (s1 , s2 , s3 , s4 , s5 )T is a dummy variable. To calculate the
probability of lineage extinction, we start with q(0) = 0 and iterate
⎛
⎜
⎝
q(t + 1) = ⎜
1 (q(t))
GX
..
.
5 (q(t))
GX
⎞
⎟
⎟
⎠
(20)
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
77
Inherent net reproductive number R 0
Inherent net reproductive number R 0
25
40
35
20
25
15
Sensitivity
Elasticity
30
10
20
15
10
5
5
0
0
s1
s2
s3
s4
s5
g1
g2
g3
g4
g5
s1
s2
s3
s4
s5
g1
g2
g3
g4
g5
Fig. 6. (left) The elasticity and (right) the sensitivity of the inherent net reproduction number R0 for a newborn calf to each of the vital rate.
n1 (0)
Q (t) = q1 (t)
n2 (0)
q2 (t)
n5 (0)
· · ·q5 (t)
.
(21)
In light of the recent oil spill in the Gulf of Mexico, we will be
concentrating on the population of sperm whale in that region.
One could also do similar analysis on other populations of sperm
whales such as on the population in the Mediterranean Sea. In
Waring et al. (2009a,b), the sperm whale population in the Gulf of
Mexico is approximately 1645. We assume the stage distribution
of the population is the stable stage distribution as shown in (14).
Therefore, we can assume the initial female sperm whale population is n(0) = (70, 171, 297, 147, 138)T . The probability of extinction
for the whole population is unlikely as long as the lineage extinction probability qi (t) is strictly less than one. To evaluate the effect
Lineage extinction probability
0.8
0.7
0.6
0.5
Calf
Immuture adult
0.3
Mature adult
Mother
0.2
Post breeding
0.1
0
0
50
100
150
200
250
300
90
80
70
60
50
40
30
20
10
0
60
80
100
120
140
160
180
200
Time (Years)
Fig. 8. Ten of the total 105 times realizations of the stochastic simulations of the
total sperm whale population size. The initial female population of each stage is
assumed to be (70, 171, 297, 147, 138)T .
of demographic stochasticity on the sperm whales, we perform
stochastic simulations and calculate the probability distribution of
times to extinction with the worst case parameters. Since sperm
whales are monovular, we consider fertility as a Bernoulli random
variable with mean F(1, 3) and treat transitions and reproduction
as independent events. The transitions are treated as multinomial
variables (Caswell, 2001). Some results of the stochastic simulation with initial female population (70, 171, 297, 147, 138)T are
shown in Figs. 8 and 9. In the worst case, the mean time to extinction under demographic stochasticity for sperm whales is about
163 years, extracted from 100,000 stochastic realizations. In the
estimated case and the best case, since the annual growth rate is greater than 1, the population extinction due to demographic
stochasticity is highly unlikely.
0.9
0.4
100
Total Population
until it converges. The limit is the extinction of the lineage. Fig. 7
demonstrates the lineage extinction probability of each stage for
the estimated model. The lineage extinction probability for sperm
whales using the best and worst case parameters from Table 1 can
also be calculated.
As in Caswell (2001), because individual sperm whales are
independent, the probability of extinction time for the whole population, given an initial population n(0), is
350
400
Time (years)
Fig. 7. The lineage extinction probability of each stage for sperm whales as a function of time t in years.
4. Discussion
Using various techniques and resources, we are able to get estimates for the vital rates needed to analyze our model (3) and
(4). Along with these estimates, we are able to get a range of
78
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
200
0.018
180
0.016
160
0.014
Extinction probability
Number of events
140
120
100
80
60
0.01
0.008
0.006
0.004
40
0.002
20
0
50
0.012
100
150
200
250
300
350
Extinction time (year)
0
50
100
150
200
250
300
350
Extinction year
Fig. 9. (left) The histogram of the extinction time of sperm whales in the worst case (105 times stochastic simulations). (right) The probability distribution of time to extinction
assuming demographic stochasticity. The initial female population of each stage used in the stochastic simulation is (70, 171, 297, 147, 138)T .
parameters representing the best and worst case parameters for
each rate in term of the survivor rates. We use these estimates
to calculate deterministic values such as the life expectancy, the
asymptotic growth rate , inherent reproductive number R0 , and
extinction time for sperm whale population.
We then perform sensitivity and/or elasticity analysis of the
model parameters. For all of the deterministic values calculated
in Section 3, we see that the survivor probabilities have the most
effect on these values, and they are much less affected by maturation. More specifically, these values are most sensitive to 3 , the
survivorship rate of the mature adult, and less to 4 , 5 , and 2 .
Thus, any increase in the mortality rate of the mature females could
have a damaging effect on the population as a whole.
The two values that really tell how potentially fragile the sperm
whale population might be are the life expectancy and the asymptotic growth rate. The life expectancy of each stage of the life cycle
of sperm whales, presented in (9), was calculated using the estimated parameters given in Table 1. For instance, a mature adult’s
life expectancy is around 45 years, and a calf’s is around 31 years.
But, if we calculate these values using the worst case parameters,
we see the number of years in each stage drop to about one third
of the value for the estimated parameters. For the calves, the life
expectancy dropped to about 11 years and the mature adults to
about 16 years.
When calculating the asymptotic growth rate for the sperm
whales with the estimated parameters in Table 1, we see that
≈ 1.0096, which indicates that the population is growing at a
rate of 0.96% per year. This is close to the maximum rate of 0.9%
per year for a sperm whale population with a stable age distribution Whitehead (2003) calculated using the population parameters
from the International Whaling Commission. Our value is also close
to the annual rate of increase of 1.1% Whitehead (2003) calculated when using the mortality schedule for killer whales and an
age-specific pregnancy rate from Best et al. (1984). Thus, using
our parameter estimations, we get a value of that is between
these two values. However, even under the best case parameters
in Table 1, the growth rate of the population is extremely slow.
Because of the slow growth under the best case parameters and
an actual decrease in population under the worst, it is possible
that any major stochastic event, such as a natural or man-made
disaster, in the ecosystem of the sperm whale may potentially
decrease the population or drive it to extinction.
One potential problem for the sperm whales is the lingering
affects of whaling. Despite the International Whaling Commission’s
adoption of a moratorium on commercial whaling in 1986, little is
known about the residual affects whaling may still have on sperm
whale populations around the world today. One study of the population of sperm whales off the Galapagos Islands by Whitehead et al.
showed about a 20% decrease each year in the population between
1985 and 1995. They suggest that the decline may be the residual
impact of the whaling industry, which ended in 1981 (Whitehead
et al., 1997). Along with the affects of whaling, sperm whale populations are now susceptible to several other threats. Some threats,
such as collisions with ships, ingestion of marine debris, and the
entrapment of the whales in fishing gear, result directly in the
killing of the sperm whales (Whitehead, 2003). Other potential
threats, such as acoustic and chemical pollution, could also have
lasting affects on sperm whale populations, particularly in areas
where the populations are small.
With the search for new petroleum deposits and in the wake of
the BP oil spill in 2010, the sperm whale population of the Northern Gulf of Mexico is one particular population that can be affected
the most by noise and chemical pollution. The Gulf sperm whale
population differs from other populations of sperm whales. On
average, they are smaller and the group size of females and immature whales is about one-third the size of populations found in other
areas. There are also significant genetic differences between sperm
whales in the Gulf compared to those for the North Atlantic Ocean
(Jochens et al., 2008; Waring et al., 2009a,b). Another major difference in the Northern Gulf of Mexico sperm whales that might make
them more vulnerable is the potential biological removal (PBR),
of the population. PBR is the maximum number of animals, not
including natural mortalities, that may be removed from a marine
mammal stock while allowing that stock to reach or maintain its
optimum sustainable population. The PBR for the Northern Gulf of
Mexico sperm whales is 2.8, compared to the PBR of 7.1 for the
Northern Atlantic Ocean sperm whales (Waring et al., 2009a,b).
Seismic vessels searching for petroleum deposits beneath the
oceans and gulfs produce one of the loudest anthropogenic sounds
(Whitehead, 2003). The Northern Gulf sperm whales have a home
R.A. Chiquet et al. / Ecological Modelling 248 (2013) 71–79
range that overlaps almost completely with areas of current and
future oil related activity and this population is living in close proximity to offshore oil and gas exploration (Jochens et al., 2008).
Although little is known about the direct effects the noise pollution produced by these seismic vessels have on the sperm whale
population, it is believed the ear damage may lead to reduction of
feeding or mating opportunities and possibly may be a cause of
ship strikes (Whitehead, 2003). Chemical pollution could also have
a long term effect on sperm whale populations. Because they feed at
high trophic levels and store the chemicals in their blubber, marine
mammals are susceptible to chemical pollution. Also, marine mammals could potentially pass these chemicals to their offspring in
their milk (Whitehead, 2003). The effects of noise and chemical
pollution on sperm whale populations just have not been studied
enough to determine exactly what long term affects might occur
to the sperm whale population. However, our sensitivity analysis
shows that if toxins, such as oil spills, reduce the survivorship rate
of the mature female sperm whales by as little as 2.2% or the survivorship rate of mothers by 4.8%, the asymptotic growth rate of the
population would drop below one. This would result in a decline
of a population that is already very fragile. Thus, to what extent
such factors affect the vital rates of sperm whales must be carefully
investigated in the future.
Acknowledgements
The authors would like to thank Dr. Hal Caswell for reading and
making useful comments on an earlier version of the manuscript,
and Dr. Hal Whitehead for providing useful information and guidance on sperm whales. Research is supported by the US National
Science Foundation under grant # DMS-1059753.
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