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Ecological Modelling
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emographic analysis of sperm whales using matrix population models
oss A. Chiqueta,∗, Baoling Mab, Azmy S. Ackleha, Nabendu Pala, Natalia Sidorovskaiac
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USADepartment of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USADepartment of Physics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA
r t i c l e i n f o
rticle history:eceived 19 February 2012eceived in revised form 30 July 2012ccepted 30 September 2012vailable online 10 November 2012
a b s t r a c t
The focus of this study is to investigate the demographic and sensitivity/elasticity analysis of the endan-gered sperm whale population. First, a matrix population model corresponding to a general sperm whalelife cycle is presented. The values of the parameters in the model are then estimated. The population’sasymptotic growth rate �, life expectancy and net reproduction number are calculated. Extinction timeprobability distribution is also studied. The results show that the sperm whale population grows slowly
eywords:perm whaleatrix models
ensitivity and elasticity analysissymptotic growth rate
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 indicatethat these survivorship rates are very delicate, and a slight decrease could result in an asymptotic growthrate below one, i.e., a declining population, leading to extinction.
The sperm whale, Physeter macrocephalus, is the largest odonto-ete, or toothed whale (Rice, 1989), and can be found throughouthe world’s oceans, gulfs, and seas (Rice, 1989; Whitehead, 2002).he studies of sperm whale populations in the literature are domi-ated by the studies of the geographic structure of sperm whaleopulations, which in recent years have mainly been throughenetic techniques and assessment of regional and global popu-ation estimates of sperm whales. The genetic studies, includingLyrholm et al., 1999; Lyrholm and Gyllensten, 1998; Richard et al.,996) and others, have used mitochondrial DNA, which is DNA
nherited from the mother, and also nuclear microsatellite. Sam-les for these genetic studies can come from commercial catches,ycatches, historical artifacts, and strandings, just to name a fewWhitehead, 2003). The genetic analyses of Lyrholm and Gyllensten1998) suggest that present-day sperm whale populations haveery low mitochondrial DNA diversity and show little geographicalifferentiation over their global range.
In the assessment of regional and global population estimations,cientist use different techniques, such as catch-per-unit-effortnalyses, length-specific techniques, mark-recapture techniques,
coustic data (Ackleh et al., 2012), and ship or aerial surveysWhitehead, 2003). These techniques were used to study popula-ions of sperm whales in the Gulf of Mexico (Fulling et al., 2003;
Waring et al., 2009a,b), in the southern Australian waters (Evansand Hindell, 2004), in the Eastern Caribbean Sea (Gero et al.,2007), in the Northeastern (Barlow and Taylor, 2005) and South-ern (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 esti-mates of the global population size and historical trajectory forsperm whales.
Even with all this research dedicated to sperm whales, very littleis known about the sperm whale’s population dynamics. There is adefinite lack of research dedicated to the study of stage-structuredpopulation models applied to sperm whales. This is probably dueto the lack of reliable estimates for the vital rates of sperm whales.Evans and Hindell (2004) suggest that techniques used to studysmaller cetaceans, such as the bottlenose dolphins and orcas, andlarge baleen whales, such as the humpback and bowhead whales,are harder to apply to sperm whales. There are very few articles inthe literature in which vital parameters are given for sperm whales,much less age-specific vital parameters needed to develop stage-structured models used to study these populations.
In this paper, we develop a stage-structured matrix popula-tion model for sperm whales. The parameters for our model wereobtained from the limited information in the literature or byconstruction using other vital rates as in Doak et al. (2006) and tech-
niques from Caswell (2001). We present the best and worst cases forthese vital rates in terms of survivorship rates, along with the val-ues that were used in the analysis of the population dynamics. Weuse our model and these parameters to construct the asymptotic
mates, for the infants especially, were based on “extremely shaky
ig. 1. Life cycle graph for female sperm whales with (right) and without (left)he consideration of death stage. Numbers represent different stages. 1: calf; 2:mmature (juvenile); 3: mature; 4: mother; 5: post breeding; 6: mortality.
rowth rate � for the population. We also calculate other deter-inistic values for the population such as life expectancy, lifetime
eproduction number, and the inherent net reproduction number0. We then perform sensitivity and elasticity analysis to each of theital rates to see which ones affect these values the most. Also, wealculate the extinction time and distribution of sperm whales fromhe matrix population model, treating transitions and reproductions independent events.
. Stage-structured model and parameter estimations
In Section 2.1, we develop a stage-structured matrix model toescribe the dynamics of the female sperm whale. The female pop-lation is divided into stages similar to those used by Fujiwara andaswell (2001) for the North Atlantic right whale. In Section 2.2,e use techniques from Caswell (2001) and Doak et al. (2006) to
onstruct vital rates for the female sperm whale. We will use thesearameters in our model to study the dynamics of the population.
.1. Stage-structured population matrix model
The life cycle of the female sperm whale is typical of most largehales. After calves are born, they are suckled by their mother
or about 2 years (Best et al., 1984) and reach maturity at the agef 9 (Doak et al., 2006). The interbirth interval for sperm whalesiffers from 3–5 years (Boyd et al., 1999; Doak et al., 2006) to–6 years (Best et al., 1984; Rice, 1989; Whitehead, 2003) which
ncludes a gestation period of 14–16 months (Evans and Hindell,004). Thus, we divide the female sperm whale population intove stages: calves (stage 1), juveniles/immature (stage 2), mature
emales (stage 3), mothers (stage 4) and post breeding femalesstage 5). The life cycle for the female sperm whales is given inig. 1.
A mature female transits from stage 3 to stage 4 each time sheroduces a calf, and takes care of the calf throughout stage 4, thushe time taken for a female on stage 4 is the same as that for a calfo grow into an immature, which again is about 2 years for spermhale (Best et al., 1984). After this 2 year period, the female spermhale then goes into stage 5 where after the interbirth interval,
he female can return to stage 3 to give birth to another calf. If theemale is no longer able to reproduce due to age or other natu-al causes, the female will remain in stage 3. For our model andhroughout the rest of the paper, we will take our time unit to bene year. Let Pi = �i(1 − � i) and Gi = �i� i, where �i is survivor proba-ility of stage i and � i is the probability of an individual in stage i to
ove onto stage i + 1 for i = 1, · · · , 4. The transition probability from
tage 5 to stage 3 is given by �5. Therefore, Pi gives the probabilityf surviving and staying in stage i, while Gi gives the probabilityf surviving and moving to stage i + 1 for i = 1, · · · , 4. G5 gives the
�4 0.4934 0.4920 0.4875�5 0.4934 0.4920 0.4875
probability of surviving and moving to stage 3 from stage 5. FromCaswell (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 asymp-totic growth rate, i.e., the growth rate of the population at the stablestage distribution. Since � i is actually used in the calculation of �,we first set � to one. Then, as described in Caswell (2001), we usean iterative process to get better estimations of � i. With � = 1, wecalculate the projection matrix for our model. The eigenvalues ofthis projection matrix will yield a second estimation for �, whichwe use to estimate � i again. We then repeat this process until weget a projection matrix whose entries are compatible with its owneigenvalues. This is how we obtain the estimated values for � i inTable 1. Now, we define the fertility number as
b1 = 0.5�3�3√
�4, (2)
which depends on the mature female survivor probability, theprobability of giving birth after survival, and the survivor prob-ability of the mother caring for the calf (Caswell, 2009). The 0.5comes from the assumption that the sex ratio is about equal atbirth (Whitehead, 2003). Thus, we obtain the following model cor-responding 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 spermwhales at each stage. The projection matrix A is given by
A =
⎛⎜⎜⎜⎜⎜⎜⎝
P1 0 b1 0 0
G1 P2 0 0 0
0 G2 P3 0 G5
0 0 G3 P4 0
0 0 0 G4 P5
⎞⎟⎟⎟⎟⎟⎟⎠
. (4)
2.2. Estimating model parameters
Despite the small number of references for vital rates of spermwhales in the literature, we estimate the birth, mortality, and tran-sition rates for model (3) and (4). Sperm whale mortality is the leastknown aspect of a sperm whale’s life history (Whitehead, 2003).In 1982, International Whaling Commission’s Scientific Committeeestimated the annual mortalities of 0.055 for female sperm whalesand 0.093 for infants. However, according to the experts, these esti-
evidence” (Whitehead, 2003). Whitehead (2001) provides an esti-mate of 0.021 for the annual mortality rate of female and immaturesperm whales in the eastern tropical Pacific. Doak et al. (2006) givea minimum of 0.02 and a maximum of 0.061 for the annual adult
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ortality of sperm whales in general. Evans and Hindell (2004)rovide estimated annual mortality rates for female sperm whales
n the southern Australian waters. The average annual mortalityate for those sperm whales is about 0.0223.
For the analysis of model (3) and (4), we consider the range of thennual female sperm whale mortality to be 0.02–0.061, and thus weave the range of the survivor rates to be 0.939 ≤ �3 ≤ 0.98. We usehe value of 0.0223 we calculated from Evans and Hindell (2004) inur initial analysis for the mortality rate or 0.977 for the survivorate. We assume that the mortality rate of the post breeding whalesnd of the mothers are the same as the adult mortality rate, i.e.,3 = �4 = �5. The mortality of juveniles and immatures is assumed
o 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
rom the International Whaling Commission.Recall from Section 2.1, the interbirth interval for sperm whales
iffers from 3–5 years (Boyd et al., 1999; Doak et al., 2006) to 4–6ears (Best et al., 1984; Rice, 1989; Whitehead, 2003). In our analy-is, we use 4 years as the interbirth interval. As in Doak et al. (2006),e take the annual fecundity rate to be one half the reciprocal of
he interbirth interval, where the one half is to account for only theemale sperm whales. Thus using (2), the value of b1 in our modelill 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 spermhale remains a juvenile until age 2 (Best et al., 1984) and reachesaturity at the age of 9 (Doak et al., 2006). Therefore, the ranges of
he vital rates for model (3) and (4) are given in Table 1. The bestase and worst case parameters are in terms of the survivorshipates.
Based on the estimated values in Table 1, the projection matrix4) is estimated to be
=
⎛⎜⎜⎜⎜⎝
0.4778 0 0.1250 0 0
0.4292 0.8339 0 0 0
0 0.1085 0.7249 0 0.4810
0 0 0.2528 0.4967 0
0 0 0 0.4810 0.4967
⎞⎟⎟⎟⎟⎠
. (5)
The dominant eigenvalue of A gives the population’s asymptoticrowth rate �. The corresponding right eigenvector provides thetable stage distribution (i.e., the proportion of individuals of eachtage within the population). Once the stable stage distribution haseen reached, the population undergoes exponential growth at rate. Throughout this paper, we analyze the population matrix model3) with the projection matrix A as given in (5).
. Model analysis
In this section, we first calculate the fundamental matrix forhe sperm whale in order to perform demographic analysis onhe model (3) and (4) with the estimated parameters presented inable 1. We calculate the life expectancy, the asymptotic growthate �, and inherent reproductive number R0 for the popula-ion. Lineage extinction time of sperm whales is then calculatedy transforming the projection matrix A, as given in (5), into aulti-type branching process (Caswell, 2001). We also obtain the
raph of the probability distribution of extinction time assumingemographic stochasticity. We perform sensitivity and/or elasticitynalysis using techniques similar to Caswell (2009) to help identifyhe life-history stage that contributes the most to the deterministicalues found for the population.
.1. Fundamental matrix and lifetime reproduction number
The fundamental matrix is one of the primary tools usedn the analysis of population models. The entries of the
odelling 248 (2013) 71– 79 73
fundamental matrix give the mean number of visits by an individualto any transient state. The first step in calculating the fundamentalmatrix for sperm whales is to decompose the projection matrix Aas given in (5), into A = T + F, where
T =
⎛⎜⎜⎜⎜⎝
0.4778 0 0 0 0
0.4292 0.8339 0 0 0
0 0.1085 0.7249 0 0.4810
0 0 0.2528 0.4967 0
0 0 0 0.4810 0.4967
⎞⎟⎟⎟⎟⎠
, (6)
and
F =
⎛⎜⎜⎜⎜⎝
0 0 0.1250 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
⎞⎟⎟⎟⎟⎠
. (7)
Here, T in (6), describes the individual transition probability and Fin (7), the individual fertility number. The fundamental matrix isthen defined by N = (I − T)−1, with I being the identity matrix. Thus,we get
N =
⎛⎜⎜⎜⎜⎝
1.9149 0 0 0 0
4.9482 6.0203 0 0 0
12.1393 14.7694 22.6203 20.6603 21.6181
6.0980 7.4193 11.3631 12.3653 10.8596
5.8279 7.0906 10.8596 11.8175 12.3653
⎞⎟⎟⎟⎟⎠
, (8)
where N(i, j) gives the expected number of visits over a life time tostage i from an individual starting at stage j.
The first column represents the female calf. On average, a femalecalf spends about 1.9 years as a calf, about 12.14 years as a maturefemale, 6.10 years as a mother, which means a calf is expectedto give birth 6.10 times on average. The value 6.10 is also knownas the expected lifetime reproduction number from the calf stage.The third column corresponds to mature females, so N(4, 3) ≈ 11.36indicates that a mature female is expected to give birth 11.36 times,on average. This number is higher than 6.10 because of the mortalitylikelihood in transiting from calf to mature female.
Fig. 2(left) indicates how the expected lifetime reproductionnumber N(4, 1) from the calf stage varies in response to changesin the vital rates (Caswell, 2009). It shows that the number ofreproduction events is most elastic to the mature female survivorprobability (�3) and very elastic to survivor probabilities of themothers (�4), post-breeding adult females (�5), and the juveniles(�2). Specifically, a 1% increase in �3 (�4 or �5) results in approx-imately a 23% (12% or 10%, respectively) increase in the breedingnumber. Surprisingly, the probability of giving birth by an adultfemale �3 has no major influence on the expected reproductionnumber. The probability of transiting from mother to post-breedingfemale �4 has a negative effect on the reproduction event. This canbe explained since the larger the value of �4 is, the shorter timethe calf is taken care by the mother, thus the less healthy or notfully developed the calf is, resulting in a smaller lifetime reproduc-tion number for the calf. The number of expected reproductionsfrom the calf stage is a random variable, and from the fundamen-tal matrix, we only get the mean value N(4, 1). To explore more of
the individual stochasticity, we calculate the elasticity of the vari-ance in the expected lifetime reproduction number (Caswell, 2009).Fig. 2(right) indicates that the vital rates having a large influenceon the reproduction events also have a large effect on the variance.
ig. 2. (left) The elasticity of the expected lifetime reproduction number to each vo each vital rates. si = � i are survival probabilities, and gi = � i are probabilities of m
.2. Life expectancy
Life expectancy, the average longevity, is one of the most impor-ant demographic characteristics of a population (Carey, 2003).sing techniques similar to those in Caswell (2009), the expectancyector for the sperm whale is
= (30.9282 35.2996 44.8430 44.8430 44.8430). (9)
is calculated by summing the columns of the fundamental matrix in (8). The entries of E represent the life expectancies of each of theve stages of sperm whales, starting with that stage. For instance,he first entry of E implies that the life expectancy for a calf is about1 years. The third entry tells us that a mature adult will live, on
verage, an additional 45 years. We see that the difference in the lifexpectancies of calves and mature sperm whales is quite significant.lso, with the assumption that the mortality rates of stages 3–5 are
he same, the life expectancies are the same for these stages.
s1 s2 s3 s4 s5 g1 g2 g3 g4 g5−2
0
2
4
6
8
10
12
14
16
18Life expectancy (female calf)
Ela
stic
ity
Fig. 3. (left) The elasticity of life expectancy for a female calf to each of the vital ra
es; (right) the elasticity of variance of the expected lifetime reproduction numberion.
Scientist are most interested in the life expectancy at birth, sowe calculate the elasticity of life expectancy of a female calf withrespect to the vital rates (Caswell, 2009). Fig. 3(left) shows thiselasticity. Interestingly, the elasticity of the life expectancy for thecalves is relatively large in terms of the survivorship probabilitiesand very small in terms of the maturation probabilities. In partic-ular, the larger the amount of time spent as a mature female, thelonger the female lives. Specifically, 1% increase in the value of �3would 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 �. Underthe assumption that the vital rates are invariant of time and envi-ronment, if � > 1, the population will grow, while for � < 1, thepopulation will decrease. Based on the estimation of vital rates
s1 s2 s3 s4 s5 g1 g2 g3 g4 g50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45Population growth rate λ
Sen
siti
vity
tes; (right) the sensitivity, to each vital rate, of the population growth rate �.
Fig. 4. The histogram of growth rate � assuming the paramete
n Table 1, the asymptotic growth rate for model (3) and (4) is ≈ 1.0096, which indicates that the population is growing at aate of 0.96% per year. Because the calculation of � involves uncer-ainties, interval estimates of � can be obtained as follows. Fromhe discussion in Section 2.1, we assume that the female adulturvival 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
ollows a uniform distribution on [3, 5]. Bootstrap resampling issed here to estimate the mean and confidence intervals of � using00,000 bootstrap samples. The histogram for the bootstrap valuesf � is shown in Fig. 4(left). The average asymptotic growth rate ispproximately 1.001. The percentile confidence interval method is
pplied to estimate the 95% confidence interval of �, where the end-oints of the 95% confidence interval are given by the 2500th and7500th sorted bootstrap values of � (Efron and Tibshirani, 1993).or these data, that interval is [0.9774, 1.0236]. Even for the best
0.5 1 1.5 2 2.5 30
500
1000
1500
R0: μ=1.1913; 95% CI [0.53252,2.4551]
Fig. 5. The histogram of the lifetime reproductive number R0 assuming the par
λ: μ=1.0011; 95% CI [0.98582,1.016]
ow uniform distributions (left) or normal distribution (right).
case parameters given in Table 1, we get � ≈ 1.0302, which is stillonly a growth rate of about 3% per year. For the worst case param-eters, � ≈ 0.9641. This means that given the worst case parameters,the population of sperm whales would eventually go to extinc-tion. If instead of uniform distributions normal distributions areassumed, the results for � are shown in Fig. 4(right). In this case,the mean asymptotic growth rate is 1.0011, while the 95% confi-dence interval is [0.9857, 1.0161]. Similar simulations are used forthe 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 calvesand juveniles, or decreasing the mortality rate of post breedingfemales, etc. We conduct sensitivity analysis on each of the model’s
parameters to identify the life-history stage that contributes mostto the population growth �. Sensitivity analysis reveals how verysmall changes in each vital rate parameters will affect � when theother elements in the matrix A are held constant. Sensitivities thus
0.5 1 1.5 2 2.5 30
500
1000
1500
R0: μ=1.084; 95% CI [0.67098,1.7658]
ameters follow uniform distributions (left) or normal distribution (right).
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6 R.A. Chiquet et al. / Ecolog
ompare the absolute effects of the same absolute changes in theital rates on �. The sensitivity of the asymptotic growth rate � tohanges in the elements aij of A are given by Caswell (2009)
∂�
∂aij
= viwj
〈w, v〉 , (10)
here w and v are the right and left eigenvectors of the projectionatrix A corresponding to �. The values we used for w and v are
= (0.0850 0.2077 0.3617 0.1783 0.1672)T
nd
= (1.000 1.2389 2.0067 1.7651 1.8820)T .
he sensitivity matrix of � to elements of A is
A =
⎛⎜⎜⎜⎜⎝
0.0501 0 0.2131 0 0
0.0620 0.1516 0 0 0
0 0.2456 0.4275 0 0.1977
0 0 0.3761 0.1854 0
0 0 0 0.1977 0.1854
⎞⎟⎟⎟⎟⎠
. (11)
lasticity analysis estimates the effect of a proportional change inhe vital rates on population growth. The elasticity of an elementij in matrix A, eij, is given by Caswell (2009)
ij = ∂ log �
∂ log aij
= aij
�
∂�
∂aij
. (12)
n essence, elasticities are proportional sensitivities, scaled sohat they are dimensionless. Elasticities thus compare the rela-ive effects on � with the same relative changes in the values ofhe demographic parameters. This allows a direct comparison ofhe effect of demographic parameters with different units on thesymptotic growth rate � . The elasticity matrix of � to elements of
is
A =
⎛⎜⎜⎜⎜⎝
0.0237 0 0.0264 0 0
0.0264 0.1252 0 0 0
0 0.0264 0.3070 0 0.0942
0 0 0.0942 0.0912 0
0 0 0 0.0942 0.0912
⎞⎟⎟⎟⎟⎠
. (13)
he elasticity matrix shows that � is most elastic to P3, the prob-bility that an adult female survives and stays as an adult, and isecond most elastic to the probability that an immature female sur-ives and stays as an immature (P2). Specifically, 1% increase in P3esults in about 30.7% increase in �. Fig. 3(right) demonstrates theensitivity of � to each of the vital rates. It can be seen that the sen-itivity of � is affected strongly by the survivor probabilities anduch less by maturation. The asymptotic growth rate � is most
ensitive to �3 and less so to �4, �5, and �2. Among all the matu-ation rates, � is most sensitive to the transition probability frommmature to mature, and from mature to mother.
The approximated stable stage distribution of the population is
0.0850 0.2077 0.3617 0.1783 0.1672)T . (14)
o when the stage distribution approaches stability, approximately6% of the population are mature adults, while 21% are immature
uveniles. The percentage of female calves (8.5%) is approximatelyne half of the percentage of mothers (17.83%), since we assumehe sex ratio at birth is 1:1 and that the female will take care of thene newborn calf for about 2 years before giving birth again. Thexpected reproductive value for each stage of the population is
1.0000 1.2389 2.0067 1.7651 1.8820)T . (15)
he mature adults contribute most to reproduction, closely fol-owed by post-breeding adults.
odelling 248 (2013) 71– 79
3.4. Inherent net reproductive number
The generation growth matrix for sperm whale is
FN =
⎛⎜⎜⎜⎜⎝
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 femalecalf is the dominant eigenvalue of FN. For (16), R0 ≈ 1.5174. Notethat only female offsprings are included in R0, while all offsprings,regardless of sex, are counted in the fundamental matrix N. Com-paring R0 ≈ 1.52 with N(4, 1) ≈ 6.10, we notice that R0 is less thana half of N(4, 1) . This is mainly due to the mortality likelihood ofcalves transiting from one stage to another.
Fig. 6 shows the elasticity (left) and sensitivity (right) of R0 toeach of the vital rates for a newborn calf. It can be seen that the sen-sitivity of R0 is roughly proportional to that of �. This indicates thatthe vital rates having large effects on � will also have large effectson R0. As with �, R0 is affected strongly by survivor probability andmuch 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. Thetransition matrix T in (6) does not include death as a state, so wecreate the matrix T̃ given by
T̃ =
⎛⎜⎜⎜⎜⎜⎜⎝
0.4778 0 0 0 0
0.4292 0.8339 0 0 0
0 0.1085 0.7249 0 0.4810
0 0 0.2528 0.4967 0
0 0 0 0.4810 0.4967
0.0930 0.0576 0.0223 0.0223 0.0223
⎞⎟⎟⎟⎟⎟⎟⎠
, (17)
which includes death. The last row of T̃ gives the death probabilityat each stage. We first consider the lineage extinction probability forthe sperm whale on each stage. As mentioned in Caswell (2001) theextinction of the lineage is founded by an individual. The probabilityq(t) = (q1(t), · · · , q5(t))T of extinction probability of each of the fivestages 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 amulti-type branching process, we obtain the probability generat-ing function of each stage for total offspring production X (Caswell,2001):
t net reproduction number R0 for a newborn calf to each of the vital rate.
udtwa
iu
Q
IcOwWMoTtft
Ft
60 80 100 120 140 160 180 2000
10
20
30
40
50
60
70
80
90
100
Time (Years)
To
tal P
op
ula
tio
n
s1 s2 s3 s4 s5 g1 g2 g3 g4 g5
Fig. 6. (left) The elasticity and (right) the sensitivity of the inheren
ntil it converges. The limit is the extinction of the lineage. Fig. 7emonstrates the lineage extinction probability of each stage forhe estimated model. The lineage extinction probability for spermhales using the best and worst case parameters from Table 1 can
lso be calculated.As in Caswell (2001), because individual sperm whales are
ndependent, the probability of extinction time for the whole pop-lation, given an initial population n(0), is
(t) = q1(t)n1(0)q2(t)n2(0)· · ·q5(t)n5(0). (21)
n light of the recent oil spill in the Gulf of Mexico, we will beoncentrating on the population of sperm whale in that region.ne could also do similar analysis on other populations of spermhales such as on the population in the Mediterranean Sea. Inaring et al. (2009a,b), the sperm whale population in the Gulf ofexico is approximately 1645. We assume the stage distribution
f the population is the stable stage distribution as shown in (14).
herefore, we can assume the initial female sperm whale popula-ion is n(0) = (70, 171, 297, 147, 138)T. The probability of extinctionor the whole population is unlikely as long as the lineage extinc-ion probability qi(t) is strictly less than one. To evaluate the effect
0 50 100 150 200 250 300 350 4000
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ig. 7. The lineage extinction probability of each stage for sperm whales as a func-ion of time t in years.
Fig. 8. Ten of the total 105 times realizations of the stochastic simulations of thetotal sperm whale population size. The initial female population of each stage isassumed to be (70, 171, 297, 147, 138)T .
of demographic stochasticity on the sperm whales, we performstochastic simulations and calculate the probability distribution oftimes to extinction with the worst case parameters. Since spermwhales are monovular, we consider fertility as a Bernoulli randomvariable with mean F(1, 3) and treat transitions and reproductionas independent events. The transitions are treated as multinomialvariables (Caswell, 2001). Some results of the stochastic simula-tion with initial female population (70, 171, 297, 147, 138)T areshown in Figs. 8 and 9. In the worst case, the mean time to extinc-tion under demographic stochasticity for sperm whales is about163 years, extracted from 100,000 stochastic realizations. In theestimated case and the best case, since the annual growth rate �is greater than 1, the population extinction due to demographicstochasticity is highly unlikely.
4. Discussion
Using various techniques and resources, we are able to get esti-mates for the vital rates needed to analyze our model (3) and(4). Along with these estimates, we are able to get a range of
ig. 9. (left) The histogram of the extinction time of sperm whales in the worst case (ssuming demographic stochasticity. The initial female population of each stage us
arameters representing the best and worst case parameters forach rate in term of the survivor rates. We use these estimateso calculate deterministic values such as the life expectancy, thesymptotic growth rate �, inherent reproductive number R0, andxtinction time for sperm whale population.
We then perform sensitivity and/or elasticity analysis of theodel parameters. For all of the deterministic values calculated
n Section 3, we see that the survivor probabilities have the mostffect on these values, and they are much less affected by matura-ion. More specifically, these values are most sensitive to �3, theurvivorship rate of the mature adult, and less to �4, �5, and �2.hus, any increase in the mortality rate of the mature females couldave a damaging effect on the population as a whole.
The two values that really tell how potentially fragile the spermhale population might be are the life expectancy and the asymp-
otic growth rate. The life expectancy of each stage of the life cyclef sperm whales, presented in (9), was calculated using the esti-ated parameters given in Table 1. For instance, a mature adult’s
ife expectancy is around 45 years, and a calf’s is around 31 years.ut, if we calculate these values using the worst case parameters,e see the number of years in each stage drop to about one third
f the value for the estimated parameters. For the calves, the lifexpectancy dropped to about 11 years and the mature adults tobout 16 years.
When calculating the asymptotic growth rate for the spermhales with the estimated parameters in Table 1, we see that
≈ 1.0096, which indicates that the population is growing at aate of 0.96% per year. This is close to the maximum rate of 0.9%er year for a sperm whale population with a stable age distribu-ion Whitehead (2003) calculated using the population parametersrom the International Whaling Commission. Our value is also closeo the annual rate of increase of 1.1% Whitehead (2003) calcu-ated when using the mortality schedule for killer whales and ange-specific pregnancy rate from Best et al. (1984). Thus, usingur parameter estimations, we get a value of � that is betweenhese two values. However, even under the best case parameters
n Table 1, the growth rate of the population is extremely slow.ecause of the slow growth under the best case parameters andn actual decrease in population under the worst, it is possiblehat any major stochastic event, such as a natural or man-made
es stochastic simulations). (right) The probability distribution of time to extinctionhe stochastic simulation is (70, 171, 297, 147, 138)T .
disaster, in the ecosystem of the sperm whale may potentiallydecrease the population or drive it to extinction.
One potential problem for the sperm whales is the lingeringaffects of whaling. Despite the International Whaling Commission’sadoption of a moratorium on commercial whaling in 1986, little isknown about the residual affects whaling may still have on spermwhale populations around the world today. One study of the popu-lation of sperm whales off the Galapagos Islands by Whitehead et al.showed about a 20% decrease each year in the population between1985 and 1995. They suggest that the decline may be the residualimpact of the whaling industry, which ended in 1981 (Whiteheadet al., 1997). Along with the affects of whaling, sperm whale popu-lations are now susceptible to several other threats. Some threats,such as collisions with ships, ingestion of marine debris, and theentrapment of the whales in fishing gear, result directly in thekilling of the sperm whales (Whitehead, 2003). Other potentialthreats, such as acoustic and chemical pollution, could also havelasting affects on sperm whale populations, particularly in areaswhere the populations are small.
With the search for new petroleum deposits and in the wake ofthe BP oil spill in 2010, the sperm whale population of the North-ern Gulf of Mexico is one particular population that can be affectedthe most by noise and chemical pollution. The Gulf sperm whalepopulation differs from other populations of sperm whales. Onaverage, they are smaller and the group size of females and imma-ture whales is about one-third the size of populations found in otherareas. There are also significant genetic differences between spermwhales in the Gulf compared to those for the North Atlantic Ocean(Jochens et al., 2008; Waring et al., 2009a,b). Another major differ-ence in the Northern Gulf of Mexico sperm whales that might makethem more vulnerable is the potential biological removal (PBR),of the population. PBR is the maximum number of animals, notincluding natural mortalities, that may be removed from a marinemammal stock while allowing that stock to reach or maintain itsoptimum sustainable population. The PBR for the Northern Gulf ofMexico 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 theoceans and gulfs produce one of the loudest anthropogenic sounds(Whitehead, 2003). The Northern Gulf sperm whales have a home
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decline of sperm whales off the Galapagos Islands. Conservation Biology 11,
R.A. Chiquet et al. / Ecolog
ange that overlaps almost completely with areas of current anduture oil related activity and this population is living in close prox-mity to offshore oil and gas exploration (Jochens et al., 2008).lthough little is known about the direct effects the noise pollu-
ion produced by these seismic vessels have on the sperm whaleopulation, it is believed the ear damage may lead to reduction ofeeding or mating opportunities and possibly may be a cause ofhip strikes (Whitehead, 2003). Chemical pollution could also have
long term effect on sperm whale populations. Because they feed atigh trophic levels and store the chemicals in their blubber, marineammals are susceptible to chemical pollution. Also, marine mam-als could potentially pass these chemicals to their offspring in
heir milk (Whitehead, 2003). The effects of noise and chemicalollution on sperm whale populations just have not been studiednough to determine exactly what long term affects might occuro the sperm whale population. However, our sensitivity analysishows that if toxins, such as oil spills, reduce the survivorship ratef the mature female sperm whales by as little as 2.2% or the sur-ivorship rate of mothers by 4.8%, the asymptotic growth rate of theopulation would drop below one. This would result in a declinef a population that is already very fragile. Thus, to what extentuch factors affect the vital rates of sperm whales must be carefullynvestigated in the future.
cknowledgements
The authors would like to thank Dr. Hal Caswell for reading andaking useful comments on an earlier version of the manuscript,
nd Dr. Hal Whitehead for providing useful information and guid-nce on sperm whales. Research is supported by the US Nationalcience Foundation under grant # DMS-1059753.
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