September 22, 2017, Friday

# MBW:Pod-Specific Demography of Killer Whales

This wiki page summarizes the paper "Pod-Specific Demography of Killer Whales (Orcinus Orca)" by Solange Brault and Hal Caswell (1993). A pdf of the paper is available here: https://www.warnercnr.colostate.edu/class_info/fw471/AdditonalReadings/BraultCaswellKillerWhales.pdf.

## Project Categorization

• Mathematics used:
• Anaylsis of Leslie matrices for population projections providing stable stage structures and reproductive values given by corresponding right and left eigenvectors respectively, as well as sensitivities and elasticities of parameters on the dominant eigenvalue; for more information on this type of model, see Stage-based population model
• Bootstrap resampling to construct confidence intervals for growth rate and its corresponding continuous-time rate using percentile methods with a bootstrap sample size of 1000
• Nonparametric randomization tests to test the statistical significance of demographic differences between sub-populations
• Type of model:
• Demographic model using biologically defined stages for a stage-structured population projection matrix which describes the dynamics of the female portion of the population
• Separate models were created for the entire population, northern vs. southern subpopulations, and individual pods
• Biological system studied:
• Demographic consequences of social stucture on killer whales (Orcinus Orca)
• Data used are the result of a long-term, longitudinal study started in 1973

## Executive Summary

In classical demography, population structure appears to be a consequence of birth and death rates, thus creating vital cost in the social structure of that population. Conversely, when looking at social animals, population structure may instead be a cause, as well as a consequence, of the vital rates. Furthermore, the rate of population growth may be determined by caste structure, presence of helpers, and the distribution of sexual selection of a family. In order to study the effects social structure has on a population, one must compare the demography of groups with different compositions. In the paper "Pod-Specific Demography of Killer Whales (Orcinus Orca)," Solange Brault and Hal Caswell take the previously mentioned approach to study the demographic consequences of the social structure in killer whales living in well-defined social groups called pods. Their study included 18 social units, known as pods, containing anywhere from 5-63 individual whales, with distinctive dialects of whistles. These pods were studied for about 15 years with no observation of pod migration and rare mating encounters; however, similar patterns in dialects between pods suggest a possible original pod later splitting. Michael Bigg and his coworkers in the coastal waters of British Columbia and Washington State obtained the data used in the analyses. These results began in 1973 and continue to this day, even though data used in Brault and Caswell’s study uses data up until 1987. The model described is used to examine demographic differences between sub-populations and tested for statistical significance, later allowing discussion of implications of the results for killer whale biology and management. These long term longitudinal studies are imperative in understanding population dynamics of long-lived organisms by providing information yielding data on individual identification, social structure and identification, and survival without relying on assumptions made about age distribution. This information is especially useful for endangered species because the data taken does not rely on destructive sampling. The value of longitudinal studies increases in time, and hopefully this one will continue to flourish.

## Background of Killer Whales and Pods

Orcinus orca, better known as the killer whale, is a mammal that is found in all of the oceans of the world. Though they are very widely distributed, orcas still seem to prefer the colder waters of the Arctic and Antarctic. Their diets range from fish to larger prey such as seals and sea lions. Though killer whales have very good vision, they can also locate prey by producing clicks and sounds, a process known as echolocation. Physically, the weight of male orcas ranges from about 3,628 to 5,442 kilograms, and their length ranges from 5.8 to 6.7 meters. Orcas can live to be 80 or 90 years of age, but the average life span is closer to 50. As orcas are a very social species, they tend to travel in packs that are known as pods. Pods can range in size from 5 to 30 whales at a time. Not too different from humans, orcas reach sexual maturity from age 10 to age 18. Females can birth one offspring about once every 4 to 6 years, and though females do not usually breed past 35 to 45 years of age, they still remain with the pod. The social structure of killer whale pods is very advanced. When there are young in the pod, both males and females help take care of and protect the calves. Offspring typically live with their mother in the pod for their entire lives, because pods are usually made up of a lineage that started with a mother. In recent years, orcas have been caught and placed in captivity for shows. Killer whales can travel up to 160 km a day, and being in captivity does not mesh with their natural traveling tendencies. Life expectancy of these whales is considerably shorter than those in the wild (about 25 years). One would think that killer whales would be better off in their natural habitats, but with human disturbances constantly changing the wild, it is becoming more difficult for species to thrive.

Figure 1: A killer whale pod in the wild

With the growing impact that humans have had on the Earth, some orca populations have been labeled as endangered or threatened. This decrease in many populations of killer whales has come about from the lack of potential food for the whales, pollution, whaling, and habitat loss, among others. With the marine ecosystems declining, it is good practice to study and understand major apex predators like the killer whale, because the more we understand about the life stages and survivability of this species, the better equipped we are to slow or stop the decline of populations of orcas in the world’s oceans.

## Experimental Methods and Methods of Model Development

### Experimental

The data for this study was found in another paper by Bigg et al. (1990). Bigg et al. studied the population of killer whales off the coast of British Columbia. This population consisted of 18 pods that were made up of between 5 and 63 whales in each pod. This study began in 1973, and the data that is used for this analysis goes through year 1987. For these years, Bigg et al. had yearly observations on almost every individual whale in the population. Bigg et al. also recorded any relationships that each whale had with other whales. In addition to the large population, there were also 2 sub-populations. These sub-populations were called the northern and southern sub-populations. The northern consisted of 16 pods with 176 whales, and the southern had 3 pods with 105 whales. The data collected by Bigg et al. was used extensively in this analysis. Please refer to the paper by Bigg et al. for more information.

### Model Development

The analysis for individual pods had to be simplified, as age life tables could not be made for each pod because some pods did not have all ages of whales represented. Instead, four life stages were used to replace the 80-90 years that would have been needed to represent all age classes. This simplification did not seem to affect the results of the model, since the analysis of the entire population mostly matched the findings in the paper by Olesiuk et al. (1990).

## Mathematical Models

### Matrix Model

This model just looks at the female part of the specific population of killer whales in British Columbia. The lives of female killer whales was broken up into four stages. Stage one was yearlings, which were whales that were between 0 and 1 years of age. Stage two was the juvenile stage. This stage was made up of individuals that were characterized as not being yearlings, but were not yet sexually or physically matured. The third stage was mature females, which were whales that were observed with a calf of their own. The fourth stage was made up of females that were no longer reproducing. This last stage was based on whether the whale was seen with a calf within the last ten years of observation. If the whale was not seen with a calf in the ten year period, the whale was assumed to be in the fourth life stage.

With these stage definitions, the model of the population is given by:

$n(t+1)=An(t)$

In this model, $A$ is the population projection matrix form that is used in all stage-based classification models and analysis. Please refer to Figure 2 for the form of the $A$ matrix. The other part of the model is n(t), which is a vector that gives the number of individuals in each life stage at each time t. Here, t is measured in years, so the term (t+1) would be the current year plus one year.

Figure 2: Life-cycle graph and corresponding stage-classified population matrix for killer whale populations.

From our $A$ matrix, the $P_{i}$ terms are the survival probabilities, and the probability of remaining in the current stage. The $G_{i}$ terms are the probabilities of survival, and moving on to the next stage of life. In the model proposed by the authors, the probability of a yearling staying in the yearling stage for one year is equal to zero, since the length of the yearling stage is one year, which is the length of the projection interval. The $F$ terms in the $A$ matrix are the fertility values for each stage, and they are described as the number of female offspring at time $t+1$ for each adult female at time $t$.

Throughout this paper, the authors use another paper by Bigg et al. on this specific killer whale population for the data. The authors used this data to calculate the elements of projection matrix $A$. To estimate these elements, estimated stage-specific survival probabilities $\sigma _{i}$ and the probability of transitioning from one stage to the next, $\gamma _{i}$, and the average reproductive rate of mature females, $m$, were used. These estimated probabilities and reproductive rates were estimated from the data collected by Bigg et al. (1990). The stage-classified birth-flow formulation, proposed by Caswell (1989), was used to create the following formulas:

$G_{1}=(\sigma _{{11}})^{{1/2}}$

$P_{1}=0$

$G_{2}=\gamma _{2}\sigma _{2}$

$P_{2}=(1-\gamma _{2})\sigma _{2}$

$G_{3}=\gamma _{3}\sigma _{3}$

$P_{3}=(1-\gamma _{3})\sigma _{3}$

$P_{4}=\sigma _{4}$

$F_{2}=(\sigma _{1})^{{1/2}}(G_{2})^{{m/2}}$

$F_{3}=(\sigma _{1})^{{1/2}}(1+P_{3})m/2$

This model uses a main assumption: that we can consider all individuals within a specific life stage to be completely identical. So, if we were curious of the probability of a juvenile advancing from this stage to the adult stage, it would be the same probability for any individual in the juvenile stage, no matter how long or short of a time they have been in the juvenile stage. Although this is a very over-simplified version of what actually occurs in nature, the authors used another study on the entire killer whale population by Olesiuk et al. (1991) to check their analysis.

Another example of stage based matrix techniques can be found in Age Structured Populations in Seasonal Environments

### Matrix Parameters

The authors of this paper used the data on the killer whale population in a paper by Bigg et al. (1990). All of the data from this paper was not used, but specific parts were. The parts that were used include:

1.) The year of birth of those born during and before the ten year study (estimated)

2.) The year that each individual reached maturity (known by the presence of a calf)

3.) The year at which individuals reached the post-reproductive stage (estimated)

4.) The year at which individuals died or disappeared

5.) The sex of each individual

6.) The total number of female calves for the duration of the study

To estimate the parameters $\sigma _{i}$, $m$, and $\gamma _{i}$, the data again was used.

To determine the mean offspring production parameter, $m$, the ratio of the number of female offspring produced by the pack to the number of female-years of exposure for the duration of the study. This “exposure” of an individual was defined to be the amount of time the individual was part of the study and was a mature, reproducing adult. The survival probabilities, $\sigma _{i}$, were determined to be one minus the ratio of deaths in each stage i to the number of years of exposure in stage i for each individual. In this case, the term “exposure” was not defined the exact same way as it was for $m$. For each stage, “exposure” was different. To begin with, the yearlings ($\sigma _{1}$) exposure was defined to be the total number of births. For the juveniles ($\sigma _{2}$), exposure was the number of years of observation of juveniles. For the adult stage ($\sigma _{3}$), exposure was defined as the number of years during which each individual was observed as a reproducing adult. For the post-reproductive stage ($\sigma _{4}$), the exposure was estimated as the number of years of observation of any post-reproductive individual.

The growth probabilities ($\gamma _{i}$) were determined to be the reciprocals of the averages of the stage durations. For the yearlings, $\gamma _{1}$ would simply be one, since the average duration of this stage is one year, and 1/1 = 1. For the juvenile stage, $\gamma _{2}$ was defined as 1 over the average time whales spend in the juvenile stage. The growth probabilities of the adult stage were estimated a bit differently, since killer whales stay in the adult stage for a long period of time (relative to the length of the study). Instead, $\gamma _{3}$ was defined as one over the average age that whales entered the post-reproductive stage minus the sum of the average lengths of the yearling and juvenile stages.

These calculations of $\sigma _{i}$, $\gamma _{i}$, and $m$ were done for the whole population, the northern and southern sub-populations, and each pod in the entire population. If a pod was too small to determine a certain parameter, the overall population values for that parameter were used.

Once these calculations were performed, the parameters for the whole population were found to be:

$\sigma _{1}=0.9554$

$\sigma _{2}=0.9847$

$\sigma _{3}=0.9986$

$\sigma _{4}=0.9804$

$\gamma _{2}=0.0747$

$\gamma _{3}=0.0453$

$m=0.1186$

These parameter values lead to the following projection matrix:

${\begin{pmatrix}0&0.0043&0.1132&0\\0.9775&0.9111&0&0\\0&0.0736&0.9534&0\\0&0&0.0452&0.9804\end{pmatrix}}$

By observing the values of $\gamma _{2}$ and $\gamma _{3}$, it was found that the mean juvenile period is about 13.4 years. This implies an age of 14.4 at the first reproduction and a reproductive adult stage of 22.1 years. These results are similar to the age-specific results of Olesiuk et al. (1990). Olesiuk et al. (1990) estimated the mean age at the first reproduction to be 14.9 years, as well as a reproductive lifespan of 21-27 years with a mean of 25.2 years.

### Matrix Analyses

By analyzing a projection matrix, the asymptotic rate of population growth can be found by determining the dominant eigenvalue $\lambda$. The corresponding continuous-time rate is $r=log\lambda$. In addition, the stable stage structure and reproductive value are determined by the corresponding right and left eigenvectors, w and v. How sensitive $\lambda$ is to changes in the elements of the projection matrix $A$ is calculated by:

${\dfrac {\partial \lambda }{\partial a_{{ij}}}}={\dfrac {v_{i}w_{j}}{}}$

where $<>$ refers to the scalar product. Also, the elasticities of $\lambda$ are given by:

$e_{{ij}}={\dfrac {a_{{ij}}}{\lambda }}{\dfrac {\partial \lambda }{\partial a_{{ij}}}}$

These elasticity values not only sum to 1, but also provide the proportional contributions of the matrix elements to $\lambda$. The sensitivity and elasticity of $\lambda$ to lower-level parameters that determine the values of $a_{{ij}}$ can be found by using the chain rule.

A bootstrap resampling procedure was employed to construct confidence intervals for $\lambda$ and r. The bootstrapping method is a modern approach to statistical inference. This method involves estimating characteristics of an estimator by measuring these characteristics when sampling from an approximating distribution. The percentile method was used with a bootstrap sample size of 1000. An individual record from a set describing a population, sub-population, or a pod was used as the resampling unit.

The growth rate of the entire population was found to be $\lambda$= 1.0254 (r = 0.0251). With 90% confidence intervals, the bootstrap estimates of $\lambda$ and r are found in the table below.

Table 1: Bootstrap estimates with 90% confidence intervals

The values in the above table correspond well with the rate of population increase that was observed, which was found to be 1.0213. Furthermore, the value found by Olesiuk et al. (1990), $\lambda$= 1.0292, is situated within this confidence interval.

The stable stage distribution and reproductive value are given by:

$w={\begin{pmatrix}0.0369\\0.3159\\0.3227\\0.3244\end{pmatrix}}$

$v={\begin{pmatrix}1.0000\\1.0491\\1.5716\\0\end{pmatrix}}$

One can determine that the stable population structure corresponds well with the observed female structure which is averaged over the study period:

$w_{{obs}}={\begin{pmatrix}0.0368\\0.3778\\0.3627\\0.2226\end{pmatrix}}$

However, this is a crude estimate as it was assumed that half of the yearlings and juveniles of unknown sex are female. In addition, the stage-specific reproductive values correspond well with the age-specific reproductive values for the ages corresponding to the beginning of each stage, as described by Olesiuk et al. (1990). This point provides a reasonable comparison as the stages in the authors’ model are memoryless.

The sensitivity matrix, including only those to non-zero transitions, is given by:

$S={\begin{pmatrix}...&0.3608&0.3686&...\\0.0443&0.3785&...&...\\...&0.5670&0.5793&...\\...&...&0&0\end{pmatrix}}$

The elasticity matrix is given by:

$E={\begin{pmatrix}0&0.0015&0.0407&0\\0.0422&0.3363&0&0\\0&0.0407&0.5386&0\\0&0&0&0\end{pmatrix}}$

The above matrices describe the sensitivity and elasticity of $\lambda$ to changes in the population projection matrix, $A$. As the entries in the $A$matrix may depend on both growth and survival, sensitivities and elasticities of $\lambda$to changes in the lower-level parameters were also calculated. Please see Table 2 below.

Table 2: Sensitivity and elasticity of population growth rate to changes in lower level demographic parameters

### Sub-Population and Inter-Pod Differences

Once the data was modeled and parameters were estimated, the demographic differences between the northern and southern sub-populations, as well as the differences among the pods must be analyzed. In addition to the analysis, the differences must be tested for statistical significance, found by using nonparametric randomization tests. Considering the two sub-populations , the observed assortment of individuals into the sub-populations will provide an observed difference $\Delta \lambda$ in growth rate, possibly reflecting structural differences or real environment differences between the sub-populations; however, the differences between sub-populations might occur because their members represent sub-samples of entire populations. Following this null hypothesis the life experience of each individual and the sub-population it belongs to remain independent of each other. Furthermore, by examining all possible permutations of individuals in the two sub-populations and calculating $\Delta \lambda$ for each, the distribution of $\Delta \lambda$ under the null hypothesis can be obtained. If $\Delta \lambda$ exceeds the observed difference, the fraction of these permutations will give the probability of obtaining such a large difference. If the probability is small, the null hypothesis can be rejected. In this analysis, the number of permutations was massive, so they settled on 1000 variations found using Algorithm P of Knuth (1981), and used a two-tailed test based on the absolute value of $\Delta \lambda$.

#### Sub-Population Comparisons

The growth rates from the matrices for the northern and southern sub-populations become :

$\lambda ^{{N}}=1.0248$

$\lambda ^{{S}}=1.0249$

The bootstrap estimates and their 90% confidence intervals, calculated for the entire population, are:

North Sub-Population:

$Lowerlimit=1.0109$

$\lambda =1.0256$

$Upperlimit=1.0349$

South Sub-Population:

$Lowerlimit=1.0129$

$\lambda =1.0250$

$Upperlimit=1.0381$

Figure 3: The randomization distribution of the absolute value of the difference in population growth rate between the northern and southern sub-populations of killer whales, under the null hypothesis of no sub-population effect. The arrow indicates the location of the observed difference, which is not significantly large.

The observed growth rate of the northern population was found by the slope of the least-squares fit of log of population size versus time and is 1.0302, and the corresponding value for the southern sub-population is 1.0070. These values slightly differ from the values found by Olesiuk et al. because these calculations are based solely on the female population. From these values, it is shown that the southern sub-population is growing less rapidly than its asymptotic rate of increase, and has an observed value slightly outside of the 90% confidence interval.

Obtained through randomization, Figure 3 above shows the distribution for the difference in growth rate under the null hypothesis. The observed difference found is .0000801 which is not significantly large. Under the null hypothesis, this significantly small of a value happens less than 1% of the time.

Randomization tests were repeated many times, randomizing only females, only males, and only juveniles of unknown sex each time. By comparing these results separately, demographic differences affecting these groups can be tested; however, in this case, none of the randomizations yielded a significant result:

Groups randomized and the probability:

$All=.9930$

$Females=.9970$

$Males=.9431$

$Unknown=.9990$

In order to look more closely at these small differences in $\lambda$, it must be decomposed into contributions from the differences in the matrix elements. Where $\Delta \lambda$ becomes:

$\Delta \lambda =\Sigma _{{ij}}{\dfrac {\partial \lambda }{\partial a_{{ij}}}}\Delta a_{{ij}}$

where $\Delta a_{{ij}}$ is the difference in $a_{{ij}}$ between the two sub-populations in which southern is subtracted from northern. The summation contains each term representing the contribution of a vital rate difference to $\Delta \lambda$. The matrix of these contributions then becomes:

${\begin{pmatrix}0&0.00017&0.0053&0\\-0.00085&-0.0062&0&0\\0&-0.00042&.0021&0\\0&0&0&0\end{pmatrix}}$

These contributions sum to an absolute value of predicted growth rate of .0000823, which is close to the observed growth rate, indicating a good approximation.

It is concluded that $\Delta \lambda$ is small because there is an approximate balance between contributions from fertility and adult survival advantages in the southern population and the contribution of a juvenile survival advantage in the northern population, not because there are no differences in the vital rates.

The predicted and observed stable stage distributions for the two populations are:

$w_{{pred}}^{{N}}={\begin{pmatrix}0.0338\\0.3099\\0.3075\\0.3487\end{pmatrix}}$

$w_{{obs}}^{{N}}={\begin{pmatrix}0.0397\\0.4131\\0.4080\\0.1391\end{pmatrix}}$

$w_{{pred}}^{{S}}={\begin{pmatrix}0.0410\\0.3200\\0.3306\\0.3086\end{pmatrix}}$

$w_{{obs}}^{{S}}={\begin{pmatrix}0.0323\\0.3221\\0.2911\\0.3544\end{pmatrix}}$

The difference between the predicted and observed structures is non-significant for the southern sub-population:

$x^{2}=0.6598,df=3,P=0.8826$

but highly significant for the northern sub-population:

$x^{2}=15.17,df=3,P=0.0017$

Because the northern population exhibited more consistent exponential growth than the southern sub-population, this discovery is strange. It is expected that the stage distribution be closer to stable; however, most of the deviation of the northern sub-population is due to a deficiency in post-reproductive females in the observed structure. This deficiency contributes to the $x^{2}$ statistic, but because these females have zero reproductive value, the deficiency will not affect the dynamics of the population.

By repeating these calculations using a predicted age structure for the first three stages affecting population dynamics, the observed and predicted structures agree almost perfectly :

$x^{2}=0.0456,df=2,P=0.9774$.

#### Inter-Pod Comparisons

Please see Table 3 for the elements of the pod-specific projection matrices and Table 4 for the resulting r values and confidence intervals.

Table 3: This table lists the pod identification number (from Bigg et al. 1990), the pod size as of 1987, and the values of the matrix elements for each of the 18 pods used in our analyses of Pacific Northwest killer whales.
Table 4: This table includes estimates of $\lambda$, pod-specific population growth rates of Pacific Northwest killer whales, and the 90% bootstrap confidence intervals of those estimates.

The differences among the pods were determined by studying the observed variance in the growth rate, $\lambda$. The authors believe that the value of the variance should reflect the differences between pods in the external environment and in social structure. The null hypothesis is that the variance in $\lambda$ reflects only sampling. The null hypothesis was tested by randomly permuting individuals among pods, but maintaining the observed pod sizes. Then, the following inter-pod variance in $\lambda$ was calculated for each permutation. Given the null hypothesis, the probability of the observed variance was estimated as the fraction of the random permutations that produced a variance greater than or equal to the observed variance. The sub-populations were not distinguished from one another in this test, as there is no significant difference between them. The probability levels that were obtained from the randomization tests are shown below in Table 5.

Table 5: Probability levels resulting from randomization tests.

As shown in the table above, the variance in $\lambda$ between pods is not significantly greater than that anticipated with the given null hypothesis. One possible reason for this result is that variation in the vital rates may fail to appear as variance in $\lambda$. This would be because of patterns of correlation among the vital rates. To test this possibility, the authors conducted a multivariate randomization test on the vital rates. This test involved defining a vector-valued observation $x_{i}$ for each pod $i$, which was composed of the non-zero entries the $A$ matrix for that pod. The following covariance matrix of the $x_{i}$ summarizes the variation in vital rates.

$x})(x_i-\bar{x$

Where ${\bar {x}}$ is the mean of $x_{i}$. Then, the magnitude of $C$ is determined by the “generalized variance,” or the square of the determinant of $C$. The randomization test was followed as before. The null hypothesis was rejected if the observed generalized variance was significantly large compared to the randomization distribution.

The authors found that the observed generalized variance was not unusually large, as shown in Table 6.

Table 6: Observed generalized variance.

Hence, there was no evidence for significant inter-pod differences in demography at either the level of vital rates or of population growth rate.

#### Decomposition of inter-pod variance in $\lambda$

Despite the fact that inter-pod variance in growth rate cannot be separated from that resulting from the variety of individuals among pods, one can determine how the variance is produced. More specifically, the authors wanted to determine to what degree the variance in each vital rate, as well as the covariance between each pair of vital rates, contributes to the overall variance in $\lambda$. A particular vital rate may make a small contribution because it does not vary much or $\lambda$ is insensitive to its variation. Therefore, both the variance-covariance structure of the vital rates and the sensitivity analysis of $\lambda$ must be found.

As before $x_{i}$ is a vector made up of all the elements of the matrix for pod $i$ and by $C$, the covariance matrix of $x$. Vector $s$ contains the sensitivities evaluated at the mean matrix.

$s_{i}={\dfrac {\partial \lambda }{\partial x_{i}}}\mid _{{\bar {x}}}$

Then, the first-order approximation to the variance in $\lambda$ is given by:

$V(\lambda )\approx \sum _{{i,j}}\sum _{{k,l}}{\dfrac {\partial \lambda }{\partial a_{{ij}}}}{\dfrac {\partial \lambda }{\partial a_{{kl}}}}Cov(a_{{ij}},a_{{kl}})=s^{T}Cs$

A graphical representation of the covariance matrix of the $a_{{ij}}$ and the matrix of contributions of those covariances to $V(\lambda )$ is shown in Figure 4.

Figure 4: Upper: Surface plot of the covariance of the matrix elements $a_{{ij}}$ among pods. Lower: Surface plot of the contributions of the matrix entry covariances to the inter-pod variance in the population growth rate, $\lambda$.

The variances coincide with the diagonal elements of the graph, while the covariances coincide with the symmetric off-diagonal elements. Variances in $G_{1}$, $P_{2}$, and $F_{3}$ correspond to peaks A, C, and D, respectively. The covariance between $G_{1}$ and $P_{2}$ is shown as the pair of peaks labeled B, indicating a positive covariance between yearling and juvenile survival. Peaks A and B make almost no contribution to the variance in $\lambda$. In fact, almost all of the variance in $\lambda$ is a result of the variance in adult fertility.

#### Further Analysis of Sensitivities

To further investigate the sensitivities and elasticities of the model, we used MatLab to change certain parameters in the $A$ matrix and look at how this affects the spread of the dominant eigenvalues. We wrote a simple for loop that replaces the most and least sensitive parameters with random values from 0.9 to 1 for ten thousand instances. This code can be seen in Figure 5. We then made histograms of the spread of the dominant eigenvalues for the most and least sensitive of the parameters in the $A$ matrix. The most sensitive parameter, based on the sensitivity and elasticity matrices, was found to be P3, which is the probability that individuals in the adult stage will survive and remain in the reproductive adult stage. Analysis with this value assigned instead to a random value between 0.9 and 1 yields Figure 6. As we can see, the range of dominant eigenvalues is about 0.0565. The least sensitive parameter was found to be G1, which is the probability of surviving in the yearling stage and moving onto the juvenile stage. When we replaced this value in the $A$ matrix with a random number from 0.9 to 1 and observed the histogram of the dominant eigenvalues, we saw a much smaller range of dominant eigenvalues compared to when we changed P3, the most sensitive parameter. This histogram is shown below in Figure 7. The range in this case was about 0.0045. This supports the given sensitivity and elasticity matrices presented by the authors.

Figure 5: Code for investigating sensitivity and elasticity of P3 and G1.
Figure 6: Histogram of dominant eigenvalues for random value of P3.
Figure 7: Histogram of dominant eigenvalues for random value of G1.

## Discussion

### Population and Demographic Results

Our analysis of the entire killer whale population in British Columbia has shown that the population is increasing by about 2.5% each year. Referring back to the elasticity matrix, it is evident that this population growth rate is the most susceptible, or sensitive to, small perturbations in the survival of adult killer whales. The most important life stage after adults is the survival in the juvenile stage of development. The next most important part of the life history of killer whales is the fertility of the reproductive stages of killer whales. Small perturbations in the survival of the life history stages of adults and juveniles lead to large changes in the mortality of killer whales. So, the more adults and juveniles that survive in the population, the more the entire population increases. Any conservation efforts towards this specific population of killer whales should focus on keeping the survival rates of adult and juvenile life stages high, while not ignoring other important factor such as calf production and fertility.

### Sub-Populations and Comparisons Between Pods

There was only a small amount of demographic differences at the pod level and at the sub-population level. The northern and southern sub-populations had different fertility and adult survival numbers. For these two categories, the northern sub-population had higher values than the southern sub-population. The southern sub-population had higher values than the northern for juvenile survival. However, the contributions of all three of these differences to $\lambda$ almost completely cancel one another out, so the variation of $\lambda$ is not even significant in the data. From this lack of variation of the value of $\lambda$, pod size, social structure, and environmental variation seemed to have not significant demographic effects on the killer whales. Density-dependent factors may have contributed to this lack of demographic effect, so it would be useful to further investigate these factors. Any variation in the $\lambda$ value may come from factors that influence reproduction, such as variation in fertility. If one wanted to further investigate the variation of $\lambda$, components affecting reproductive success would be the place to start.

## Conclusions

Species like the killer whale, which have long life spans, benefit greatly from long-term studies that aim to further understand the population and sub-population dynamics of the species. Since the observational methods employed by Bigg et al. (1990) do not require capture or killing of the whales, this makes the experimental methods very applicable to endangered or threatened species. The more people know about species like the killer whale, the better equipped people will be in the future if the populations show any sign of declining. By knowing what aspects of the population are most sensitive in causing a dramatic increase or decrease in the entire population, conservation efforts will be aimed at survivability of a certain life stage, or at the fecundity of a certain stage. This type of analysis will hopefully benefit the conservation of many species in the future.

## Discussion of a Recent Paper Citing Brault and Caswell

Link, William, and Paul Doherty, Jr. "Scaling in Sensitivity Analysis." Ecology. 83. (2002): 3299-3305. Web. 28 Feb. 2012. <http://www.jstor.org/stable/3072080>.

In this paper, Link and Doherty, Jr. offer a solution to the apparent contradiction in conclusions made in the study by Brault and Caswell vs. the conclusions of a study of the same data done by Olesuik et al in 1990. As shown above, Brault and Caswell conclude that λ is more sensitive to changes in survival than fertility – the opposite of what Olesuik et al conclude. Brault and Caswell address this contradiction and make an argument for why their results are more relevant biologically. However, Link and Doherty, Jr. give several arguments against Brault’s and Caswell’s reasoning. They subsequently describe a scaling method that considers the demographic parameters as random variables and scales them so that the variation of the parameter is independent of its mean. Link and Doherty, Jr. claim that the choice of scaling should change depending on the parameter that is being scaled.

The choice for scaling and hence the method for doing sensitivity analysis offered by Link and Doherty, Jr. uses “variance stabilizing transformations.” Furthermore, their scaling choice uses a quantity, variance-stabilized sensitivity, they define as:

${\textrm {VSS}}_{q}(\lambda ,\theta )={\textrm {Sensitivity}}(ln(\lambda ),q(\theta ))$

where q(.) is a variance-stabilizing transformation for θ. More specifically, they offer an arcsine square-root transformation which creates equal rankings of demographic parameters and their complements (fixing a shortcoming, at least according to Link and Doherty, Jr., in Brault and Caswell’s reasoning). The paper continues with a reconstruction of the sensitivity analysis using the variance-stabilized sensitivity defined earlier in the paper. This analysis implies λ is most sensitive to the rate at which whales leave the adult reproductive stage and enter the post-reproductive stage, which is different from the conclusions of both Brault and Caswell as well as Olesuik et al.

Gleason, John R. "Algorithms for Balanced Bootstrap Simulations." The American Statistician 42.4 (1988): 263-66. JSTOR. Web. 19 Apr. 2010. <http://www.jstor.org/stable/2685134?cookieSet=1>.

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