
MBW:PodSpecific Demography of Killer WhalesFrom MathBioThis wiki page summarizes the paper "PodSpecific 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. ContentsProject Categorization
Executive SummaryIn 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 "PodSpecific 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 welldefined social groups called pods. Their study included 18 social units, known as pods, containing anywhere from 563 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 subpopulations 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 longlived 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 PodsOrcinus 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. 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 DevelopmentExperimentalThe 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 subpopulations. These subpopulations were called the northern and southern subpopulations. 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 DevelopmentThe 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 8090 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 ModelsMatrix ModelThis 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:
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 . To estimate these elements, estimated stagespecific survival probabilities and the probability of transitioning from one stage to the next, , and the average reproductive rate of mature females, , were used. These estimated probabilities and reproductive rates were estimated from the data collected by Bigg et al. (1990). The stageclassified birthflow formulation, proposed by Caswell (1989), was used to create the following formulas:
Another example of stage based matrix techniques can be found in Age Structured Populations in Seasonal Environments Matrix ParametersThe 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:
2.) The year that each individual reached maturity (known by the presence of a calf) 3.) The year at which individuals reached the postreproductive 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 determine the mean offspring production parameter, , the ratio of the number of female offspring produced by the pack to the number of femaleyears 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, , 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 . For each stage, “exposure” was different. To begin with, the yearlings () exposure was defined to be the total number of births. For the juveniles (), exposure was the number of years of observation of juveniles. For the adult stage (), exposure was defined as the number of years during which each individual was observed as a reproducing adult. For the postreproductive stage (), the exposure was estimated as the number of years of observation of any postreproductive individual. The growth probabilities () were determined to be the reciprocals of the averages of the stage durations. For the yearlings, would simply be one, since the average duration of this stage is one year, and 1/1 = 1. For the juvenile stage, 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, was defined as one over the average age that whales entered the postreproductive stage minus the sum of the average lengths of the yearling and juvenile stages. These calculations of , , and were done for the whole population, the northern and southern subpopulations, 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:
Matrix AnalysesBy analyzing a projection matrix, the asymptotic rate of population growth can be found by determining the dominant eigenvalue . The corresponding continuoustime rate is . In addition, the stable stage structure and reproductive value are determined by the corresponding right and left eigenvectors, w and v. How sensitive is to changes in the elements of the projection matrix is calculated by:
where refers to the scalar product. Also, the elasticities of are given by:
These elasticity values not only sum to 1, but also provide the proportional contributions of the matrix elements to . The sensitivity and elasticity of to lowerlevel parameters that determine the values of can be found by using the chain rule. A bootstrap resampling procedure was employed to construct confidence intervals for 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, subpopulation, or a pod was used as the resampling unit. The growth rate of the entire population was found to be = 1.0254 (r = 0.0251). With 90% confidence intervals, the bootstrap estimates of and r are found in the table below.
The stable stage distribution and reproductive value are given by:
The sensitivity matrix, including only those to nonzero transitions, is given by:
SubPopulation and InterPod DifferencesOnce the data was modeled and parameters were estimated, the demographic differences between the northern and southern subpopulations, 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 subpopulations , the observed assortment of individuals into the subpopulations will provide an observed difference in growth rate, possibly reflecting structural differences or real environment differences between the subpopulations; however, the differences between subpopulations might occur because their members represent subsamples of entire populations. Following this null hypothesis the life experience of each individual and the subpopulation it belongs to remain independent of each other. Furthermore, by examining all possible permutations of individuals in the two subpopulations and calculating for each, the distribution of under the null hypothesis can be obtained. If 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 twotailed test based on the absolute value of . SubPopulation ComparisonsThe growth rates from the matrices for the northern and southern subpopulations become :
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:
where is the difference in between the two subpopulations in which southern is subtracted from northern. The summation contains each term representing the contribution of a vital rate difference to . The matrix of these contributions then becomes:
It is concluded that 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:
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 : . InterPod ComparisonsPlease see Table 3 for the elements of the podspecific projection matrices and Table 4 for the resulting r values and confidence intervals.
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The authors found that the observed generalized variance was not unusually large, as shown in Table 6.
Decomposition of interpod variance inDespite the fact that interpod 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 . A particular vital rate may make a small contribution because it does not vary much or is insensitive to its variation. Therefore, both the variancecovariance structure of the vital rates and the sensitivity analysis of must be found. As before is a vector made up of all the elements of the matrix for pod and by , the covariance matrix of . Vector contains the sensitivities evaluated at the mean matrix.
Then, the firstorder approximation to the variance in is given by:
A graphical representation of the covariance matrix of the and the matrix of contributions of those covariances to is shown in Figure 4.
Further Analysis of SensitivitiesTo further investigate the sensitivities and elasticities of the model, we used MatLab to change certain parameters in the 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 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 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.
DiscussionPopulation and Demographic ResultsOur 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. SubPopulations and Comparisons Between PodsThere was only a small amount of demographic differences at the pod level and at the subpopulation level. The northern and southern subpopulations had different fertility and adult survival numbers. For these two categories, the northern subpopulation had higher values than the southern subpopulation. The southern subpopulation had higher values than the northern for juvenile survival. However, the contributions of all three of these differences to almost completely cancel one another out, so the variation of is not even significant in the data. From this lack of variation of the value of , pod size, social structure, and environmental variation seemed to have not significant demographic effects on the killer whales. Densitydependent factors may have contributed to this lack of demographic effect, so it would be useful to further investigate these factors. Any variation in the value may come from factors that influence reproduction, such as variation in fertility. If one wanted to further investigate the variation of , components affecting reproductive success would be the place to start. ConclusionsSpecies like the killer whale, which have long life spans, benefit greatly from longterm studies that aim to further understand the population and subpopulation 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 CaswellLink, William, and Paul Doherty, Jr. "Scaling in Sensitivity Analysis." Ecology. 83. (2002): 32993305. 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, variancestabilized sensitivity, they define as:
where q(.) is a variancestabilizing transformation for θ. More specifically, they offer an arcsine squareroot 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 variancestabilized 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 postreproductive stage, which is different from the conclusions of both Brault and Caswell as well as Olesuik et al. References and External LinksGleason, John R. "Algorithms for Balanced Bootstrap Simulations." The American Statistician 42.4 (1988): 26366. JSTOR. Web. 19 Apr. 2010. <http://www.jstor.org/stable/2685134?cookieSet=1>. "Killer Whale." Wikipedia, the Free Encyclopedia. Web. 17 Apr. 2010. <http://en.wikipedia.org/wiki/Killer_whale>. "Killer Whales." SeaWorld/Busch Gardens ANIMALS  HOME. Web. 17 Apr. 2010. <http://www.seaworld.org/infobooks/killerwhale/home.html>. "Surveying Killer Whale Abundance and Distribution In the Gulf of Alaska and Aleutian Islands." Alaska Fisheries Science Center Homepage. Web. 17 Apr. 2010. <http://www.afsc.noaa.gov/Quarterly/ond2003/featurelead.htm>. "Whale Chat." British Columbia Wild Killer Whale Adoption Program. Web. 17 Apr. 2010. <http://www.killerwhale.org/fieldnotes/chat.html>. Wikipedia contributors. "Population ecology." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 8 Feb. 2012. Web. 29 Feb. 2012.<http://en.wikipedia.org/wiki/Population_ecology>. Statsoft, "Nonparametic Statistics." Statsoft.com. Statsoft, n.d. Web. 29 Feb 2012. <http://www.statsoft.com/textbook/nonparametricstatistics/>. 