Linking Reproduction and Survival Can Improve Model Estimates of Vital Rates Derived from Limited Time-Series Counts of Pinnipeds and Other Species

<div><p>We propose a method to model the physiological link between somatic survival and reproductive output that reduces the number of parameters that need to be estimated by models designed to determine combinations of birth and death rates that produce historic counts of animal populations. We applied our Reproduction and Somatic Survival Linked (RSSL) method to the population counts of three species of North Pacific pinnipeds (harbor seals, <i>Phoca vitulina richardii</i> (Gray, 1864); northern fur seals, <i>Callorhinus ursinus</i> (L., 1758); and Steller sea lions, <i>Eumetopias jubatus</i> (Schreber, 1776))—and found our model outperformed traditional models when fitting vital rates to common types of limited datasets, such as those from counts of pups and adults. However, our model did not perform as well when these basic counts of animals were augmented with additional observations of ratios of juveniles to total non-pups. In this case, the failure of the ratios to improve model performance may indicate that the relationship between survival and reproduction is redefined or disassociated as populations change over time or that the ratio of juveniles to total non-pups is not a meaningful index of vital rates. Overall, our RSSL models show advantages to linking survival and reproduction within models to estimate the vital rates of pinnipeds and other species that have limited time-series of counts.</p></div

Final transition matrices for the Reproduction and Somatic Survival Linked (RSSL) model.

<p>A. The multiplication of Fertility <i>F</i> and Survival <i>S</i> values from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone-0077389-t001" target="_blank">Table 1</a> and the initial stationary population adjustment parameter <i>d</i>. B. Final RSSL Leslie matrix for Steller sea lions (<i>Eumetopias jubatus</i> (Schreber, 1776)).</p

Estimation of survival, maturity and pregnancy rates for Steller sea lions (<i>Eumetopias jubatus</i> (Schreber, 1776)).

<p>A. Adult female survival data from the York <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-York1" target="_blank">[7]</a> Weibull model, using age 3 to 30 y with the fitted Weibull curve (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389.e004" target="_blank">Eq. 2c</a>). B. Female maturity rate based on ovulation data with the fitted logistic curve (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389.e001" target="_blank">Eq. 1</a>). C. Female pregnancy rates from the result of fitting the multiplication of the survival curve and the maturity curve by adjusting the scalar of the logistic maturity function, here, only late term pregnancy data to age 20 was used to fit the model to the data, as the data from age 21 to 30 y is represented by only 3 individuals.</p

Numbers of estimated parameters for the fits to population data.

<p>RSSL is the Reproductive and Somatic Senescence Linked model, HFYS is Holmes et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Holmes1" target="_blank">[5]</a>, NFS is northern fur seals (<i>Callorhinus ursinus</i> (L., 1758)), HS is harbor seals(<i>Phoca vitulina richardii</i> (Gray., 1864)), Stnd is the standard models from the literature, J/T is the proportion of juveniles to total non-pups data, <i>σ</i> is the variance of the data and <i>p₂</i> and <i>p₃</i> are parameters in the Holmes et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Holmes1" target="_blank">[5]</a> model that relate the number of animals visually detected in counts to the actual numbers in the population.</p

Parameter summary for survival and maturity curves.

<p>The adjustment to <i>b_max</i> of the maturity logistic curve to fit the growth × maturity curve to pregnancy data (in parenthesis).</p

Survival, maturity and fertility parameter values for harbour seals.

<p>Adult and juvenile survival scalar, maturity scalar and fertility rates derived from all the harbor seal (<i>Phoca vitulina richardii</i> (Gray., 1864)) models. Italicized values are parameters that the models estimated. For the % change multipliers under the Reproduction and Survival Senescence Linked (RSSL) models, we used mean fertilities and adult survival over ages and time periods because these were non-linear functions of age and time. 95% confidence limits calculated via the hessian are in parenthesis. The non-RSSL models multiplied percentage change scalars directly with the Leslie matrix elements to affect changes over time while the RSSL changed <i>S_x</i> which consequently changed the Leslie matrix elements.</p

Estimation of survival, maturity and pregnancy rates for Harbor seals (<i>Phoca vitulina richardii</i> (Gray., 1864)).

<p>A. Survival data for both sexes and all ages showing the fitted polynomial for females age 4 to 27(<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389.e002" target="_blank">Eq. 2a</a>). Male survival rates were close enough to females from age 4 to 17 y that we used the female survival curve for males of those ages. B. Female maturity rate based ovulation data from ages 1–10 first age classes with the fitted logistic curve (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389.e001" target="_blank">Eq. 1</a>). C. Female pregnancy rates from the result of fitting the multiplication of the survival curve and the maturity curve extended out to age 27 y, by adjusting the scalar of the logistic maturity function, in this case up. All data are from Pitcher <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Pitcher1" target="_blank">[2]</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Pitcher2" target="_blank">[22]</a>.</p

Model AIC_c values.

<p>HFYS is Holmes et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Holmes1" target="_blank">[5]</a>, RSSL is Reproduction and Somatic Survival Linked, “w J/T” indicates that the Juvenile/(Juvenile+Adult) data was included in the Steller sea lion model and Standard indicates models created from unmodified Lander <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Lander1" target="_blank">[31]</a> or Pitcher <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Pitcher2" target="_blank">[22]</a> birth rates. The most parsimonious model based on AIC_c in each section is highlighted in bold.</p

Steller sea lion (<i>Eumetopias jubatus</i> (Schreber, 1776)) model fits.

<p>Reproduction and Somatic Survival Linked (RSSL) initial stationary population (A1–A2) and RSSL initial increasing population (B1–B2).</p

Time series counts for north pacific pinnipeds.

<p>A. Northern fur seals (<i>Callorhinus ursinus</i> (L., 1758)) at St. Paul Island, Alaska, from the National Marine Fisheries Service Annual Fur Seal Investigation Reports and Lander <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Lander2" target="_blank">[43]</a>, (B&C) Steller sea lions (<i>Eumetopias jubatus</i> (Schreber, 1776)) in the Central Gulf of Alaska from Holmes et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Holmes1" target="_blank">[5]</a> and (D) harbor seals (<i>Phoca vitulina richardii</i> (Gray., 1864)) at Tugidak Island, Alaska, from Jemison et al. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0077389#pone.0077389-Jemison1" target="_blank">[44]</a>.</p