This type of ecological control is called "bottom-up" control. Sandeels may induce bottom-up control on harbour porpoises. The large fishing industry that developed during the 1980s contributed to the decline of the sandeel stock in the North Sea (Furness 1999). Furthermore, the recruitment of sandeels in the North Sea has decreased since the 2000s, with a possible combined influence of fishing and climate change (ICES 2006) through an impact on plankton communities (Beaugrand et al. 2002, 2003; Frederiksen et al. 2006). The decrease in sandeel stocks and quality has had an impact on harbour porpoises, with greater starvation and lower survival rates in the North Sea (MacLeod et al. 2007). It is possible that this limited access to the resource has led to a migration of harbour porpoise from the North Sea to adjacent waters (Hammond et al. 2017, 2002, 1995).

The bottom-up of sandeels over har-
bour porpoises, combined with a de-
cline in the sandeels stock, may have
caused a change in the distribution of
harbour porpoises in the North-East
Atlantic.

<figure><img src="image_130.png" /><figcaption>**Figure 7.9: Abundance estimates of common dolphin by ICES divisions.**
The abundance estimates seem to be slightly higher for SCANS III than for SCANS II + CODA. Estimates from Hammond et al. (2017).</figcaption></figure>

It is possible that the common dolphin population is at a stationary abundance throughout the Northeast Atlantic and that the demographic rates estimated in this project are related to part of the population sampled in the Bay of Biscay. There may be more prey of interest available in the Bay of Biscay than in the past. Fishing mortality pressure in the Bay of Biscay would be higher than in other parts of the North-East Atlantic, but population abundance would be essentially stationary on a large scale. Abundance surveys in the North-East Atlantic of SCANS II + CODA and SCANS III seem to show that the abundance of the population is almost stationary or perhaps increasing very slightly. Recent works by Evans and Waggitt (2020) also supports an increasing trend in abundance observed in the North-East Atlantic using Waggitt et al. (2020). Recently, Astarloa et al. (2021) also highlighted a trend of increasing abundance of common dolphin in the Bay of Biscay and the correlation of this trend with environmental and biological

The demographic rates obtained in
the Bay of Biscay may not be repre-
sentative of the whole North-East At-
lantic. It is possible that bycatch pres-
sure is responsible for the observed
values.
variables using the Vector-Autoregressive Spatio-Temporal (VAST) model (Thorson 2019). The authors point out that the increase in estimated abundance is not necessarily a sign of a growing population but may reflect migration from unsampled areas. The correlation between common dolphin abundance and the North Atlantic Oscillation Index, prey abundance and chlorophyll is studied (Figure 7.10). Some correlations are highlighted but it is difficult to attribute the possible change in distribution of common dolphins to these variables. It is difficult to link variations in abundance or changes in distribution of common dolphins to environmental and prey data. This is mainly due to the scarcity of data and the rare co-occurrence of the two predator-prey system data in the large scales of the phenomena of interest (Gilbert et al. 2021; Astarloa et al. 2021).

<figure><img src="image_131.png" /><figcaption>**Figure 7.10:** Correlation between common dolphin abundance and ecosystem variables. Abundance estimates predicted by the baseline spatio-temporal model (black line) and by the covariates-based model (with no random effects, colored line) so that the contribution made by each variable (A–F) can be visualized.
Figure from Astarloa et al. (2021).</figcaption></figure>
**Abundance estimates: an incomplete indicator**

Abundance is an indicator of interest to assess the "Good Environmental Status" of marine European waters (European Parliament 2008) but it may be incomplete to assess the viability of megafauna populations. This indicator is considered to fulfil Descriptor 1 (D1²) and Descriptor 4 (D4³). Abundance assessment at the scale of a particular population is very instructive since it provides an initial assessment of the population's state (Laran et al. 2017). At the scale of a community (e.g. cetaceans in the Bay of Biscay) it can also inform the biodiversity state of the cetaceans community which is an important aspect of the D4 criterion (Azzellino et al. 2014; Lauriano et al. 2014).

<figure><img src="image_132.png" /></figure>

2: https://mcc.jrc.ec.europa.eu/main/dev.py?N=19&0=118& titre_chap=D1%20Biological%20diversity

3: https://mcc.jrc.ec.europa.eu/main/dev.py?N=22&0=133& titre_page=& titre_chap=D4%20Food%20webs

**Figure 7.11: Percent of stocks within four ranges of statistical power.** Statistical power is the probability of correctly rejecting the null hypothesis that a population is not declining (i.e., one-tailed t-test) when the stock is experiencing a precipitous decline (50% over 15 yr). Results are summarized for six categories of marine mammals: large whales, beaked whales, dolphins and porpoises, pinnipeds on ice, pinnipeds on land, and polar bears and sea otters.

Figure from (B. L. Taylor et al. 2007).

The ability to detect population declines in abundance below a critical threshold remains poor for many marine mammal species. Abundance trend cannot be effective in identifying populations at risk because the statistical power to resolve abundance trends is sometimes very low if the time series is sparse (B. L. Taylor et al. 2000). The statistical power to detect a precipitous decline through abundance studies was calculated by B. L. Taylor et al. (2007) (Figure 7.11). A precipitous decline is defined by the authors as a "decline of 50% over a 15-yr monitoring period". They justify this choice by the fact that such a decline would lead to the classification of a stock as "depleted" under the MMPA and as "vulnerable" or "endangered" under the IUCN Red List guidelines (B. L. Taylor et al. 2007). In Figure 7.11, we can see that the statistical power (probability of correctly rejecting the null hypothesis that a population is not declining) is very low for dolphins and porpoises population (the common dolphin was not included in their analyses). These results indicate that the use of abundance estimates to detect precipitous declines is not appropriate for small cetacean populations.

Using abundance trends alone to define the status of small cetacean populations can be dangerous due to low statistical power.
## Examples of delays between the onset of extrinsic mortality and the implementation of management measures

Uncertainty in abundance indicators may limit the conservation of small cetaceans affected by bycatch (B. L. Taylor et al. 2000). There may be a delay between the onset of pressure and the implementation of management measures. This delay may be detrimental to the viability of populations. Two examples will follow to support this point.

### Harbour porpoise in California

The first example concerns the harbour porpoise in California. In the mid-1980s, an increasing number of porpoises were found stranded on the shoreline, as well as an expansion of coastal gillnet fishing in central California. These two observations led to the assumption that the population was threatened by fishing. To monitor this population, mortality was assessed by an observer program and porpoise abundance was estimated by surveys and back-calculation from fishing effort and mortality rates (B. L. Taylor et al. 2000). Uncertainty in the parameters tended to make it impossible to determine the level of depletion and the MNPL. The MMPA was not able to conserve this species and its preservation was achieved indirectly through fishery closures associated with the capture of sea otters (B. L. Taylor et al. 2000). This example shows that the mere availability of some abundance and bycatch data does not allow the implementation of conservation measures at the right time.

The harbour porpoise in California and the Tuna-Dolphin problem are two representative examples of the delay between the onset of depletion, its recognition and the implementation of management measures.

### Tuna-Dolphin problem

The second example involves many steps over 60 years, encompassing many areas of expertise (biology, conservation policy, legislation), media coverage and the creation of a label for consumers (Supplementary-Figure 1). It is known as the "Tuna-Dolphin Problem" and has recently been the subject of a review made by Ballance et al. (2021). The history of this problem is extensive and is provided in Appendix 7.4. In the East Pacific Ocean, large-bodied yellowfin tuna (*Thunnus albacares*, Bonnaterre 1788) associate with several species of dolphins such as the pantropical spotted (*Stenella attenuata*, Gray 1846), the spinner (*Stenella longirostris*, Gray 1828) and the short-beaked common dolphin. These dolphins interact together with birds in a beneficial way (Scott et al. 2012; Ballance, Pitman, and Reilly 1997). These associations are readily visible at the surface and provide some indication to purse seine fisheries targeting tuna. Purse-seine are deployed near dolphin schools,

<figure><img src="image_133.png" /><figcaption>**Figure 7.12:** Aerial photograph of a purse-seine set on a school of tuna and dolphins. The purse-seine vessel is deploying the net in a large circle around the entire school while a skiff holds the end of the net in place. In this photograph the net is not yet closed; four speedboats are driving in tight circles near the opening to prevent the dolphins (and tuna) from escaping.
Photo from Ballance et al. (2021).</figcaption></figure>
resulting in high levels of bycatch (Figure 7.12). All three dolphin species were recognised as bycaught in the purse-seine Tuna fishery (Perrin 1969; Figure 7.13). Catches were in the hundreds of thousands between 1960 and 1995. Between 1995 and 2020, they fell to around a thousand.

<figure><img src="image_134.png" /></figure>

**Figure 7.13: Estimated number of dolphins killed annually in the eastern tropical Pacific tuna purse-seine fishery.** Total for all dolphins and separately for the stocks of the two dolphin species with the highest number killed. The inset graph has an expanded vertical scale to show details from 2000 to 2019 (change of scale on the Y-axis).

Figure from Ballance et al. (2021).

The observation of bycatch numbers from 1960 to 1970, and in particular the outcry over the magnitude of dolphin mortality, prompted the establishment of the MMPA in 1972, which aims to reduce bycatch mortality to a level close to zero. Given the degree of uncertainty regarding the status of the population, numerous data acquisition efforts have been made to justify its depleted status. In total, "nine abundance surveys over 12 years, 17 years of relative trend indices from data collected on the tuna vessels, 25 years of observer data on dolphin mortality rates in the fishery, and 34 years of data on fishing effort" have been required. From the first abundance estimate to the recognition of the depleted status of the population, 23 years of inaction were reported (Wade 1994). By the late 1970s, it was clear that the dolphin mortality was too high relatively to the populations abundances estimates. However, it was only after sufficient data had been collected and analytical methods developed, *i.e.* until 1993, that it was possible to prove that the dolphin populations were reduced to 44% of its total abundance (Wade 1993a,b). At this time, the populations were declared depleted under the MMPA (depletion being defined as stock abundance below 60% of carrying capacity; Ballance et al. 2021).

This example is similar to the case of the common dolphin in the North-East Atlantic. There is currently uncertainty about the true status of the population, given the abundance estimates and the poor usability of on-board observer data. Although we have developed new methods and provided new bycatch estimates from a biased observer scheme, it is not possible to estimate the total
bycatch mortality of the population due to the lack of adequate data (and not because of a lack of methodology, e.g. see Chapter 5). Using only abundance estimates, observer programmes and stranding mortality monitoring, it is possible that the mismatch between the actual time of depletion and the recognised time of depletion is dangerous for the viability of the population.

## 7.3 Population dynamics of elusive species

### Life-history traits variations

<figure><img src="image_135.png" /></figure>

**Figure 7.14:** raphical representation of predicted changes in optimal reproductive effort (RE) of a generalized life history. RE is defined as the proportion of resources allocated to reproduction as opposed to investment in growth or survival. The age at which RE increases from zero indicates age at maturity. The linearly increasing depiction of RE with age is arbitrary, and plot A is the baseline iteroparous life history. Plots B–D give predictions for unregulated populations growing exponentially after: (B) mortality increases only for older age classes (earlier age at maturity and higher RE at each age), (C) mortality increases only for younger age classes (later age at maturity and lower RE at each age), and (D) mortality increases uniformly across all ages (no change from baseline). Plots E and F give predictions for regulated populations subjected to density dependence: the effect of a uniform increase in mortality when density dependence affects (E) only juveniles (note the increase in RE over baseline and earlier age at maturity) vs. (F) all ages (no change from baseline). Figure from Reznick, Bryant, and Bashey (2002).

We found a possible decrease in the age at sexual maturity of females compared to the methodology commonly used in the literature. The age at sexual maturity of females decreases by about one year according to these results (8 to 7 years). Using the joint model and random effects approach, it appeared that the age at sexual maturity of females also appeared to decrease over the period considered. It can be assumed that this possible change is part of a life history optimisation (Ferriére and J. Clobert 1992). Age at sexual maturity should respond differently depending on the mortality experienced by the population and on the presence of density-dependent mortality (Reznick, Bryant, and Bashey 2002;
Figure 7.14). It is possible that increased adult mortality in the population favours genotypes that reproduce earlier. This assumption has also been made by Perrin, J. R. Henderson, et al.; Barlow; Chivers and Myrick Jr for the North-East Pacific dolphins (Perrin, J. R. Henderson, et al. 1979; Barlow 1989; Chivers and Myrick Jr 1993). The survivorship also seemed to decrease according to the results of the joint modelling approach. There may be a link between age at sexual maturity of females and survivorship of adults. This link could depend on the extrinsic mortality rate (Ricklefs 2010). It is possible that the extrinsic mortality experienced by adults decreases their maximum potential longevity and thus their age at sexual maturity.

The population may undergo a pro-
cess of variation in vital rates with
a shorter life history than before
(shorter life expectancy and advanced
reproduction).

Loss of interest in cross-sectional monitoring

<figure><img src="image_136.png" /><figcaption>**Figure 7.15:** Cumulative number of studies published prior to 27.07.2015 containing animal matrix population models (MPMs). Some Important events in the development of animal MPMs: (a, b) applications of matrix population models in demography, (f) nonlinear, density-dependent MPMs for animals, (g) sensitivity analysis for stage-classified MPMs and calculation of selection gradients for animals, (i) calculation of the stochastic growth rate from an animal MPM, (k) application of elasticity analysis to conservation biology and Life Table Response Experiment analysis, (n) presentation of multistate mark–recapture methods for estimating stage-structured MPMs in animals, (o) development of MPM from photograph identification data.
Figure from Salguero-Gómez et al. (2016).</figcaption></figure>

Longitudinal monitoring is now favoured to study the de-
mography of long-lived megafauna species. Monitoring of these
species traditionally used age-at-death data or hunting bags to
obtain vital rates (cross-sectional monitoring). The most important
assumption regarding these data is that the population must have
a stationary age-distribution (Caughley 1966). In the wild, this
assumption is rarely met (McCullough et al. 1994; Menkens and
Boyce 1993) and it is even more difficult to achieve it with the
large marine mammals whose strandings are used (Barlow and
Hohn 1984; Stolen and Barlow 2003). Demographic rates estimated
by cross-sectional monitoring may not be reliable in these cases.
Longitudinal monitoring (when feasible) and the associated analy-
ses, have therefore been favoured and have undergone significant
development. Since the Cormack-Jolly-Seber (Seber 1965; Cormack
1964; Jolly 1965) approach, numerous developments in modelling
and sampling protocols have taken place (Lebreton et al. 1992).
The 1990s saw many developments in the use of CMR and radio
telemetry protocols to estimate survival rates (Lebreton, Pradel,
and J. Clobert 1993; Gaillard, Festa-Bianchet, and Yoccoz 1998).
Survival analyses based on longitudinal monitoring is now an
entire well developed scientific field in its own with a firm statistic
basis (Gaillard, Festa-Bianchet, and Yoccoz 1998) and can be used to
feed demographic databases such as COMADRE (Salguero-Gómez
et al. 2016; Figure 7.15).

Cross-sectional monitoring remains the most suitable ap-
proach for some marine megafauna species that may be elusive. It is
also the only possible approach for fish populations and exploited
stocks. For some species of marine megafauna, there is a lack of
demographic information (Heppell, Caswell, and Crowder 2000;
Moore and A. J. Read 2008) and it may be necessary to conduct
population viability analyses (W. F. Morris et al. 2002). For marine
mammals associated with bycatch, population managers are tasked
to make decision (e.g. fisheries closures) despite a large amount of
uncertainty (Thompson et al. 2000; Harwood 2000). Since the devel-
opment of Siler (1979) and Heligman and Pollard (1980) approaches,
there has been few methodological development associated with
cross-sectional monitoring. A recent improvement has been the
development of the package strandCet (Saavedra 2018) which is
interesting for taking into account additional mortality and the
treatment of sampling biased age classes (e.g. due to bycatch). A
question arises: why is the community so reluctant to produce vital
rates from cross-sectional monitoring?

The contribution of this project to cross-sectional approaches

Obtaining vital rates from cross-sectional monitoring of
long-lived megafauna species is limited by the lack of sampling
control and the limited possibility to measure variability in vital
rates (Gaillard, Festa-Bianchet, and Yoccoz 1998). One of the points
of this work was to partially address these limitations.

With regard to the first item, the use of stratified random sampling
is intended to allow for better control of the sampling. This control
takes place prior to the analysis of the data and aims to avoid
introducing additional uncontrolled bias.

The use of post-stratification to quantify human pressures (e.g. bycatch) is also a type of correction. Here, it is carried out during the analysis, in order to take into account possible biases in the

Longitudinal monitoring has been
favoured for the study of long-lived
megafauna species for which such
monitoring is feasible. The associ-
ated survival analyses have devel-
oped strongly since the 1990s and
longitudinal analysis is now a field of
statistical analysis in its own right.

Cross-sectional monitoring has not
received the same attention from the
community. Despite the fact that it
is the only feasible monitoring for
elusive species and fish stocks.
representativeness of the data. This is of interest for elusive species, such as the common dolphin, whose bycatch is not well informed. The inclusion of covariates and random effects in the joint model to quantify vital rates is also an improvement in approaches. With this development it is possible to quantify the effect of different factors on vital rates and to highlight variations at the population level.

## Application of the demographic approach to other megafauna species

When available, age-at-death data should be used to their fullest potential (Ferguson 2002). Here I propose an analytical and a conceptual framework which incorporates important elements of this thesis project (Figure 7.16). This diagram incorporates both elements of the literature and elements developed within the framework of the project for illustrative purpose.

<figure><img src="image_137.png" /><figcaption>**Figure 7.16:** Diagram of the proposed framework along with the example in our case. The steps are defined to allow for a better consideration of representativeness bias. This framework is primarily a point of discussion and not definitive.</figcaption></figure>
Odontocete species

This framework is transposable to other odontocete species
and can serve different purposes depending on the problems
associated with the population. It is preferable that data be collected
by a stable and centralised stranding network. The more stable
the stranding network is over time and centralised, the longer and
more representative the period covered can be. This reduces the
bias associated with collecting stranded individuals.

Harbour porpoise in North-East Atlantic

<figure><img src="image_138.png" /><figcaption>Figure 7.17: Age-frequency distribution and sample sizes of female and male harbor porpoises sampled within the North Sea MU and Celtic and Irish seas MU during the two time periods, 1990–1999 and 2000–2012. These data could be analysed with our joint modelling approach. This would highlight possible differences in vital rates values depending on area and/or sex.</figcaption></figure>

Figure from Murphy et al. (2020)

The harbour porpoise is also a species for which there are
strong conservation issues in European waters (ICES 2020b). The
Baltic Sea population is now listed as critically endangered (Benke
et al. 2014). In the North-East Atlantic the Harbour population
is distributed from Norway to southern Spain. Two genetically
distinct stocks are recognised, the first in Spain, the second between
France and Norway (Fontaine et al. 2007). The population has been
divided into several management units based on genetic differenti-
ation but also on morphological differences (Evans and Teilmann
2009; JNCC 2015; Margin-Figure 7.18). However, it appears that
the overall population is subject to all types of pressure (Murphy
et al. 2020), notably primary (bycatch) (ICES 2020b), secondary
(pollution disruption) (Murphy et al. 2015) and tertiary (noise
disturbance) (Brandt et al. 2011). It is also supposed that climate
change influences the physical condition of harbour porpoises
(Heide-Jørgensen et al. 2011). With regard to population trends,
declining trends have been observed in the Celtic and Irish Seas
(NAMMCO, IMR 2019).

Some biological parameters are well known for the overall pop-
ulation and the different stocks (e.g. age-at-death in Figure 7.17)
but there are still uncertainties about the population dynamics of
these stocks (Murphy et al. 2020). It could be of interest to apply
the framework used here to produce stock-specific population
rates (via covariates). This will also allow the identification of
possible changes in life history traits over time as a function of
stock to explore possible exchanges between sinks and sources. his
would help to refine and prioritise the management interest of the
different units recognised by the MSFD.

Hector’s dolphin

Dolphins of the genus *Cephalorhynchus* are distributed in the Southern hemisphere in South-America, South Africa and New Zealand. The Hectors'dolphin (*Cephalorhynchus hectori*, van Beneden 1881) is the only species of this genus to occur in New Zealand waters (A. N. Baker 1978) has recently been estimated to be around 15,000 individuals (Annex D 2017). The biological parameters of this species were determined with an average lifespan of 20 years and an age at sexual maturity between 6 and 9 years for females (Slooten 1991). From the age distribution of the dolphins obtained from standings, the survivorship of the population and its growth rate were determined using the Siler model and a Leslie matrix approach (Slooten and Lad 1991). The population appears to have a low growth rate and entanglement in gillnet fisheries is suspected to be an important source of additional mortality for this population, as is predation by sharks (Dawson and Slooten 1993; Slooten, Fletcher, and B. L. Taylor 2000). A sub-species of Hector's dolphin, the Maui's dolphin (*Cephalorhynchus hectori maui*, Baker 2002) lives in the north of New Zealand (Margin-Figure 7.19). The population of this subspecies has fallen to about 60 individuals (C. S. Baker et al. 2016). It might be interesting to use a similar methodology to that employed in this project on the Hectors dolphin population. The population size of 15,000 individuals could provide enough strandings to achieve vital rates. The advantage could be to use the long age-at-death data time serie (without using only females samples) and discriminate the impact of predation and bycatch on the value of survival rates as covariate effect. The use of the year of stranding could be of interest to identify a possible shift in life history-traits.

<figure><img src="image_139.png" /><figcaption>Figure 7.18: Map showing Recom-
mended Management Units for Har-
bour Porpoise in the ASCOBANS
Agreement Area and Environ.

Figure from Evans and Teilmann
(2009).</figcaption></figure>

Figure from Evans and Teilmann (2009).

<figure><img src="image_140.png" /><figcaption>Figure 7.19: Hector's dolphin draw and distribution. Hector's dolphins are found along the east coast of the South Island. They are found down the West Coast to Jackson Bay and Fiordland. On the south coast, two bays east and west of Invercargill have Hector's dolphins.</figcaption></figure>

Draw from https://www.fisheries.noaa.gov/species/hectors-dolphin.

Distribution from https://teara.govt.nz/en/map/4675/distribution-of-hectors-and-mauis-dolphins.
## River dolphins

River and coastal dolphin populations can be monitored both on the basis of individuals found stranded and from capture-recapture programmes. It is the case for the population of Irrawaddy dolphins (*Orcaella brevirostris*, Owen 1866) that lives in the Mekong river (Cambodia) and is currently classified as Critically Endangered by the IUCN (Beasley et al. 2013). It was possible to conduct the two surveys and compare their results. Survival rates were both estimated by a carcass recovery programme and a capture-recapture programme. The authors concluded that the comparison was interesting and provided more information on the bias that could impact the survival estimate for the two monitoring (Beasley et al. 2013). Coastal dolphins species can be followed through capture-recapture studies such as the Guiana dolphins (*Sotalia guianensis*, van Beneden 1864) (Cantor et al. 2012). It could be of interest to use age-at-death data to compare between the estimates made through CMR monitorings and carcass recoveries for similar species.

<figure><img src="image_141.png" /><figcaption>Pink dolphin in colombia © Jaime Rojo.</figcaption></figure>

## Mammal and/or megafauna species

The current challenges in terms of the biodiversity crisis linked to direct anthropogenic impact or indirectly through climate change, require a global cooperative effort to acquire biological data (Urban et al. 2016). Vital rates are an important source of information for understanding and predicting the effects of anthropogenic impacts and climate change (Paniw et al. 2021). Demographic databases are particularly interesting given the issues outlined above and allow both sharing and a global effort to produce demographic indicators (Salguero-Gómez et al. 2016). These databases also provide good sources of information for generating life history knowledge and are important today in a context of digitisation of ecology that requires good and reproducible research practices.

The European wild boar (*Sus scrofa*, Linnaeus 1758) is studied using several forms of monitoring (hunting bags, CMR monitoring, age or stage matrix models): (Sáaez-Royuela and Telleriia 1986; Keuling et al. 2013; Massei et al. 2015). For this species, it might be interesting to compare age-at-death estimates and individual long-term monitoring.

The Arctic fox (*Vulpes lagopus*, Linnaeus 1758) is a mammalian species for which little demographic information exists (Tannerfeldt and Angerbjörn 1996; Meijer, Norén, and Angerbjörn 2011). The analysis of the teeth of this species allows the determination of variations in their feeding ecology in addition to age (Ungar et al. 2021). With the approach of this thesis, it would be possible
to link ecological variations and vital rates of threatened Arctic fox populations.

Crocodilians are an order of sauropsids for which little informa-
tion is available on population trends. They may be sensitive to
pressures such as mercury contamination. The few demographic
studies focus mainly on the Mississippi alligator (Alligator mississip-
piensis, Daudin 1802; D. Taylor and Neal 1984). Age is approximated
by body size (Hutton 1986; Eaton and Link 2011). Although it ap-
pears that age-based matrix models do not necessarily capture
the effect of factors on populations for crocodilians, it may be of
interest to provide baseline estimates of survival rates.

Some species are important to indigenous human populations. The S'ami people depended on the presence of reindeer (Rangifer tarandus, Linnaeus 1758), which they partially domesticated. This is no longer the case, but there is still a strong link between this people and the presence of reindeer in the Arctic. Other peoples such as the Inuit still hunt reindeer. This species seems to be impacted by climate change: (Heggberget, Gaare, and Ball 2002; Moen 2008). Recent progress has made it possible to determine the age at death of this species (Van den Berg, Loonen, and Çakırlar 2021). It could be interesting to link the survival of this species to variations in the environment and food resources in order to better understand its dynamics and to favour the conservation of the culture of certain indigenous peoples.

<figure><img src="image_142.png" /><figcaption>**Arctic fox** © Marco Gaiotti.</figcaption></figure>
## 7.4 Conclusion and perspectives

This thesis project has provided a new and more flexible approach to survival modelling based on linear regression. This approach allowed the inclusion of covariates and random effects. It seems appropriate to model the survival of elusive species taking into account covariates and random effects. This project allowed a new assessment of the common dolphin population in the Bay of Biscay to be conducted. The stratified random sampling developed in the demographic study allowed a better representation of the stranded population and avoided potential selection biases that could affect the survival estimate. The survival modelling approach developed in the project allowed the variation in vital rates over time to be modelled. It also highlighted the effect of bycatch or sex. The associated population projections showed that the viability of the population has potentially deteriorated over the time period considered. The viability of the population appears to be strongly threatened by bycatch as an external mortality force. The multilevel regression with post-stratification approach adapted to estimate bycatches allowed for the first time to use an on-board observer dataset despite its known biases. It also provided an estimate of bycatch and associated parameters at a spatial and temporal scale never achieved before for common dolphins in the Bay of Biscay. Ultimately, the management strategy evaluation approach allows the combination of demographic and bycatch estimates to define sustainable removal limits for the Bay of Biscay common dolphin population. This is primarily a feasibility exercise as the study area considered does not take into account the entire management unit currently recognised for the common dolphin. This approach is the first to take into account several types of data specific to this population in order to define management strategies with regard to the conservation objective proposed by OMMEG in European waters.

This work is important from both a fundamental and an applied point of view. Elusive species are defined in this project as species that may have one or more of the following characteristics: difficult to monitor individually, sparsely distributed, difficult to estimate anthropogenic mortality and difficult to implement management policies. The demographic approach developed in this project is an interesting contribution to the literature on these species. It addresses some of the issues inherent to cross-sectional monitoring and analyses. It shows that the sampling of individuals is important to minimise bias in the estimation of vital rates. It provides a method for estimating vital rates that allows the effect of different variables on vital rates to be estimated. It could be very interesting to apply this modelling approach to other elusive species such as the harbour porpoises in the North-East Atlantic. This population presents major conservation challenges. The number of strandings seems to be sufficient for stratified random sampling. We also have long time series that would allow us to test for variation in vital rates over time. The covariates to be tested are also multiple. We have data on contamination by pollutants. We also have strandings in several management units, which would allow us to highlight differences in viability between populations.

The work carried out in this project is mainly associated with the Bay of Biscay. However, the current management unit of the common dolphin is much larger and includes a large part of European Atlantic waters. It is difficult to generalise the demographic results and removal thresholds obtained in this project to the whole management unit. It might be interesting to apply the project's approach using further data covering more of the European coastline. But for the time being, the demographic estimates produced can directly feed into the MSFD. It would even be possible and interesting to revise the classification of demographic indicators within the framework of the MSFD. The work carried out could provide an operational indicator that would make it
possible to highlight an alteration in viability before it is translated into a depletion that could be
detected by abundance surveys.
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<figure><img src="image_143.png" /></figure>
# APPENDIX
<figure><img src="image_144.png" /></figure>
Appendix 1: Abundance estimates and MSFD
Marine regions

<table><caption>Figure 1: Estimated abundances of common dolphin and stripped dolphin for the SCANS II, CODA, SAMM and SCANS III surveys. Abundances are provided by survey cruise, by block and by species. Table formatted by Olivier Van Canneyt, Observatoire Pelagis.</caption><thead><tr><th colspan="10">European atlantic waters - summer</th></tr><tr><th colspan="10">2005-2007</th></tr><tr><th></th><th></th><th>Common dolphin</th><th>Animal Abundance</th><th>CV</th><th>CL low</th><th>CL high</th><th>Animal Density</th><th>Surface (km²)</th></tr></thead><tbody><tr><td>SCANS 2 Summer 2005</td><td></td><td>Common or striped dolphin</td><td>63 366</td><td>0.46</td><td>26 973</td><td>148 865</td><td>0.05</td><td>1 370 114</td></tr><tr><td>CODA Summer 2007</td><td></td><td>Common or striped dolphin</td><td>89 404</td><td>0.35</td><td>46 110</td><td>173 349</td><td>0.07</td><td></td></tr><tr><td></td><td></td><td>Common or striped dolphin</td><td>162 266</td><td>0.46</td><td>65 990</td><td>399 001</td><td>0.17</td><td>967 538</td></tr><tr><td></td><td></td><td>Common or striped dolphin</td><td>282 749</td><td>0.38</td><td>134 189</td><td>595 780</td><td>0.29</td><td></td></tr><tr><td><strong>Total SCANS 2 + CODA</strong></td><td></td><td>Common dolphin + common or striped</td><td><strong>597 785</strong></td><td><strong>0.20</strong></td><td><strong>273 262</strong></td><td><strong>1 316 995</strong></td><td><strong>0.26</strong></td><td><strong>2 337 652</strong></td></tr><tr><th colspan="10">2016</th></tr><tr><td>SCANS 3 Summer 2016</td><td></td><td>Common dolphin</td><td>467 673</td><td>0.26</td><td>281 129</td><td>777 998</td><td>0.26</td><td>1 816 137</td></tr><tr><td></td><td></td><td>Common or striped dolphin</td><td>158 167</td><td>0.19</td><td>109 689</td><td>228 069</td><td>0.09</td><td></td></tr><tr><td>ObServe Summer 2016</td><td></td><td>Common dolphin</td><td>13 632</td><td>0.86</td><td>5 213</td><td>35 645</td><td>0.02</td><td>820 000</td></tr><tr><td></td><td></td><td>Common or striped dolphin</td><td>33 214</td><td>0.44</td><td>19 331</td><td>57 069</td><td>0.04</td><td></td></tr><tr><td><strong>Total SCANS 3 + Observe</strong></td><td></td><td>Common dolphin + common or striped</td><td><strong>672 686</strong></td><td><strong>0.08</strong></td><td><strong>415 362</strong></td><td><strong>1 098 781</strong></td><td><strong>0.26</strong></td><td><strong>2 636 137</strong></td></tr><tr><td><strong>Estimation WGMME*</strong></td><td></td><td>Common dolphin*</td><td><strong>634 286</strong></td><td><strong>0.31</strong></td><td><strong>352 227</strong></td><td><strong>1 142 213</strong></td><td><strong>0.24</strong></td><td></td></tr><tr><td colspan="9">* estimate derived from common dolphin and a proportion of common/striped dolphins (precautionary abundance of common dolphin is estimated at 481,306)</td></tr><tr><th colspan="10">Bay of Biscay (Bob) - summer</th></tr><tr><td>SCAMM BOB Winter 2012</td><td></td><td>Common and common or striped dolphin</td><td>284 894</td><td>0.73</td><td>174 423</td><td>480 612</td><td>0.74</td><td>282 141</td></tr><tr><td><strong>SCAMM BOB Summer 2012</strong></td><td></td><td>Common dolphin + common or striped</td><td><strong>493 591</strong></td><td><strong>0.17</strong></td><td><strong>342 094</strong></td><td><strong>718 593</strong></td><td><strong>1.45</strong></td><td><strong>282 141</strong></td></tr><tr><td>SCANS3 vs SAMM</td><td></td><td>Common dolphin</td><td>92 893</td><td>0.27</td><td>52 766</td><td>149 494</td><td>0.78</td><td>118 471</td></tr><tr><td>SCANS 3 Block B (french shelf)</td><td></td><td>Common or striped dolphin</td><td>61 741</td><td>0.21</td><td>38 143</td><td>90 843</td><td>0.52</td><td></td></tr><tr><td>SCANS 3 Block 9 (Oceanic North)</td><td></td><td>Common dolphin</td><td>150 208</td><td>0.72</td><td>39 190</td><td>575 716</td><td>1.04</td><td>144 352</td></tr><tr><td></td><td></td><td>Common or striped dolphin</td><td>3 377</td><td>0.66</td><td>956</td><td>11 932</td><td>0.02</td><td></td></tr><tr><td>SCANS 3 Block 13 (Oceanic south)</td><td></td><td>Common dolphin</td><td>3 110</td><td>0.65</td><td>861</td><td>11 236</td><td>0.05</td><td>59 340</td></tr><tr><td>SCANS 3 Block AC (Iberian shelf north)</td><td></td><td>Common or striped dolphin</td><td>13 414</td><td>0.40</td><td>5 948</td><td>30 255</td><td>0.23</td><td></td></tr><tr><td></td><td></td><td>Common dolphin</td><td>71 082</td><td>0.31</td><td>36 898</td><td>124 302</td><td>2.02</td><td>35 180</td></tr><tr><td><strong>Total SCANS 3 (Bob)</strong></td><td></td><td>Common dolphin + common or striped</td><td><strong>5 504</strong></td><td><strong>0.84</strong></td><td><strong>0</strong></td><td><strong>16 252</strong></td><td><strong>0.15</strong></td><td><strong>357 343</strong></td></tr><tr><td colspan="9">Bay of Biscay (Bob) - shelf - winter</td></tr><tr><td>SAMM vs SPEE</td><td></td><td>Common dolphin + common or striped</td><td>179 421</td><td>0.32</td><td>119 638</td><td>269 079</td><td>1.17</td><td>282 141</td></tr><tr><td><strong>SPEE Winter 2019</strong></td><td></td><td>Common dolphin + common or striped</td><td><strong>7 877</strong></td><td><strong>0.27</strong></td><td><strong>4 676</strong></td><td><strong>13 271</strong></td><td><strong>0.53</strong></td><td><strong>2 447</strong></td></tr><tr><td>SPEE Spring 2019</td><td></td><td>Common dolphin + common or striped</td><td>1 906</td><td>0.26</td><td>1 158</td><td>3 138</td><td>0.13</td><td>2 632</td></tr><tr><td>SPEE Summer 2019</td><td></td><td>Common dolphin + common or striped</td><td>2 008</td><td>0.35</td><td>1 021</td><td>3 949</td><td>0.13</td><td>2 457</td></tr><tr><td>SPEE Autumn 2019</td><td></td><td>Common dolphin + common or striped</td><td>761</td><td>0.38</td><td>368</td><td>1 573</td><td>0.05</td><td>2 279</td></tr></tbody></table>
<figure><img src="image_145.png" /><figcaption>Figure 2: Marine regions as identified by the European Marine Strategy Framework Directive. Figure from http://www.eea.europa.eu/about-us/countries-and-eionet/marine-regions.</figcaption></figure>
# Appendix 2: Flexible regression approach simulation study

Received: 5 February 2020

Revised: 14 June 2020

Accepted: 4 August 2020

DOI: 10.1002/1438-390X.12069

<figure><img src="image_146.png" /></figure>

Check for
updates

ORIGINAL ARTICLE

Population
Ecology

WILEY

Flexible parametric modeling of survival from age at death data: A mixed linear regression framework

Etienne Rouby<fnref n="1" />,<fnref n="2" /> |

Vincent Ridoux<fnref n="1" />,<fnref n="2" /> |

Matthieu Authier<fnref n="2" />,<fnref n="3" />

Correspondence

Etienne Rouby, Centre d'études
Biologiques de Chizé, UMR 7372, Villiers-
en-Bois, France.

Email: etienne.rouby@univ-lr.fr

Abstract

Many long-lived vertebrate species are under threat in the Anthropocene, but their conservation is hampered by a lack of demographic information to assess population long-term viability. When longitudinal studies (e.g., Capture-Mark-Recapture design) are not feasible, the only available data may be cross-sectional, for example, stranding for marine mammals. Survival analysis deals with age at death (i.e., time to event) data and allows to estimate survivorship and hazard rates assuming that the cross-sectional sample is representative. Accommodating a bathtub-shaped hazard, as expected in wild populations, was historically difficult and required specific models. We identified a simple linear regression model with individual frailty that can fit a bathtub-shaped hazard, take into account covariates, allow goodness-of-fit assessments and give accurate estimates of survivorship in realistic settings. We first conducted a Monte Carlo study and simulated age at death data to assess the accuracy of estimates with respect to sample size. Secondly, we applied this framework on a handful of case studies from published studies on marine mammals, a group with many threatened and data-deficient species. We found that our framework is flexible and accurate to estimate survivorship with a sample size of 300. This approach is promising for obtaining important demographic information on data-poor species.

KEYWORDS

age at death, Monte Carlo study, regression, survival analysis, survivorship

1 | INTRODUCTION

The protection and conservation of wildlife is a salient
challenge in the Anthropocene, where human-induced
pressures are both far-reaching in their geographic and
temporal scope, and acute in their impact on biodiversity
(Bongaarts, 2019). Long-lived species are particularly sen-
sitive to these pressures: they can act as sentinel species
for whole ecosystems (Heithaus, Frid, Wirsing, &
Worm, 2008). Their demographic trajectory (e.g., extinc-
tion) can also affect the functioning of ecosystems, espe-
cially in the case of top predators (Beschta &

Ripple, 2009; Ritchie et al., 2012). Assessing the
demographic viability of long-lived species in the face of
current and future pressures is required for their pro-
active conservation; but such an endeavor may be ham-
pered in practice because of logistical challenges in moni-
toring and collecting relevant data at the relevant
spatiotemporal scales.

The classical way to assess a populations viability is to
first construct a static life-table where mortality and
fecundity rates for a cohort of individuals are tallied at
each age/stage of their life cycle (Caughley, 1966).
Knowledge of mortality at each age/stage allows to

<footnote n="1">¹Centre d'études Biologiques de Chizé,
UMR 7372, Villiers-en-Bois, France</footnote>

<footnote n="2">²Observatoire PELAGIS, UMS-CNRS
3462, La Rochelle Université, La Rochelle,
France</footnote>

<footnote n="3">³ADERA, La Rochelle Université, Pessac
Cedex, France</footnote>
investigate the associated hazards, estimate vital rates, such as (cumulative) survival; and to project the population trajectory forward in time using matrix models (Caswell, 2001; Leslie, 1945). Historically, the age at death data used to construct life-tables were cross-sectional (e.g., Gompertz, 1825): a sample of the population is taken at a specific time and the observed age structure is assumed similar to that of a cohort of individuals had we been able to monitor them from birth to death. This cross-sectional design allows for a rapid demographic assessment (Boyd, Bowen, & Iverson, 2010; Margules & Austin, 1990, pp. 126-127.) whose accuracy however hinges on the assumption that the cross-sectional design approximates well a longitudinal one.

Longitudinal designs, whereby an individual is monitored from birth to death are, in wildlife ecology, a byproduct of capture-recapture studies. The initial motivation for the latter was the accurate estimation of abundance in open populations when detectability is less than perfect (Cormack, 1964; Jolly, 1965; Seber, 1965). Models (e.g., the Cormack-Jolly-Seber model) tailored for these life history data can provide accurate estimates of survival that can be fed into matrix population models (e.g., Fujiwara & Caswell, 2001). Since the 1980s, the emphasis of capture-recapture studies in wildlife ecology shifted from abundance to survival estimation (Lebreton, Burnham, Clobert, & Anderson, 1992) and the inclusion of individual-level covariates, which paved the way for a better understanding of evolution and natural selection in the wild (Cam, 2009). Concomitant with this change in focus was the less frequent reporting of life-tables in publications, an omission partly mitigated by the rise of open online databases such as COMADRE/COMPADRE (Salguero-Gómez et al., 2015; Salguero-Gómez et al., 2016). From an applied perspective in conservation, the obvious drawback of capture-recapture studies is the time and manpower required to collect data, especially for long-lived species. White, Nagy, and Gruber (2014) recently proposed a new development that aims to avoid birth to death monitor. From both juvenile mortality and fecundity, it is possible to infer adult mortality. However, it requires monitoring the juvenile part of the whole population in order to model the overall population dynamics. The mismatch between the necessary delay in data acquisition and the urgency of mitigation on conservation decisions can be acute, suggesting interest for cross-sectional data that have the additional benefit of being applicable on species where the identification of individuals is difficult (Williams, Nichols, & Conroy, 2002). One alternative to species identification, for vital rates estimation, is the use of count data. This approach may involves the building of an N-mixture model relying on the individual-state knowledge. Zipkin et al. (2014) proposed

a development that allows taking into account data even if the individual stage is unknown. Another way to obtain vital rates with count data, as time-series, is to conduct an inverse modeling approach: this method also presents the advantage to consider individual-level state data (González, Martorell, & Bolker, 2016). Even if these approaches are promising for hard to monitor species, they remain difficult to apply on some species, especially very mobile ones.

Assuming that cross-sectional data on age at death are available and representative, conservationists now face some further choices, such as non-parametric (e.g., Kaplan-Meier survivorship curves; Clark, Bradburn, Love, & Altman, 2003; Kaplan & Meier, 1958) versus parametric modeling. The latter presents the advantage to provide smoothing and to summarize the data in a handful of parameters. This is small sample attractive: non-parametric approaches such as the Kaplan-Meier estimators (also known as Product Limit Estimators) for survivorship is piecewise constant with discontinuities at the time of death of individuals in the sample. While assumption-lean, this estimator may yield from small samples rough survivorship curves shaped like staircases that can be difficult to compare. Even with little sample size (i.e., few age at death data regarding the overall time series length), parametric modeling aims to estimate smooth survivorship and allow the statistical comparison of parameters values (Kleinbaum & Klein, 2010). It is important to notice that the estimation hinges on the correct specification of the underlying model. More realistically, the challenge is to find a parsimonious model that will nevertheless accommodate the available data without shoehorning the analysis into a convenient, but not necessarily realistic, template. For long-lived animal species, a realistic model should provide the so-called bathtub-shaped curve whereby there is a high juvenile mortality, followed by a lower and rather constant adult mortality and, finally, a late increase due to senescence (Choquet, Viallefont, Rouan, Gaanoun, & Gaillard, 2011; Siler, 1979). Although there are models to accommodate this bathtub pattern (Heligman & Pollard, 1980; Siler, 1979), they can be difficult to fit (but see Saavedra, 2018), their goodness of fit can be hard to measure; and they usually do not allow for the inclusion of individual-level covariates such as sex.

Our aims are to investigate a parametric approach for the construction of life-tables in a data-poor context, with long-lived vertebrate species in mind. We will first present a simple regression modeling framework for the analysis of age at death data (with or without right-censoring) that can accommodate mortality patterns such as constant, increasing/decreasing, unimodal or bathtub-shaped. Importantly, this framework allows the seamless inclusion
of individual-level covariates. We then conduct a Monte Carlo simulation study with five biological scenarios to assess our framework, focusing on the accurate estimation of survivorship and mortality. Finally, we illustrate our methodology on a handful of real case studies, with a focus on marine mammals. This group includes many elusive and long-lived species that are threatened (Avila, Kaschner, & Dormann, 2018), data-deficient (Parsons, 2016) and difficult to study with a longitudinal approach. The most common age at death data for marine mammals are obtained from strandings (i.e., cross-sectional monitoring) which may be associated to selection biases. However, stranding samples are a source of age data from which it is possible to obtain vital rates.

## 2 | MATERIALS AND METHODS

### 2.1 | Definitions

Survival analysis deals with the analysis of the timing of death (Clark et al., 2003; Kaplan & Meier, 1958; Lesaffre & Lawson, 2012; Tanner & Wong, 1984). We assume data $y \ge 0$ to be time-to-event data, for example longevity, survival time or age at death data. Such data can be collected from stranding of animals (that is marine animals that are washed ashore), or any recovery of dead specimens (e.g., Lepus europaeus, Pallas 1778; Ovis dalli, Nelson 1884) when aging is possible (e.g., from teeth growth layers). Let the index $i$ denotes the $i^{th}$ individual, and $N$ the sample size. The equation $y_i \sim D(\theta)$ reads as datum $y_i$ follows statistical distribution $D$ of parameters $\theta$ and with probability density function $f(y; \theta)$ and cumulative density function $F(y; \theta) = \text{Pr}(y \le t) = \int_{0}^{t} f(y; \theta)dy$.

The survival function $S(t; \theta)$ gives the probability of being alive at time $t$, that is $\text{Pr}(y > t) = 1 - \text{Pr}(y \le t) = 1 - F(y; \theta)$. At the population level, this quantity correspond to the fraction of the population that is still alive at age $t$, that is cumulative survival or simply survivorship. The hazard rate, or age-specific mortality rate, is the instantaneous probability of dying at time $t + dt$ given that and individual $i$ has survived until time $t$.

$$h(t) = \lim_{dt \to 0} \frac{\text{Pr}(t \le y_i < t + dt; y_i > t)}{dt} \quad (1)$$

With parametric models, the hazard rate can be expressed in terms of the probability density and survivorship functions:

$$h(t; \theta) = \frac{f(t; \theta)}{S(t; \theta)}, \text{with } S(t; \theta) > 0 \quad (2)$$

Estimation of the hazard rate function is the goal of parametric survival analysis (Lesaffre & Lawson, 2012).

### 2.2 | Data simulation scenarios

We considered five biological scenarios corresponding to different patterns in survivorship and the underlying hazard (Figure 1, see Text S1):

1. A unimodal distribution of age at death corresponding to a unimodal hazard, whereby mortality risk increases in early ages, peaks and decreases in late life.

2. A mixture of two hazards corresponding to either a unimodal (a) or bimodal (b) distribution of age at death.

3. A bathtub-shaped hazard due to individual frailty, that is individual-specific risk of mortality.

4. A bathtub-shaped hazard with an additional bump in early life due, for example, to an additional source of mortality.

Each scenario is associated to survivorship and hazard functions (Figure 1).

### 2.3 | Statistical analysis of age at death data

We used parametric models $\mathcal{M}$ of age at death data $y$ to estimate hazard and survivorship rates (see Data S1 for model code): the latter are especially of interest to conservationists. We assumed that the exact timing of death is available, but our framework can easily accommodate censoring (that is cases when death is known to have occurred before or after measurement). Our framework consists in modeling the logarithmic transform of $y_i$ in a regression framework (location-scale model):

$$\text{logy}_i = \mu + \sigma \times \varepsilon_i - \frac{Z_i}{\beta} \quad (3)$$

where $\mu$ is a location parameter; $\sigma$ and $\frac{1}{\beta}$ are (positive) scale parameters; and $Z_i$ are independent standard exponential deviates. The parameter $\frac{1}{\beta}$ quantifies individual frailty (Kannisto, 1991; Reed, 2011) or persistent demographic heterogeneity sensu Cam, Aubry, and Authier (2016). The choice of the statistical distribution for the residuals $\varepsilon_i$ determines the shape of the underlying hazard rate. We considered three different choices, each corresponding to a model (see Text S2 for the associated survivorship and hazard functions).

$$\bullet \mathcal{M}_1: \varepsilon_i \sim \mathcal{N}(0,1) \text{ and } \frac{1}{\beta} = 0 (\beta = +\infty)$$
<figure><img src="image_147.png" /><figcaption>**FIGURE 1** The five biological scenarios considered in the Monte Carlo simulation study. Scenario 1 corresponds to a low juvenile mortality, a high adult mortality and no senescence. Scenario 2a corresponds to a low juvenile mortality and a higher adult mortality plateau. Scenario 2b corresponds to a high juvenile mortality and a lower adult mortality that decreases in a linear fashion. Scenario 3 corresponds to a high juvenile mortality, a lower adult mortality and senescence (bathtub shaped hazard). Finally, Scenario 4 corresponds to a high juvenile mortality and a high adult mortality, with a transient dip in mortality risk between these two life-stages</figcaption></figure>

This model assumes a log-normal distribution for $y_i$, which corresponds to a unimodal hazard curve.

$$ \bullet \mathcal{M}_2: \varepsilon_i \sim \mathcal{G}(0,1) \text{ and } \frac{1}{\beta} = 0 (\beta = +\infty) $$

This model assumes a Gumbel distribution for the residuals $\varepsilon_i$, which corresponds to a Weibull distribution for $y_i$. It is known as the Accelerated Failure Model. The hazard rate is monotonic: it can be constant, increasing or decreasing depending on the value of $\sigma$.

$$ \bullet \mathcal{M}_3: \varepsilon_i \sim \mathcal{N}(0,1) \text{ and } \frac{1}{\beta} > 0 $$

This choice leads to assume a normal-Laplace distribution for $\log y_i$, which induces a flexible hazard curve depending on the value of $\beta$ (Reed, 2011). In particular, the hazard can be a bathtub-shaped, as expected for example for long-lived species of vertebrates in the wild (Choquet et al., 2011).

Our framework given by Equation (3) is a flexible location-scale model and boils down to a generalized linear mixed model (GLMM; Bolker et al., 2009) with a handful of parameters $\theta = (\mu, \sigma, \beta)$ to accommodate a large diversity of survivorship and hazard curves (see Text S2 for equations). One attractive feature of this framework is the seamless incorporation of $p$ individual-level covariates $x_{ip}$ in Equation (3) (Reed, 2011):

$$ \log y_i = \mu + \sum_{j=1}^{p} \gamma_p x_{ip} + \sigma \times \varepsilon_i - \frac{Z_i}{\beta} \quad (4) $$

An important restriction of our approach is that only time-invariant individual-level covariates can be included. This covers however interesting cases such as sex-differences in survival or differences due to geography (that is comparing different populations of the same species).

## 2.4 | Monte Carlo study

Our aim is to carry a Monte Carlo study (e.g., Morris, White, & Crowther, 2019) to investigate whether our modeling framework can provide accurate survivorship estimates from age at death data. We considered five biological scenarios to cover a diversity of realistic mortality patterns. For each scenario, we simulated 100 data sets of sample size 100, 200, 300, 400, 500 and 1000 to provide recommendations on the minimum sample size required for accurate estimation. Data simulation was carried out in R version 3.6.0 (R Core Team, 2019) using base
<table><caption>**TABLE 1** Design of the Monte Carlo study: model *M*₂ acts as a negative control as it was never used to simulate data, hence it should not be selected as the best model as it is misspecified</caption><thead><tr><th>Model/scenario</th><th>1</th><th>2a</th><th>2b</th><th>3</th><th>4</th></tr></thead><tbody><tr><td>M<sub>1</sub></td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>M<sub>2</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>M<sub>3</sub></td><td>-</td><td>-</td><td>-</td><td>+</td><td>-</td></tr></tbody></table>

*Note:* Likewise, scenarios 2a, 2b and 4 corresponds to data generated from mixture models not included in the set: they serve as tests of the performance of our framework to obtain accurate estimates with misspecified models. Finally, scenarios 1 and 3 act as positive control as they correspond to data simulated under models *M*₁ and *M*₃, respectively.

functions such as *rnorm* and *rexp*. Each simulated data set was then analyzed with our framework that considered three parametric models. Crucially, we considered scenarios for which the true model was not among the set (Table 1). In other words, we assessed the performance of our framework under the possibility that none of the candidate models is correctly specified with respect to the data at hand. For example *M*₂ is a Weibull model, which is widely used in survival analysis (Kleinbaum & Klein, 2010) it is biologically unreasonable for marine megafauna and top predators as a model over their entirely lifetime because it cannot accommodate the expected bathtub-shape hazard. Model fitting was done with software Stan version 2.18 (Carpenter et al., 2017) called from *R* via the library *rstan* (Stan Development Team, 2018). Three chains were run with a warm-up of 500 iterations, followed by an additional 1000 iterations. No thinning was performed but the delta parameter of the NUTS algorithm was increased from 0.80 to 0.95 to avoid divergent transitions and the maximum tree depth increased to 15. Parameter convergence was assumed when its *R* statistics was lower than 1.10. Upon convergence, the three chains were pooled to obtain a sample of ≈1000 values from the posterior distribution.

Model fit was assessed with the widely applicable information criterion (WAIC, Gelman, Hwang, & Vehtari, 2014), computed with *R* package *loo* (Vehtari, Gelman, & Gabry, 2017). Survivorship estimates from each model were computed from the posterior distribution of parameters, and visually compared to the true survivorship curve and a non-parametric (Kaplan-Meier) estimate. Because we carried out a simulation study, the true survivorship curve was known and could have been used to compute the root-mean-squared error (RMSE). However, in practice, this is not the case and we chose instead to compute RMSE with respect to the non-parametric Kaplan-Meier survivorship estimates. The latter were thus assumed to represent the best estimates

available to researchers, and the aim was to assess whether a parametric model could provide a fit as good as that from a non-parametric approach.

$$RMSE_M = \sqrt{\mathbb{E}[(\hat{S}_{KS}(t) - \hat{S}_M(t))^2]} \quad (5)$$

where $\hat{S}_{KS}(t)$ is the Kaplan-Meier estimate of survivorship at age *t*, $\hat{S}_M(t)$ is the corresponding estimate from parametric model *M*, and the expectation is taken over a sequence of values of *t*.

In order to assess model selection and model check, it is necessary to determine if the models represent well the data. Conn, Johnson, Williams, Melin, and Hooten (2018) recently reviewed some ways to proceed. We choose to do a prior predictive check to test for adequacy between models and time to event data (Text S3 and Figure S1). We also did a posterior predictive check (Text S3 and Figure S2) to see whereas data simulated through the fitted models are similar to that observed from the Barlow and Hohn (1984) data set. The comparison is done with both the true Kaplan-Meier survivorship curve and the ones from posterior simulated data sets (Text S3).

Our focus was on accurate estimation of survivorship, and thus we assessed goodness-of-fit by comparing the expected mean survivorship under each model to the observed Kaplan-Meier estimates. This focus was in line with downstream use of such estimates in matrix population models: here the salient statistics we want our model to reproduce (Gelman, 2003) is the survivorship function.

Our study design is summarized in Table 1 and Figure 2. It consisted in a comprehensive factorial design crossing (a) sample size (100, 200, 300, 400, 500 and

<figure><img src="image_148.png" /><figcaption>**FIGURE 2** Monte Carlo study flowchart. We considered five biological scenarios, six different sample sizes for data, and three models for analysis. For each combination, we assessed model fit and the accuracy of parameter estimates. Model selection is done with Watanabe-Akaike information criterion (WAIC) and root-mean-square error (RMSE) [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
<table><caption>TABLE 2 Case studies</caption><thead><tr><th>Data set</th><th>Taxon</th><th>Species</th><th>Sample size</th><th>Covariate</th></tr></thead><tbody><tr><td>Murphy et al. (2012)</td><td>Pinniped</td><td>Monk seal (Monachus monachus, Hermann 1779)</td><td>28</td><td>Sex</td></tr><tr><td>Slooten (1991)</td><td>Cetacean</td><td>Hector's dolphin (Cephalorhynchus hectori, Van Beneden 1881)</td><td>60</td><td>None</td></tr><tr><td>Rodríguez-Caro et al. (2019)</td><td>Chelonian</td><td>Spur-thighed tortoise</td><td>154</td><td>None</td></tr><tr><td>Murphy et al. (2009)</td><td>Cetacean</td><td>Common dolphin (Delphinus delphis, Linnaeus 1758)</td><td>515</td><td>None</td></tr><tr><td>Kesselring, Viquerat, Brehm, and Siebert (2018)</td><td>Cetacean</td><td>Harbor porpoise (Phocoena phocoena, Linnaeus 1758)</td><td>561</td><td>Area</td></tr><tr><td>Saavedra (2018)</td><td>Cetacean</td><td>Delphinidae</td><td>579</td><td>None</td></tr><tr><td>Barlow and Hohn (1984)</td><td>Cetacean</td><td>Pantropical spotted dolphin (Stenella attenuata, Gray 1846)</td><td>1892</td><td>Sex</td></tr></tbody></table>

Note: Data sets were selected based on the availability of published (raw) data, species and sample size.

1000), (b) mortality patterns (five scenarios), (c) parametric models (three models) and (d) individual covariate inclusion. Two covariates ($x_1, x_2$) were generated by sampling from a Bernoulli distribution with probability 0.5. These covariates could represent for example sex or two sub-populations in different geographic areas.

## 2.5 Applications

We analyzed real data sets from published case studies (Table 2) within our framework, and compared the estimated parametric survivorship curves with the non-parametric Kaplan-Meier one. All species in Table 2 are marine mammals except the spur-thighed tortoise (Tectudo graeca, Mertens 1946) which was included to compare our approach with that of Rodríguez-Caro et al. (2019) in a data-poor context for conservation. It is important to precise that data from Rodríguez-Caro et al. (2019) were obtained from live animals, still alive at the time of measurement (Rodríguez-Caro, Graciá, Anadón, & Gimenez, 2013; Sanz-Aguilar et al., 2011). However, we used these data as if they were age at death data and ignored right-censoring.

## 3 RESULTS

Across all scenarios and sample size, parameter convergence (assessed with $\hat{R}$) was not equivalently reached depending on the model and scenario (see Figures S3 and S4a). $M_1$ always converged very easily. In contrast, $M_2$ and $M_3$ were not as good as $M_1$ to converge with 1000 iterations (500 as warm up). More precisely, convergence for $M_3$ was difficult for $\beta$ given this configuration (see

Figures S5 to S10). An increased in the number of iterations per chains (here from 1,000 to 2000) solved the problem (Figure 3).

### 3.1 Hazard rate estimation

Estimated hazard curves are shown in Figure 4 for each combination of scenario, model and sample size. Models $M_1$ and $M_3$ were the best fitting ones in the first and third scenarios respectively (i.e., positive control, Table 1): estimates were accurate and precision increased with sample size. For scenarios 2a, 2b and 4, all estimates were biased, and precision increased with sample size (Figure 3). In contrast, model $M_2$ never provided accurate hazard estimates (i.e., negative control). The same results were obtained when covariates were included (not shown).

### 3.2 Survivorship estimation

Estimated cumulative survival curves are shown in Figure 5 for each combination of scenario, model and sample size. Across all scenarios and sample sizes, estimates from model $M_3$ were the most accurate (Figure 5, see Text S4 and Figure S11 for RMSE results). Precision increased with sample size. In particular, survivorship rates estimated with $M_3$ were very close to Kaplan-Meier estimates for sample size $\ge$300. This sampling size also provided a good confidence interval precision with a maximum width of 0.04% while estimating survivorship (Figure S12). Predictive ability, as measured with WAIC, was the greatest for model $M_3$; it was consistently ranked first across each combination of scenario and sample size,
<figure><img src="image_149.png" /><figcaption>**FIGURE 3** Models parameters convergence assessment for each sampling size with *n* (warm-up) = 1000 iterations and a total of N = 2000 iterations [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

even for scenario 1 where model *M*₁ was the true data-
generating mechanism (see Text S4 and Figure S13).

3.3 | Covariate effects

Covariate effect estimation is summarized on Figure 6 as
a difference between survivorship rates. Model *M*₂ could
not estimate difference in survivorship. Model *M*₁ is only
effective for both scenario 1 and 2a. *M*₃ could estimate
accurately covariate effects when the difference is
expressed as an unimodal pattern (i.e., scenario 1, 2a and
3). The precision of the estimated effect increased with
sample size for models *M*₁ and *M*₃. With small sample
size, sign errors on the effect of covariate was possible
but disappeared with sample size ≥300.

3.4 | Case studies

Estimated survivorship curves from published data sets
are plotted against the Kaplan–Meier curves in Figure 7
for each model. Model *M*₃ was the most flexible: it pro-
vided the most accurate estimates for each data set.
Uncertainty, as measured with 80% credible intervals

were narrower with model *M*<sub>3</sub>, and overlapped most with
Kaplan–Meier estimates. Model *M*<sub>3</sub> consistently had the
lowest WAIC.

In the handful of case studies where covariates were
available (Table 2), a similar pattern arose. Including
covariate can be expected to account for more variation
in the data, and a lower WAIC. For both M₁ and M₂, this
was indeed the case. For M₃ it was only true with the
Murphy et al., 2012 data set (see Text S4 and Table S1).
The estimated covariate effect for both Barlow and
Hohn (1984) and Kesselring et al. (2018) case studies is
summarized on Figure 8 (see also Text 5). Models M₁
and M₃ both estimated a survivorship difference between
each covariate. As it is the case without covariates
(Figure 7), the M₃ curve fits the Kaplan–Meier estimate
better. However, there is a discrepancy between both, for
each covariate. M₃ is able to spot a difference depending
on covariate, but is not able to perfectly fit the Kaplan–
Meier estimate.

4 | DISCUSSION

We assessed the ability of a simple linear mixed model
to estimate hazard and survivorship rates with
<figure><img src="image_150.png" /><figcaption>**FIGURE 4** Estimated hazard curves confidence intervals for each combination of scenarios (rows), models (columns), and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the true hazard curves (in black). Even-numbered scenarios (2a, 2b and 4) are on the left, and odd-numbered ones (1 and 3) on the right. [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

cross-sectional age at death data. We used Monte Carlo simulations to investigate the accuracy and precision of estimates across a diversity of mortality patterns and several sample sizes. We contrasted the performance of three different models and found one model, the model $M_3$; that was consistently better with respect to prediction of survivorship, even when it was misspecified.

**4.1 | Age at death data and sampling bias**

A crucial but implicit assumption of the cross-sectional approach to survival analysis is that the sample is representative of the larger population, especially with respect to age structure (Caughley, 1966). This assumption is on the data set, not on the modeling. The way of collecting age at death data is therefore determinant to support this assumption. Transerval designs involving, for example, recording of dead carcasses, hunting bags or population census must match the representativity premise. However, it is possible that an implicit process (e.g., bycatch)

shapes the observed age frequency (e.g., from stranded animals). In this case, it is necessary to explore some potentials age-dependent selection biases in the population (Barlow & Hohn, 1984). With stranding data, the stationary age distribution assumption needs to be substantiated with auxiliary data, but ultimately, it is likely to remain a working hypothesis on which any cross-sectional method will lean on. In the case of cetaceans, many species are not amenable to study with a longitudinal design, and strandings remain an important source of information (e.g., Murphy et al., 2009), and demographic information in particular (Ferguson, Stirling, & McLoughlin, 2006; Saavedra, 2018). When longitudinal studies are not possible, there are very limited options to obtain demographic information. In some cases, a comparative approach may be possible where information on species with similar life histories can be leveraged (Caswell, Brault, Read, & Smith, 1998; Hashimoto, Shirakihara, Shirakihara, & Hiramatsu, 2013). This choice relies crucially on the assumption of between-species similarity. Even if comparative data are available,
<figure><img src="image_151.png" /><figcaption>**FIGURE 5** Estimated survivorship confidence intervals for each combination of scenarios (rows), models (columns) and sample size (facets). Estimated confidence intervals are depicted in each panel and compared to the truth (in black). Even-numbered scenarios (2a, 2b and 4) are on the left, and odd-numbered ones (1 and 3) on the right [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

the ecological context may be too different to justify this approach in some instances. Strandings may be the only available source of data, and even if they are suspected to suffer from some selection bias, a pragmatic approach to conservation requires to use them (Boyd et al., 2010), keeping in mind the inherent limitations of these data.

Many conservation instruments specifically requires to use the best available science (e.g., the Marine Strategy Framework Directive EC 2008/56 in Europe). The operative expression “the best available science” may be understood broadly as what lies at the intersection of state-of-the-art methods, good data and accurate knowledge. In practice, there may be a hiatus between the canonical approach that should be, and the pragmatic one that can be pursued at the time conservation actions need to be decided. A conclusion of a species being data-deficient often leads to the doldrums with respect to conservation decisions (Parsons, 2016). Likewise, the many uncertainties that can affect any scientific studies can easily lead to inactions (Ascher, 2004). It is because we are starkly aware of these limitations that we carried out this study to identify a pragmatic approach to estimate survivorship rates from age at death data. Crucially, all the

models we used are conditional of the underlying sample being representative of the population it is taken from. Granting this assumption, we identified a simple model from Reed (2011) to obtain accurate estimates.

## 4.2 | Prediction accuracy and models' goodness of fit

Traditionally, estimating survivorship rates of marine mammals with cross-sectional data was done with parametric models such as the Siler or Heligman-Pollard models (Heligman & Pollard, 1980; Siler, 1979). These models aim at reproducing a bathtub-shaped hazard curve, but need several parameters to do so. Furthermore, they can be difficult to fit although new tools have been developed to use these models (Saavedra, 2018). A remaining challenge with these models is to assess goodness of fit, to incorporate individual-level covariates and perform model selection. The simple parametric form of Reed (2011) for analyzing the logarithm of age at death data provides a bathtub-shaped hazard with a linear mixed model, the current workhorse of ecologists (Bolker
<figure><img src="image_152.png" /><figcaption>**FIGURE 6** Estimated survivorship difference ($(\hat{S}_M(t)|x=1) - (\hat{S}_M(t)|x=0)$) under each scenario. The covariate effect could correspond to that of sex (e.g., $x=1$ for females and $x=0$ for males). Ten estimated difference curves are depicted in each panel and compared with the truth [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

et al., 2009). This linear mixed modeling framework allows for a seamless incorporation of covariates, and to use standard tools for model selection and goodness-of-fit assessments. In other words, model $M_3$ brings back survival analysis within the comfort zone of ecologists. We harnessed the simplicity of Reed (2011) to conduct our Monte Carlo study, and found that the model suggested by Reed (2011), our model $M_3$, was very accurate in predicting survivorship (i.e., it consistently had the lowest RMSE), even in cases when it was not the true model behind the data (see Text S6 to have details about priors). This result is important as it suggests to start building model of increasing complexity from $M_3$ and use tools such as WAIC (Gelman et al., 2014) to balance model complexity with prediction accuracy.

From our simulations, we can recommend a sample size of at least 300 individuals to obtain accurate and precise estimates of survivorship, from which age-specific survival estimates can be derived (see Figure S14 for parameters estimates). This sample size recommendation is practical and realistic (e.g., Kesselring et al., 2018; Mannocci et al., 2012; Murphy et al., 2009). This recommendation is mostly to obtain precise estimates, but may

be relaxed in some cases where only sparse data may be available (data-poor context hereafter). Estimates from a simple linear model with an individual frailty term, were accurate, if imprecise, with a sample size as small as 100. The possibility to use sparse data is critical as it can help conservation of marine mammals, many of which being classified as data-deficient (Parsons, 2016; Schipper et al., 2008). Some of the cases studies presented in this paper support this statement, although, with so few data, including covariates in the model will be difficult or will require great care (see, e.g., Cox, Authier, Orgeret, Weimerskirch, and Guinet (2020)). Rodriguez-Caro et al. (2019) recently provided an approach to estimate survival in data-poor settings using inverse modeling, also to obtain accurate estimates of demographic rates. With the latter, population matrix models can then be used to assess population dynamics and the fate of populations over time (Caswell, 2001).

5 | HAZARD AND FRAILTY

A linear mixed model can fit age at death data very well: this ability comes for the individual frailty term, which
<figure><img src="image_153.png" /><figcaption>**FIGURE 7** Survivorship curves estimated from published data sets. Each subpanel corresponds to a case studies in Table 2. Data set size is represented horizontally. Panels are associated to the following data sets: (a) Murphy et al., (2012); (b) Slooten, (1991); (c) Rodriguez-Caro et al., (2019); (d) Murphy et al., (2009); (e) Kesserling et al., (2018); (f) Saavedra, (2018); (g) Barlow and Hohn, (1984) [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

corresponds to an individual random effect in the mixed
modeling framework. Individual frailty in statistical
models translate the empirical observation that two simi-
lar individuals (with respect to observable features of
their phenotypes) can nevertheless differ markedly in
their longevity (Cam et al., 2016; Kannisto, 1991). The
parametric form (i.e., exponential) for individual frailty
gives extra flexibility to the model, and can accommodate
a bathtub-shaped hazard curve (Reed, 2011). However, in
our simulations, we found that estimating hazard rates
was more difficult than estimating survivorship rates
(Figures 3 and 4). Thus, even though we identified a
model (M₃) for reliable and accurate estimation of survi-
vorship, the same model was less reliable with respect to
hazard. In other words, the individual frailty term in our
model M₃ should not be over-interpreted, and is probably
best seen as a statistical device for robust estimation. Haz-
ard estimation is a difficult statistical problem (Watson &
Leadbetter, 1964), for which there are better tools avail-
able, especially non-parametric ones (see, e.g., Hanson &
Jara, 2013), than the simple parametric approach we con-
sidered in this study. Non-parametric approaches to infer

the shape of the hazard curve are data-hungry: Hanson
and Jara (2013) using Bayesian non-parametric, which is
better described as a model with a massive number of
parameters (Hoff, 2013). The traditional Kaplan-Meier
approach is truly a non-parametric approach but it does
not give access to the underlying hazard, and give rough
(i.e., non-smooth; Figure 5) survivorship rates with small
sample size. In data-poor settings, parametric modeling
remains attractive because it has interpretable parameters
(e.g., individual frailty) and because these parameters can
smooth out noise in data, yielding more precise estimates
if the model is at least approximately correct, or more
pragmatically, if it is grounded in theory (e.g., bathtub-
shaped hazard for natural populations) and cannot be
rejected from a goodness-of-fit test. It is precisely in this
data-poor setting that we envision our parametric model-
ing approach to be most useful. Because the approach
boils down to linear mixed effects modeling, great flexi-
bility in model specification of additional random effects
(e.g., year effects, sex-specific frailties) is possible pro-
vided there are enough data to offset the increase in
complexity.
<figure><img src="image_154.png" /><figcaption>**FIGURE 8** Estimated survivorship difference (($\hat{S}_M(t)|x=1$) - ($\hat{S}_M(t)|x=0$)) for both case studies [Color figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

# 6 | BENEFITS AND LIMITS

We believe our approach is very valuable to estimate survivorship from sparse data but may be inadequate for bimodal age at death data distribution. A bimodal distribution of age at death data may be a sign of selection in data collection Barlow & Hohn, 1984. Rather than consider the sampling as biased, our approach can accommodate the selection phenomenon through two ways. Firstly, it is possible to consider the selection bias as a covariate in the study if one such covariate is available (e.g., Bycatch index). If no such covariate is available, it may be possible to build a mixture of two models (e.g., $M_1$ and $M_3$) in order to take into account an additional mortality on some age classes. However, such a development requires to conduct its own simulation study which is beyond the scope of this study. Since our approach is based on linear regression, it is straightforward to expand the model (e.g., random year effects, mixtures) while using familiar and well-established methods

for model selection (e.g., WAIC) and assessment of model fit (e.g., posterior predictive checks, $R^2$ statistics). These features are assets, and suggest that our approach is complementary to existing ones (e.g., Heligman & Pollard, 1980; Saavedra, 2018; Siler, 1979) to estimate vital rates in some data-poor species.

# 7 | CONCLUSION

Survivorship and hazard estimation are the goals of survival analysis but they remain difficult to achieve, even for some long-lived and charismatic vertebrate species such as cetaceans. User-friendly modeling methods are of primary interest to leverage the demographic information available in sparse, but previous, data that field ecologists have collected. We think that our framework can facilitate the difficult statistical problem of survival analysis in data-poor context by providing ecologists with a flexible method to obtain accurate survivorship estimates from
age at death data. A simple linear mixed-model can
accommodate various mortality patterns without drasti-
cally increasing the number of estimated model parame-
ters. All the methods developed to assess fit quality
within models can be applied since our framework is
based on a mixed linear regression approach. This frame-
work is also very convenient to deal with small sample
size. With as few as 100 data points, survivorship may be
estimated to conduct exploratory analysis. With 300 data
points, survivorship estimates can be precise enough to
build life-tables and project populations trajectories tak-
ing into account covariates. Covariate inclusion can allow
to distinguish different causes of mortality (e.g., pathol-
ogy, bycatch, collision and so on) and to quantify their
respective influence on survival. This is valuable when
the studied population is under various pressures as it
may allow to identify the most threatening ones, and to
design efficient and relevant conservation policies
accordingly.

ACKNOWLEDGMENTS

We would like to thank Pr. Elisabeth Slooten, Dr. Aleta
Hohn, Dr. Ursula Siebert, Dr. Sinéad Murphy, Pr. Jay
Barlow, Dr. Leszek Karczmarski and Dr. Camilo
Saavedra for sharing their data. Also, we would like to
thanks the two anonymous reviewers for their critical
comments that led to many improvements in the
manuscript.

AUTHOR CONTRIBUTIONS

Matthieu Authier, Vincent Ridoux and Etienne Rouby
conceived the ideas; Matthieu Authier and Etienne
Rouby chose the methodology; Etienne Rouby carried
out the analyses; Etienne Rouby and Matthieu Authier
led the writing of the manuscript. All authors contributed
critically to the drafts and gave final approval for
publication.

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## SUPPORTING INFORMATION

Additional supporting information may be found online in the Supporting Information section at the end of this article.

**How to cite this article:** Rouby E, Ridoux V, Authier M. Flexible parametric modeling of survival from age at death data: A mixed linear regression framework. *Population Ecology*. 2020; 1–15. https://doi.org/10.1002/1438-390X.12069
<figure><img src="image_155.png" /></figure>
# Appendix 3: Insights of the demographic approach

## Histology

<figure><img src="image_156.png" /><figcaption>Figure 1: Diagram of the ovulation process with the important stages of the corpus.</figcaption></figure>

## Convergence

To estimate vital rates from previously acquired data, we built 8 models combining both a survival and a reproduction part. The survival part was built from (Rouby, Ridoux, and Authier 2021) framework and the reproduction component was built from an Accelerated failure time framework. The two components may share a part of the overall variance. Estimation was carried out in a Bayesian framework using programming language Stan (Carpenter et al. 2017b) called from R v.4.0.1 (R Core Team 2020) with library Rstan (Stan Development Team 2020). Stan uses Hamiltonian dynamics in Markov chain Monte Carlo (MCMC) to sample values from the joint posterior distribution (Carpenter et al. 2017b). Four chains were initialised from diffuse random starting points and run for a total of 20,000 iterations, discarding the first 10,000 as warm-up. Default settings for the No-U-Turn Sampler (NUTS) were changed to 0.90 for adapt delta and 15 for max treedepth (Hoffman and Gelman 2014).
## Siler and Heligman-Pollard

Complementary to the use of this model, we also used the models of Siler (Siler 1979) and Heligman-Pollard (Heligman and Pollard 1980) which are the conventional methods in cross-sectional survival analysis on delphinids (Mannocci et al. 2012; Barlow and Hohn 1984; Stolen and Barlow 2003; Barlow and Boveng 1991; Secchi and Fletcher 2004). The Siler model is a five-parameter model ($a_1, b_1, a_2, a_3$ and $b_3$). Siler parameter estimations are realised using the Nelder & Mead optimization function (Nelder and Mead 1965) method and optim function of the package stats in R as detailed in the strandCet package (Saavedra 2018). The survivorship $l(x)$ and the total mortality $q(x)$ are expressed as the product of the three competing risks processes:

$$l(x) = l_j(x) \times l_c(x) \times l_s(x) \qquad (1)$$

$$q(x) = q_j(x) + q_c(x) + q_s(x) \qquad (2)$$

with the exponential decreasing risk due to the juveniles:

$$l_j(x) = \exp\left(\left(\frac{-a_1}{b_1}\right) \times (1 - \exp(-b_1 \times x))\right) \qquad (3)$$

$$q_j(x) = a_1 \times \exp(-b_1 \times x) \qquad (4)$$

with the constant risk experienced by each age-class:

$$l_c(x) = \exp(-a_2 \times x) \qquad (5)$$

$$q_c(x) = a_2 \qquad (6)$$

with the exponential risk due to senescent ages:

$$l_j(x) = \exp\left(\left(\frac{a_3}{b_3}\right) \times (1 - \exp(b_3 \times x))\right) \qquad (7)$$

$$q_j(x) = a_3 \times \exp(b_3 \times x) \qquad (8)$$

The Heligman-Pollard model used here is suited to admit cetacean bycatch at age 0 as it is explained in (Saavedra 2018). The mortality function at age $x$ is expressed as it follows:

$$q(x) = A^{(x+B)^c} + \left(I + D \times \exp\left(-E\left(\ln\left(\frac{x}{F}\right)\right)^2\right)\right) + \frac{GH^x}{(1 + GH^x)} \qquad (9)$$
It considers three mortality components: the young mortality curve $A^{(x+B)^c}$, the "accident hump" in adult life (for example due to bycatch) $D \times \exp(-E(\ln(\frac{x}{F}))^2)$ and the adult mortality curve $\frac{GH^x}{(1+GH^x)}$. The parameter *I* allows to control the starting point of the mortality curve, making possible to represent bycatch at age 0.

Both Siler and Heligman-Pollard models were computed using the `strandCet` R package and are used for comparison with the Reed survival model (Reed 2011). The use of Siler and Heligman-Pollard is more consistent with the literature and acts here as a sensibility analysis on the survival estimation for the common dolphin population.

We applied the Heligman-Pollard model on the dataset used in this study to compute mortality depending on the three life-components detailed in (Heligman and Pollard 1980). By using the methodology developed by (Saavedra 2018), it was possible to take into account a bycatch effect on 0 age-class. The overall mortality curve showed an increasing mortality from 0 to 21 years (figure 2) with a "plateau" from 5 to 8 years. The hump from 0 to 6 is associated to the additional mortality model component whereas the increasing along the lifetime is due to the senescent component of the model.

<figure><img src="image_157.png" /><figcaption>**Figure 2:** Mortality estimation for each age-class realised with the Heligman-Pollard adaptation from the StrandCet package that allows to take into account a bycatch effect on 0 age-class. The black solid line represents overall mortality. The blue dashed line is the mortality part associated with senescent individuals. The green dashed line is the mortality part due to juveniles. The red dotted line corresponds to the additional mortality part on the natural mortality. The white circles corresponds to the observed mortality rates computed with the data.</figcaption></figure>

## Logistic regression for Age at Sexual Maturity

To be consistent with the literature, we also used a logistic regression approach to compute the proportion of mature females and the age at sexual maturity (ASM) along the ages (Mannocci et al. 2012; Murphy et al. 2009; Danil and Chivers 2007). The logistic regression was fitted using a generalized linear model approach with the `glm()` function of the package `stats` on R. It is also possible to use the `dose.p()` function of the package `MASS` on R to determine the ASM using the previously fitted GLM.
## Leslie Matrixes

Both Leslie matrixes and elasticities were computed using the demogR package on R. To compute the matrix, we assumed a female population taking into account the age-specific cumulative survival $l(x)$ (as it is detailed in the package options) and the age-specific fertility rates $f(x)$. The multiplier for fertility equals 1 (since all age-class are of length one) and the infant.class parameter is equal to TRUE since there are three infant age-class.

## Life-tables building

Life-tables are useful tools to explore the demographical strategy of a particular species or of a population during its lifetime. We created North-East common dolphin life-tables on a period-based approach (i.e. vertical life-tables). We built life-tables depending on each modelling approach (i.e. simple; with covariate; with year effect). Here we present the basic structure of our life-table approach.

For each age $x$, several rates are computed. The first rate to be computed is $Q_x$ which is the probability of dying between age $x$ to $x + 1$: $Q_x = 1 - (\frac{l_{x+1}}{l_x})$ with $l_x$ as the survivorship rate to age $x$ (probability to reach the age $x$). From $Q_x$ it is possible to compute $D_x$ which is the number dying in the interval $x$ to $x + 1$: $D_x = N_x \times Q_x$ with $N_x$ the number of individuals alive in the middle of the age class $x$. The number surviving to at least age $x$, $L_x$, can be deducted as: $L_x = N_x - D_x$. From $L_x$ we can compute the total number of person-years lived by the cohort from age $x$ until all members of the cohort have died $T_x$ (i.e. the sum of $L_x$ from $x$ onwards): $T_x = \sum_{a=x}^{\infty} L_x$. Finally, it is possible to derive the life expectancy at age $x$: $e_x$ as $e_x = \frac{T_x}{L_x}$. It is possible to add the age-specific fertility rates obtained with pubescents $f_p(x)$ which are used to build the Leslie-Matrix.

## WAIC and LOOIC of models

<table><caption>**Table 1:** Model selection. Models are ordered in increasing order of WAIC (the smaller, the better the fit). *se* stands for 'standard error'.</caption><thead><tr><th>Model</th><th>Specification</th><th>Joint modelling</th><th>WAIC<sub>se</sub></th><th>LOOIC<sub>se</sub></th></tr></thead><tbody><tr><td>8</td><td>cov + years + trend + cor</td><td>Yes</td><td>733<sub>13</sub></td><td>1169<sub>16</sub></td></tr><tr><td>7</td><td>cov + years + trend</td><td>Yes</td><td>730<sub>13</sub></td><td>1150<sub>16</sub></td></tr><tr><td>6</td><td>cov + years + cor</td><td>Yes</td><td>782<sub>13</sub></td><td>1195<sub>16</sub></td></tr><tr><td>5</td><td>cov + years</td><td>Yes</td><td>780<sub>13</sub></td><td>1194<sub>16</sub></td></tr><tr><td>4</td><td>years + cor</td><td>Yes</td><td>781<sub>13</sub></td><td>1192<sub>16</sub></td></tr><tr><td>3</td><td>years</td><td>Yes</td><td>777<sub>13</sub></td><td>1190<sub>16</sub></td></tr><tr><td>2</td><td>cov</td><td>No</td><td>922<sub>14</sub></td><td>1307<sub>17</sub></td></tr><tr><td>1</td><td>simple</td><td>No</td><td>925<sub>14</sub></td><td>1318<sub>17</sub></td></tr></tbody></table>
# Appendix 4: Explain and quantify the incidental captures of common dolphins

Vol. 679: 195-212, 2021

https://doi.org/10.3354/meps13894

MARINE ECOLOGY PROGRESS SERIES

Mar Ecol Prog Ser

Published November 25

<figure><img src="image_158.png" /></figure>

## Spatiotemporal models highlight influence of oceanographic conditions on common dolphin bycatch risk in the Bay of Biscay

Lola Gilbert<fnref n="1" />,<fnref n="2" />,*, Etienne Rouby<fnref n="1" />,<fnref n="2" />, Emilie Tew-Kaï<fnref n="3" />, Jérôme Spitz<fnref n="1" />,<fnref n="2" />,
Hélène Peltier<fnref n="1" />,<fnref n="4" />, Victor Quilfen<fnref n="3" />, Matthieu Authier<fnref n="1" />,<fnref n="4" />

**ABSTRACT:** The population of short-beaked common dolphins *Delphinus delphis* of the Bay of Biscay (northeast Atlantic) has been subjected to potentially dangerous levels of bycatch since the 1990s. As the phenomenon intensifies, it represents a potent threat to the population. Here, we investigated the relationship between bycatch mortality and oceanographic processes. We assumed that oceanographic processes spatiotemporally structure the availability and aggregation of prey, creating areas prone to attract both common dolphins and fish targeted by fisheries. We used 2 datasets from 2012 to 2019: oceanographic data resulting from a circulation model and mortality data inferred from strandings. The latter allows location of mortality areas and quantification of the intensity of mortality events at sea. We fitted a series of spatiotemporal hierarchical Bayesian models using integrated nested Laplace approximations (INLA). Results provided first insights on how bycatch of common dolphins in the Bay of Biscay might be related to key seasonal and dynamic oceanographic features. We showed that from a statistical predictive point of view, the monthly trend of 2019 bycatch mortality could be predicted with few oceanographic covariates. This study highlights how gaining knowledge about environmental influences on interactions between short-beaked common dolphins and fisheries could have great conservation and management value. Identified relationships with oceanographic covariates were complex, as expected given the dynamic aspects of oceanographic processes, dolphins and fisheries distributions. Further research focusing on smaller time scales is needed to elucidate proximal drivers of common dolphin bycatch in the Bay of Biscay.

**KEY WORDS:** Spatiotemporal modelling · Reverse drift modelling · Prey–predator relationship · Sea surface temperature · Thermal fronts · Eddies

Resale or republication not permitted without written consent of the publisher

## 1. INTRODUCTION

Incidental catch, or bycatch (i.e. the undesired catch of nontargeted species in fishing gear), represents the most potent and well-identified threat to small cetaceans' populations (Reeves et al. 2005). Because they are long-lived species, cetaceans can-

not sustain high rates of depletion (Lewison et al. 2004, Mannocci et al. 2012). Bycatch has already led populations or species to serious reduction (Reeves et al. 2005, Brownell et al. 2019) or even extinction (T revenue et al. 2007, Brownell et al. 2019), the vaquita por- poise *Phocoena sinus* Norris and MacFarland, 1958 (Norris & MacFarland 1958) being a potentially immi-

<footnote n="1">¹Observatoire Pélagis, UMS 3462 Université de La Rochelle - CNRS, 17000 La Rochelle, France</footnote>

<footnote n="2">²Centre d'Etudes Biologiques de Chizé-La Rochelle, UMR7372 Université de la Rochelle - CNRS, 17000 La Rochelle, France</footnote>

<footnote n="3">³Shom, CS 92803-29228 Brest, France</footnote>

<footnote n="4">⁴ADERA SAS, CS 60040-33608 Pessac Cedex, France</footnote>

<footnote>*Corresponding author: lola.gilbert@univ-lr.fr</footnote>