A new perspective on the management of wild boar populations, based on a state-space model

Abstract

Wild boars easily adapt to highly diverse habitats. As a consequence, these animals currently represent the most widespread ungulate species worldwide, with many publications attesting to the impressive growth in numbers and spatial spread over the last few decades. Most wild boar populations are being managed by harvesting, not least to reduce the damage caused to agriculture and natural habitats. However, management success seems limited at best: hunting bag numbers continue soaring in Europe and elsewhere, and the lack of abundance estimates is a major factor hampering efficient and effective management actions.
In this study we propose a new perspective on wild boar state-dependent management based on a state-space model, and this model is presented by analysing time series data from a local population in Switzerland. We show how to estimate past and present population abundances while explicitly accounting for mast dependencies in model components. We further show how to use parameter estimates and abundance forecasts to set management targets to efficiently regulate a population.
For the analysed study population as one result we show that the discrepancy between the harvesting effort required and the one realized in mast years potentially hampers population regulation. This is caused by mast intensity having a positive effect on population growth while negatively affecting harvesting success. This result and the strong mast dependency of the population’s dynamics found in this study suggest setting management targets on a yearly basis as well as adjusting harvesting strategies in a mast-dependent way.
The approach we present is suitable to generate abundance estimates and set state-dependent management targets on a yearly basis while accounting for mast effects. The model can be used as a template for wild boar populations inhabiting dis-/similar habitats to the one analysed or for other species with similar factors affecting population dynamics. Only widely available data such as hunting bag and agricultural damage costs are needed, and we hope that the approach presented will help to better understand local conditions shaping the dynamics of wild boar populations and other over-abundant wildlife species.

Full text

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A new perspective on the
management of wild boar populations,
based on a state-space model
Technical Report
Claudio Bozzuto
Wildlife Analysis GmbH
Hannes Geisser
Natural History Museum Thurgau
Zurich, 21 September 2019

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A new perspective on the management of wild boar populations,
based on a state-space model
Technical report
Authors
Claudio Bozzuto Hannes Geisser
Wildlife Analysis GmbH Natural History Museum Thurgau
Oetlisbergstrasse 38 Freie Strasse 24
8053 Zurich 8510 Frauenfeld
Switzerland Switzerland
bozzuto@wildlifeanalysis.ch hannes.geisser@tg.ch
Date published | 21 September 2019.
How to cite this document | Bozzuto, C., Geisser, H. (2019): «A new perspective on the
management of wild boar populations, based on a state-space model». Technical Report
Wildlife Analysis GmbH, Zurich, Switzerland. DOI: 10.13140/RG.2.2.29713.79206/1.
Copyright notice | The authors are the copyright holders, licence CC-BY-NC-ND 4.0.
Cover illustration | Andrea Klaiber, Doppelkopf Grafik & Illustration, Schaffhausen.

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Acknowledgements | We are grateful to Roman Kistler and Michael Vogel (Hunting and
fishing administration, Canton of Thurgau, Switzerland) for their dedicated interest in and
support of this new perspective on wild boar management. We further thank Stefano
Canessa (Ghent University), Anthony R. Ives (University of Madison-Wisconsin), and Heinz-
Uli Reyer (University of Zurich) for their comments that greatly improved this report.
Author contributions | CB set up and analysed all mathematical models; HG provided wild
boar-related information; both authors wrote the report.

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Contents
Abstract ………………….………………….………………….………………….……………
4
Introduction …………….………………….………………….………………….…………….
5
Materials and methods …………….………………….………………….…………………..
8
Study area …………….………………….………………….……………………..
8
Available data …………….………………….………………….…………………
8
State-space model: presentation …………….………………….……………….
9
State-space model: estimation, filtering, and smoothing ……………………...
13
Forecasting and state-dependent management ……………………………….
14
Alternative population reconstruction ……………………………………………
14
Results …………….………………….………………….………………….…………………..
15
Discussion …………….………………….………………….………………….……………...
20
References …………….………………….………………….………………….……………..
23
Supplementary material …………….………………….………………….…………………
28

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Abstract
Wild boars easily adapt to highly diverse habitats. As a consequence, these animals currently
represent the most widespread ungulate species worldwide, with many publications attesting
to the impressive growth in numbers and spatial spread over the last few decades. Most wild
boar populations are being managed by harvesting, not least to reduce the damage caused
to agriculture and natural habitats. However, management success seems limited at best:
hunting bag numbers continue soaring in Europe and elsewhere, and the lack of abundance
estimates is a major factor hampering efficient and effective management actions.
In this study we propose a new perspective on wild boar state-dependent management based
on a state-space model, and this model is presented by analysing time series data from a
local population in Switzerland. We show how to estimate past and present population
abundances while explicitly accounting for mast dependencies in model components. We
further show how to use parameter estimates and abundance forecasts to set management
targets to efficiently regulate a population.
For the analysed study population as one result we show that the discrepancy between the
harvesting effort required and the one realized in mast years potentially hampers population
regulation. This is caused by mast intensity having a positive effect on population growth
while negatively affecting harvesting success. This result and the strong mast dependency of
the population’s dynamics found in this study suggest setting management targets on a
yearly basis as well as adjusting harvesting strategies in a mast-dependent way.
The approach we present is suitable to generate abundance estimates and set state-
dependent management targets on a yearly basis while accounting for mast effects. The
model can be used as a template for wild boar populations inhabiting dis-/similar habitats to
the one analysed or for other species with similar factors affecting population dynamics. Only
widely available data such as hunting bag and agricultural damage costs are needed, and we
hope that the approach presented will help to better understand local conditions shaping the
dynamics of wild boar populations and other over-abundant wildlife species.

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Introduction
Wildlife and humans often live in close neighbourhood, and many species suffer from human
activities: for example habitat loss and fragmentation, disturbance, or poaching, all incur a
considerable toll on many wildlife species worldwide (Soulé, 1991). Some wildlife species,
however, not only have successfully adapted to the ‘human factor’, but even profit from
habitats shaped by human activities. Such species reach high population densities and
continually colonize new areas, at the expense of potentially causing human-wildlife conflicts
(Conover, 2001): examples include damages caused to agricultural areas or natural habitats,
the spread of diseases, overharvesting of other wildlife populations as their prey, or causing
traffic collisions (Fattebert, Baubet, Slotow, & Fischer, 2017). To resolve, or at least to
minimize, the outcome of these conflicts, efficient wildlife management strategies are needed.
The wild boar (Sus scrofa) can be considered as one of those species that has successfully
adapted to human-influenced landscapes. At present, the species thrives on all continents
except Antarctica (Carpio, Hillström, & Tortosa, 2016) and is regarded as one of the most
widely distributed large mammal species in the world (Genov & Massei, 2004). In the last five
decades the species has increased in numbers and spread considerably in many countries
worldwide (Kopij & Panek, 2016; Vetter, Ruf, Bieber, & Arnold, 2015). For example, Massei et
al. (2015) estimated a minimum of 2.2 million wild boars harvested across 18 European
countries in 2012, compared to an estimated 864’000 animals harvested two decades earlier.
At the same time these authors found a stable or declining harvesting pressure over that
period, thus suggesting that the increasing hunting bag data reflect a continued growth of the
analysed wild boar populations.
An important factor determining the species’ success is its high reproductive potential
(Frauendorf, Gethöffer, Siebert, & Keuling, 2016; Servanty et al., 2011 and references
therein), comparable to small mammals (Gaillard & Yoccoz, 2003). Environmental conditions
strongly affect wild boar population growth (Gamelon et al., 2017; Vetter et al., 2015), survival
and fecundity rates being particularly sensitive to food availability (Herrero, García-Serrano, &

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García-González, 2008). Fruits of deciduous trees in forests (the species’ primary habitat),
such as beech, oak or chestnut, constitute the most important natural food resource in many
areas (Servanty, Gaillard, Toïgo, Brandt, & Baubet, 2009). These tree species intermittently
produce disproportionately high amounts of fruits (full mast), while in other years an almost
complete lack of fruits is observed (Burschel & Huss, 1987; Gamelon et al., 2017).
Expectedly, the frequency and intensity of mast events strongly affect wild boar population
growth rates, driving large inter-annual fluctuations in population abundance (Bieber & Ruf,
2005; Gamelon et al., 2017). For example, Bieber and Ruf (2005) found a population’s
multiplicative growth rate (!) to vary from !=0.85 in years with a lack of mast up to
!=1.63 in years with full mast. Mast of forest trees is of course not the only environmental
factor influencing a wild boar population. Nonetheless, many studies have shown that food
conditions influence wild boar populations far more than any other environmental factor
(Frauendorf et al., 2016; Geisser & Reyer, 2005; Holland, Burrow, Dytham, & Aegerter, 2009;
Vetter et al., 2015). It is therefore reasonable to focus on this key environmental factor when
modelling the dynamics of a wild boar population.
When natural habitats are in close vicinity of farmland, wild boar activity frequently extends
into the latter (e.g. Kopij & Panek, 2016). Conflicts due to agricultural damages or the wildlife-
livestock transmission of diseases like African swine fever have caused increased public and
political discussions in many countries (Baldwin, Salmon, Schmidt, & Timm, 2014; Frauendorf
et al., 2016; Gethöffer, Sodeikat, & Pohlmeyer, 2007; Holland et al., 2009). Meanwhile, the
IUCN’s Invasive Species Specialist Group considers the wild boar among the 100 of the
world’s worst invading species (Lowe, Browne, Boudjelas, & De Poorter, 2000).
Although considerable knowledge of the species’ biology and ecology has accumulated
during the last decades (e.g. Melletti and Meijaard 2017), judging by the ongoing
development worldwide it seems that this knowledge has only partly aided wild boar
management, and this is probably unlikely to change in the near future without substantial
changes to management-related hunting practices (Massei et al., 2015; Vetter et al., 2015).
Several reasons might contribute to this limited success. A main reason is that abundance

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estimates are rare (Ebert, Knauer, Spielberger, Thiele, & Hohmann, 2012; Engeman, Massei,
Sage, & Gentle, 2013), not least because wild boars are difficult to count (ENETWILD
consortium et al., 2018). Consequently, harvest targets and other management decisions are
often based on hunting bag statistics or road kill numbers (Imperio, Ferrante, Grignetti,
Santini, & Focardi, 2010; Vetter et al., 2015). However, these approaches might misrepresent
the true dynamics of wild boar populations (e.g. ENETWILD consortium et al., 2018). For
example, this is true in cases where mast intensity negatively affects hunting success while
positively affecting population growth: in such years hunting bag data will underestimate
population growth if used as a proxy for abundance. More fundamentally, for a growing
population it is difficult to assess the efficacy of past management actions based on hunting
bag data alone. A further reason hampering wild boar management might be incomplete
knowledge of mast events affecting population dynamics and hunting success (Cutini et al.,
2013 and references therein). Besides, such knowledge would require adapting harvesting
targets annually based on current mast conditions, a practice highly recommended in several
previous studies (e.g. Bieber & Ruf, 2005; Cutini et al., 2013). However, the harvesting effort
needed in some years might simply not be feasible.
In light of the mentioned difficulties surrounding current wild boar management practices, as a
new perspective on the efficient control of wild boar populations we present a state-
dependent management approach based on a state-space population model to determine
management targets. The statistical dynamical model allows estimating past, present, and
forecasted population abundances, based on time series data that are very often available to
wildlife managers. At the same time the method allows estimating mast-dependent
parameters, such as harvesting proportions and the population’s growth rate, that aid setting
management targets. We base the presentation on the analysis of the wild boar population in
the north-eastern Swiss Canton of Thurgau. This population has grown considerably over the
last three decades and continues causing considerable agricultural damage (Geisser &
Reyer, 2004, 2005), thus paralleling the development in many other European regions and
elsewhere.

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Materials and methods
Study Area
The Canton of Thurgau in north-eastern Switzerland is an administrative entity and spans an
area of roughly 86’000 ha, with elevation ranging from 400 to 1000 meters above sea level.
Mainly mixed beech forests and farmland characterize the landscape. While several
European regions saw a sharp increase in wild boar numbers after Second World War
(Massei et al., 2015), the animals already present in the Canton of Thurgau were mainly
roaming over a larger area until the early 1990s, without appreciable local reproduction.
Natural predators of wild boars such as wolves continue to be absent, but wild boars have
been harvested since their arrival. Despite harvesting, the agricultural damage caused by the
growing population continues remaining on a high level, in terms of compensation costs as
well as of public and stakeholder awareness, calling for a deeper understanding of the
population’s dynamics and for innovative methods to plan management actions.
Available data
Our main motivation to formulate our state-space model was to use data very often available
to wildlife managers. Hunting bag data, often by legal requirement, are collected for most
harvested wild boar populations. Some authorities might possess age-structured data, but
maybe only spanning a short period, compared to time series of yearly total numbers of
harvested animals potentially spanning several decades.
In addition to hunting bag data (Fig. S1a), where a hunting season starts in April and ends in
March of the following year, we used data on yearly compensation costs for agricultural
damage caused by wild boars (Fig. S1b); these data, too, were organized according to
hunting seasons. The idea behind using this additional data source is that harvesting data are
most useful when corrected for a potentially time-dependent effort (Skalski, Ryding, &
Millspaugh, 2005). However, if working only with uncorrected data, the model runs into an
identifiability problem: a yearly proportion removed from a population that depends on its
current abundance and on a time-dependent effort is very difficult to separate using hunting

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bag data alone. Thus, including compensation cost data offers additional information on the
population state variable. Note that compensation cost data are a proxy for the caused
damage, because of fluctuations in crop assemblages or compensation valuation; see
subsection State-space model: presentation for the inclusion of this variation into our model.
Hunting bag data and compensation cost data were made available by the Hunting and
fishing administration of the Canton of Thurgau.
Finally, given the sensitivity of wild boars to mast events of forest trees, we gathered mast
data assuming that in most countries a forestry agency monitors mast conditions. We used
yearly indexed beech mast data (Fig. S1a-b; Burkart, 2018): the index !!∈0,1,2,3 covers
the range ‘lack of mast’ (0) up to ‘full mast’ (3), similar to other studies where yearly mast
production is categorized (e.g. Gamelon et al., 2017; Geisser & Reyer, 2005). Mast data were
available for the years 1982-2017, and consequently we analysed the study population for
this period.
State-space model: presentation
The application of state-space models in wildlife ecology and management has been gaining
momentum lately (e.g. Newman et al., 2014, and references therein). A state-space model is
a statistical dynamical model that offers the possibility to explicitly account for measurement
error, thus leading to the conceptual separation of a process of interest – assumed not
observable – from observations, i.e. data that are linked to this process, albeit with
uncertainty. For example, if the task is to model how a population changed over time and
abundance counts were available for this population, these data most probably would contain
measurement error. Measurement error modelling allows even ‘looser’ connections between
unobservable state variables and data. For instance, in our study we use hunting bag data
that can be, in a statistically appropriate way, linked to the modelled, but unobservable,
abundance (see below). In addition to the separation of data and unobservable processes,
further advantages of working with a state-space model are its iterative nature, allowing

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trends in the data to be explicitly modelled and estimated, as well as the ease to produce
forecasts, i.e. estimating future population abundances.
We first state the full model, consisting of a measurement sub-model (eq. 1a-b) and an
unobservable process sub-model (eq. 1c-d), and we then present the rationale behind this
formulation.
!!=!!!1−!!(!!!!!!)!!!
(eq. 1a)
!!=!!!!!−!!!!!!!
(eq. 1b)
!!=!!!!!!+!!!!!!+!!!!+log 1−!!!!!!!!!!+!!+!!
(eq. 1c)
!!=!!!!+!!
(eq. 1d)
Our model will be fit to yearly data, and thus time !−1 and ! refer to two adjacent years. As
a consequence, data and estimated process states mirror one time point per year. We chose
to set up a post-breeding model, meaning that we track yearly changes in population
abundance after reproduction, July say, assuming that wild boar mainly produce offspring
once per year in spring (Briedermann, 2009). A post-breeding model is useful to efficiently
capture mast effects: mast intensity mainly affects winter survival and reproduction, and
therefore both effects fall into the same modelled year using a post-breeding model.
In eq. 1a we relate harvesting data, !!, to the unobservable log-transformed population
abundance !!=log !!, where !! is the (approximately mid-year) post-breeding population
abundance. The idea behind eq. 1a is to assume a time- and mast-dependent binomial
process: every year a proportion !!=1−exp −ℎ!+!!! of the population is harvested,
and this proportion changes through time and depends on mast conditions. The inclusion of
mast-dependent harvesting success stems from the hypothesis that in poor mast years wild
boars are easier to hunt because they visit agricultural fields and hunters’ baiting sites more

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often. Therefore, in eq. 1a the instantaneous harvesting mortality rate, ℎ!, is adjusted based
on mast conditions, !!, where parameter !>0 (to be estimated) is a scaling factor: with full
mast wild boars have the highest, and with lack of mast the lowest (deterministic) survival
probability. As detailed below, we connected the state variable !! (eq. 1d) to the population’s
proportion surviving harvesting in eq. 1a using the logit transformation !!=logit !!!!.
Finally, the random variable !!! accounts for the fact that even if the harvesting proportion
were fixed, hunting bag would nonetheless vary through time by chance alone. Given the
assumed binomial process, on a log scale the random variable can be approximated by
!!~!0,log !!
!!!!!+1, and thus !! is approximately normally distributed with zero mean
and the reported state-dependent variance; see Bozzuto and Ives (2019, under review) for
further details. To reduce the total number of parameters to be estimated, we approximated
this variance by setting !!
!=1−!!!!
!! (Fig. S2).
In addition to harvesting data, we used compensation cost data related to agricultural
damage, !! (eq. 1b). The idea behind eq. 1b is similar to the previously presented harvesting
equation: yearly compensation costs depend linearly on population abundance with a per
capita effect of !!−!!!!, where !! and !!>0 are parameters to be estimated. Thus, we
assumed that in years lacking mast, damages are highest compared to years when food
sources in forests are more abundant. The random variable !!~!0,!!
!, with variance !!
! to
be estimated, captures damage-related measurement error: because we used yearly total
damage costs disregarding fluctuations in crop assemblages or compensation valuation, this
accounts for yearly fluctuations related to these factors.
The unobservable process sub-model (eq. 1c-d) consists of two state variables capturing
changes in population abundance and changes in harvesting effort, respectively. Most wild
boar populations seem to experience a very weak negative density-dependent regulation, if at
all (Choquenot, 1998; Frauendorf et al., 2016; Vetter et al., 2015). However, wild boars are
strongly regulated by changing environmental conditions, especially available food sources

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(see Introduction). Thus, based on previous findings (Bieber & Ruf, 2005) we included, on the
log scale of eq. 1c, a density-independent population growth rate that depends linearly on
mast intensity, !
!=!!+!!!!, where !! and !!>0 are parameters to be estimated;
therefore, the highest population growth rate is achieved in full mast years. The indicator
variable !!={0,1} allowed separating years up to 1991 (the authors’ estimate) when the
population was roaming (see sub-section Study area), from the following years when it
became established and reproduced locally; for the roaming years we excluded local
reproduction (!!=0). A further important factor affecting population dynamics is harvesting,
and we included hunting bag data as a covariate, where the term log 1−!!!!!! in eq. 1c is
the log-transformed proportion surviving harvesting in year !. Finally, we included two
different sources of variability affecting population changes, demographic stochasticity (!!)
and environmental stochasticity (!!). The random variable !! models state-dependent
variability stemming from reproduction at low abundance, and we defined
!!~!0,log !!"#
!!!!!+1 (Bozzuto & Ives, 2019, under review). Environmental
stochasticity, on the other hand, is independent of the mean and captures year-to-year
variation in population growth not accounted for by the other model components in eq. 1c,
and we assumed !!~!0,!!"#
!; both variances !!"#
! and !!"#
! will be estimated.
Finally, the second state variable, !!, tracks changes over time in harvesting effort (eq. 1d).
Lacking effort-related data, we modelled the logit-transformed harvesting survival probability,
!!=logit !!!!, as a random walk, assuming !!~!0,!!
! and the variance !!
! to be
estimated. The logit-transformation allows modelling proportional data as normally distributed
data. Using such a random walk formulation is useful because the fitting routine will deduce
the yearly changes directly from data (e.g. Ives & Dakos, 2012). In sum, including this state
variable allows capturing potential changes over time in effort, including a trend.

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State-space model: estimation, filtering, and smoothing
We used the extended Kalman filter (EKF; chapter 10 in Durbin & Koopman, 2012) to fit our
model (eq. 1a-d) to hunting bag, compensation cost, and mast data; all analyses were
performed using MATLAB (MathWorks 2015). We log-transformed eq. 1a-b, along with the
respective data, and in addition to the previously presented parameters and variances we
also estimated the initial log-transformed abundance, !!, and the initial transformed
harvesting effort, !!, both for year 1982. To initialize the EKF, we used the respective
variances in eq. 1c-d for the parameterization of the process (co-)variance matrix. We first
fitted the model (filtering) and at the same time estimated parameters (maximum likelihood).
Filtering is based on iteratively using the adjacent time steps in the model, and this is
adequate for parameter estimation. To estimate the whole time-evolution of the state
variables, here including abundance, it is advisable to subsequently apply a state smoother,
where the state estimates are ‘adjusted’ by considering all available data (chapter 4 in Durbin
& Koopman, 2012).
Using the full model (eq. 1a-d) for the EKF rarely converged to produce robust estimates and
abundance reconstructions. We thus adapted the EKF in an admittedly ad hoc way as
follows. The standard EKF procedure is based on the linearization of all equations with
respect to the state variables, in our case !! and !!. Our adaptation consists in linearizing the
process equations (eq. 1c-d) in a standard fashion with respect to !! and !!, whereas we
linearized the measurement equations (eq. 1a-b) with respect to !! and the proportion
surviving harvesting, !!!!, instead of !!, recalling that logit!!!!=!!!!; see the
Supplementary Material for the linearized equations (eq. S1a-b). This approach allowed for
robust, reproducible results.

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Forecasting and state-dependent management
A state-space model approach to a state-dependent wild boar management is helpful
because of the ease to produce forecasts. For example, with the most recent data from
hunting season 2017/18, the last possible abundance reconstruction reflects the population’s
abundance in mid-2017 (see above). Nonetheless, given the already realized hunting bag
data and mast index for that season, the current abundance (mid-2018) can be forecasted
using the state-space model: this simply means treating the current year as a year with
missing data; otherwise nothing else has to be changed, and this also includes calculating
confidence intervals for the forecasted abundance. To this end, we used smoothed state
variables and variances; for further details see chapter 4 in Durbin and Koopman (2012).
Alternative population reconstruction
To qualitatively validate the reconstructed population trajectory from the EKF, we additionally
used an age-at-harvest reconstruction from a previous project (unpublished results), following
Gulland (1965) with minor modifications; we will refer to this approach as the Gulland method.
To this end, we used stage-structured data spanning 14 years (2002/03 – 2015/16, Fig. 1),
containing yearly counts of harvested piglets (0-12 months), yearlings (13-24 months), and
adults (25+ months). Further, we used life history data from the literature (Briedermann,
2009), partly adjusted by one of us (HG) to the study area environment (Table S1); further
details can be found in the Supplementary Material.

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Results
We start with the reconstructed population trajectory based on the state-space model and, in
comparison, the reconstruction based on age-at-harvest data (Fig. 1). Over the shared
timeframe, the EKF reconstruction is in good agreement with the Gulland method
reconstruction. Note that we do not expect a perfect match: although both methods share the
same hunting bag data, albeit very differently employed, our implementation of the Gulland
method for example does not account for mast-dependent variation in vital rates.
The maximum likelihood point estimates for the EKF suggest two mast dependencies
affecting changes in population abundance: population growth as well as the proportion
harvested change according to mast intensity; all estimated model parameters are reported in
Table S2, and the estimated multiplicative growth rate !!!=exp !!+!!!! is
additionally reported in Table 1. The estimated mast-dependent multiplicative growth rate is
comparable to results based on data from Eastern Germany and Poland (Bieber & Ruf,
2005); the quantitative difference here is probably driven by a more severe climate in the
latter regions. The proportions harvested, modelled as a latent state variable, were negatively
related to mast intensity (Fig. 2): the highest values are achieved in years lacking mast (index
0). Further, the EKF results also indicate a substantial increase in effort over the last two
decades, albeit with the tendency to levelling off (Fig. 2). The modelled harvesting proportions
are in good agreement with the realized proportions (Fig. S3), i.e. hunting bag data (Fig. S1a)
proportionally related to the reconstructed abundances (Fig. 1). But there are years in the
1990s with noticeable deviations (Fig. S3) probably caused by exceptional weather conditions
and/or transient dynamics when the population became stationary, and whose effects are not
duly captured by the model components (eq. 1a-d). Moreover, the modelled harvesting
proportions are accompanied by high uncertainty in the early years, whereas this uncertainty
is reduced in later and recent years (Fig. S3). Finally, in contrast to growth rate and
harvesting success, agricultural damage was deemed to lack a mast dependency (Table S2).
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!
Figure 1 | Reconstructed population trajectories and one-step-ahead forecast. The x-axis gives the hunting
season, where e.g. 2010 is brief for the season 2010/11. The dark green lines show the reconstructed abundance
with 95% confidence limits (solid and dotted lines, respectively) based on the EKF; in comparison, the red dashed
line shows the reconstructed abundances based on the Gulland method (using age-at-harvest data). The vertical
line with associated probability distribution (far right) gives the projected abundance for mid-2018, where the
marker shows the expected abundance (the distribution’s mean) and the line itself spans the 95% confidence
interval. The underlying bars show the scaled mast index values, originally ranging from 0 (lack) to 3 (full).

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!
!
Figure 2 | Reconstructed expected mast-dependent proportional hunting bag. The x-axis gives the hunting
season, where e.g. 2010 is brief for the season 2010/11. For every mast intensity (legend) the reconstructed
average, statistically expected harvested proportions are shown; years with identical mast index (dots) are
connected. The underlying bars show the scaled mast index values, originally ranging from 0 (lack) to 3 (full).

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Given our hypothesized lack of density-dependent population regulation, the trajectory (in
principle without harvesting) follows exponential growth (Fig. 1). In such a case, to
deterministically keep the abundance at a current level, the necessary proportion to be
harvested is density-independent and should equal 1−1/!!!. In Table 1 we report these
mast-dependent necessary harvesting proportions, along with a comparison (column
Difference) with the realized proportions harvested (see also Fig. S3). Here, a clear pattern
emerges: When mast is lacking (index 0), the proportion harvested is more than sufficient to
keep a current abundance (positive difference), even leading to a deterministic net
abundance reduction. However, when mast is present the proportions harvested are
insufficient, and this insufficiency worsens with increasing mast index. For example, this
means that in a year with full mast a population of 1000 animals can nonetheless increase to
approximately 1250 animals, harvesting included. Interestingly, the reconstructed
abundances over the last five years (Fig. 1) suggest successful regulation, in the sense of a
population fluctuating around a stable mean. This short-term stationarity is most probably the
result of alternating years with full mast and lack of mast: even if hunting effort is insufficient
in full mast years, it is more than sufficient in years lacking mast and effectively helps
reducing population abundance. In fact, Bieber and Ruf’s (2005) analysis of mast data
showed that full mast years are almost always followed by years with a complete lack of
mast.

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Table 1 | Comparison of required vs. realized proportions harvested, ordered by the mast intensity index
!!. !!!: estimated multiplicative, mast-dependent population growth rate. Proportion harvested required (in %),
i.e.!100 1−1/!!!: applying these proportions deterministically leads to an unchanged abundance in a
particular year. Proportion harvested realized (in %): most recent, realized proportions (Fig. S3). Difference (in
percent points): difference between realized and required proportions.
!!
!!!
(!!!)
Proportion harvested
required (%)
Proportion harvested
realized (%)
Difference
(%-points)
0 (lack)
1.259
21.0
35.2
14.2
1
1.357
26.3
22.5
- 3.8
2
1.463
31.7
16.7 1
- 15.0
3 (full)
1.577
36.6
20.5
-16.1
1!Most!recent!data!from!hunting!season!2006/07;!today!most!probably!higher.!
Finally, to gauge the number of animals to be harvested during the current season in order to
achieve a management target, we need to forecast the population’s abundance under mast-
dependent effects on population growth and harvesting success; we will elaborate on the
envisioned state-dependent management approach in more detail in the Discussion. In the
far right of Fig. 1 we show the forecasted abundance for mid-2018, including its 95%
confidence limits. Further, we supplement it with the associated probability density function
showing how likely a particular projected abundance is, with the marker as the distribution’s
mean. In sum, because the season 2017/18 was characterized by a lack of mast (index 0), it
seems highly likely, with a probability of 83%, that the abundance mid-2018 will be below that
of mid-2017: the expected forecasted abundance is 2250 animals, implying an expected
(deterministic) reduction of approximately 510 animals.

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Discussion
Through the analysis of the wild boar population in the Swiss Canton of Thurgau we have
presented a state-space model that allows reconstructing abundances and at the same time
estimating mast-dependent parameters driving the dynamics. In turn, estimated parameters
and the current, forecasted population abundance can be used for a state-dependent
management approach to regulate a population on a yearly basis, thus fully accounting for
mast effects.
A successful regulation of a wild boar population highly profits from having an estimate of the
current population abundance (see Introduction) – to calculate a required hunting bag in
absolute terms – as well as knowing how population growth and harvesting depend on mast
intensity. We thus recommend that wild boar managers explicitly embrace mast conditions as
follows. We suggest the implementation of a state-dependent management strategy in three
steps; in Fig. 1, !−1 refers to the season 2017/18, ! to the season 2018/19, and !+1 to the
season 2019/2020:
1. Reconstruct population abundance for year !−1 (mid-year).
2. With hunting bag data and mast intensity of year !−1 now at hand (i.e., in year !),
calculate the expected population abundance (forecast, incl. 95% confidence interval)
for year ! (mid-year).
3. With a reliable guess of the mast index for year ! and target abundance set by
management for mid-year !+1, calculate the necessary hunting bag to achieve the
management target.
For the analysed population, a reliable guess of the current mast intensity is made available
in late summer / early autumn. Thus, for the remaining second half of the hunting season,
managers have the possibility to potentially increase harvesting success. For example, an
obvious decision is to concentrate the deployable effort on wild boar harvesting. Further, the

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additional recruitment of personnel in case of a full mast year could be an option: this would
generally increase hunting pressure, but also allow additional and (more) efficient driven hunt
events. As a last example, Bieber and Ruf (2005) showed that the sensitivity of a wild boar
population’s growth rate to vital rates is mast-dependent. Thus, managers could preferentially
target (st)age classes according to current mast conditions, as opposed to the present study
where we assume untargeted harvesting with respect to sex and (st)age.
How well the presented approach will perform in other regions remains to be seen and tested.
Estimated parameter values (Table S2) and the comparison with an alternative abundance
reconstruction method (Fig. 1) both suggest reliable, i.e. realistic, results. Nonetheless,
correlations in data and/or model parameters could potentially hamper estimating realistic
values of parameters or abundances. For example, a high growth rate estimate could be
accompanied by a high harvesting proportion estimate counteracting the former estimate, but
together they might nonetheless produce realistic abundance estimates. Further, our
admittedly ad hoc adaptation of the EKF (see Materials and methods, subsection State-space
model: estimation, filtering, and smoothing) deserves critical scrutiny. In general, the
presented model (eq. 1a-d) should be regarded as a template to be adapted to local habitat
or agricultural conditions and management practice or could be expanded to include
additional data sources such as wildlife vehicle collision data (Gren, Häggmark-Svensson,
Andersson, Jansson, & Jägerbrand, 2016).
Geographical location does seem to be one factor contributing to differences in wild boar
population growth over a large scale (Holland et al., 2009; Keuling et al., 2013; Massei et al.,
2015). We presented our state-space model by analysing a single wild boar population in a
Swiss canton. Although the characteristics and correlations of the data used are likely to be
country-specific in detail, they have been reported by numerous studies (e.g. Briedermann,
2009; Frauendorf et al. 2016; Gamelon et al. 2017; Gethöffer et al. 2007; Howells & Edwards-
Jones, 1997), indicating that the analysed Swiss population probably mirrors populations in
other regions in their dynamics.

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We included several mast dependencies in our model, based on prior knowledge or
hypotheses (see Materials and methods). Note, however, that we did not force the fitting
routine to consider these dependencies. In fact, the maximum likelihood results entail no
mast dependency of agricultural damage (Table S2), while both the multiplicative growth rate
and harvesting proportion show such a dependency (Table 1 and S2). While the multiplicative
growth rate’s mast dependency has been reported and discussed in prior studies (e.g. Bieber
& Ruf, 2005; Geisser & Reyer, 2005; Holland et al., 2009; Massei et al., 2015; Vetter et al.,
2015), our finding of a mast-dependent harvesting proportion, although intuitive, is less well
understood. In terms of adapting the presented state-space model to local conditions, it might
well be that this mast dependency is an idiosyncrasy pertaining to the analysed Swiss wild
boar population.
The current high numbers of harvested wild boars in many countries worldwide likely indicate
inadequate and/or inefficient management strategies for wild boar populations, and this
tendency is probably being exacerbated by a changing climate (Massei et al., 2015; e.g.
Melis, Szafrańska, Jędrzejewska, & Bartoń, 2006; Övergaard, Gemmel, & Karlsson, 2007;
Vetter et al., 2015). As an important result we showed that harvesting effort in (full) mast
years is too low to stop population growth or even reduce population size. Annually adjusted
management targets might attenuate this discrepancy. Moreover, even if harvesting effort in
full mast years is not increasable, management targets could explicitly ‘track’ expected mast
events, such as the above-mentioned lack of mast following a full mast with a high probability
(Bieber & Ruf, 2005; see also Fig. 1). To conclude, we think that our model offers a new
perspective on the management of wild boar populations that we deem worthy of being
pursued and applied in other regions.

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References
Baldwin, R. A., Salmon, T. P., Schmidt, R. H., & Timm, R. M. (2014). Perceived damage and
areas of needed research for wildlife pests of California agriculture. Integrative
Zoology, 9(3), 265–279. doi: 10.1111/1749-4877.12067
Bieber, C., & Ruf, T. (2005). Population dynamics in wild boar Sus scrofa: ecology, elasticity
of growth rate and implications for the management of pulsed resource consumers.
Journal of Applied Ecology, 42(6), 1203–1213.
Bozzuto, C., & Ives, A. R. (2019). Inbreeding depression and the detection of changes in the
intrinsic rate of increase from time series.
Briedermann, L. (2009). Schwarzwild. Kosmos.
Burkart, A. (2018). Langzeitbeobachtung der Samenmast im Schweizer Wald, inkl.
Versuchsgarten WSL. Eidg. Forschungsanstalt WSL.
Burschel, P., & Huss, J. (1987). Grundriss des Waldbaus: ein Leitfaden für Studium und
Praxis. Retrieved from https://catalog.hathitrust.org/Record/007879712
Carpio, A. J., Hillström, L., & Tortosa, F. S. (2016). Effects of wild boar predation on nests of
wading birds in various Swedish habitats. European Journal of Wildlife Research,
62(4), 423–430. doi: 10.1007/s10344-016-1016-y
Choquenot, D. (1998). Testing the Relative Influence of Intrinsic and Extrinsic Variation in
Food Availability on Feral Pig Populations in Australia’s Rangelands. Journal of
Animal Ecology, 67(6), 887–907.
Conover, M. R. (2001). Resolving Human-Wildlife Conflicts!: The Science of Wildlife Damage
Management. doi: 10.1201/9781420032581
Cutini, A., Chianucci, F., Chirichella, R., Donaggio, E., Mattioli, L., & Apollonio, M. (2013).
Mast seeding in deciduous forests of the northern Apennines (Italy) and its influence
on wild boar population dynamics. Annals of Forest Science, 70(5), 493–502. doi:
10.1007/s13595-013-0282-z
Durbin, T. late J., & Koopman, S. J. (2012). Time Series Analysis by State Space Methods
(Second Edition). Oxford, New York: Oxford University Press.

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Ebert, C., Knauer, F., Spielberger, B., Thiele, B., & Hohmann, U. (2012). Estimating wild boar
Sus scrofa population size using faecal DNA and capture-recapture modelling. Wildlife
Biology, 18(2), 142–152. doi: 10.2981/11-002
ENETWILD consortium, Keuling, O., Sange, M., Acevedo, P., Podgorski, T., Smith, G., …
Vicente, J. (2018). Guidance on estimation of wild boar population abundance and
density: methods, challenges, possibilities. ESFA Supporting Publications, 15(7).
Engeman, R. M., Massei, G., Sage, M., & Gentle, M. N. (2013). Monitoring wild pig
populations: a review of methods. Environmental Science and Pollution Research,
20(11), 8077–8091. doi: 10.1007/s11356-013-2002-5
Fattebert, J., Baubet, E., Slotow, R., & Fischer, C. (2017). Landscape effects on wild boar
home range size under contrasting harvest regimes in a human-dominated agro-
ecosystem. European Journal of Wildlife Research, 63(2), 32. doi: 10.1007/s10344-
017-1090-9
Frauendorf, M., Gethöffer, F., Siebert, U., & Keuling, O. (2016). The influence of
environmental and physiological factors on the litter size of wild boar (Sus scrofa) in
an agriculture dominated area in Germany. Science of The Total Environment, 541,
877–882. doi: 10.1016/j.scitotenv.2015.09.128
Gaillard, J.-M., & Yoccoz, N. G. (2003). Temporal variation in survival of mammals: a case of
environmental canalization? Ecology, 84(12), 3294–3306.
Gamelon, M., Focardi, S., Baubet, E., Brandt, S., Franzetti, B., Ronchi, F., … Gaillard, J. M.
(2017). Reproductive allocation in pulsed-resource environments: a comparative study
in two populations of wild boar. Oecologia, 183, 1065–1076. doi: 10.1007/s00442-
017-3821-8
Geisser, H., & Reyer, H.-U. (2004). Efficacy of Hunting, Feeding, and Fencing to Reduce
Crop Damage by Wild Boars. The Journal of Wildlife Management, 68(4), 939–946.
doi: 10.2193/0022-541X(2004)068[0939:EOHFAF]2.0.CO;2
Geisser, H., & Reyer, H.-U. (2005). The influence of food and temperature on population
density of wild boar Sus scrofa in the Thurgau (Switzerland). Journal of Zoology,
267(1), 89–96. doi: 10.1017/S095283690500734X
Genov, P. V., & Massei, G. (2004). The environmental impact of wild boar. Galemys: Boletín
Informativo de La Sociedad Española Para La Conservación y Estudio de Los
Mamíferos, 16(Extra 1), 135–145.

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Gethöffer, F., Sodeikat, G., & Pohlmeyer, K. (2007). Reproductive parameters of wild boar
(Sus scrofa) in three different parts of Germany. European Journal of Wildlife
Research, 53(4), 287–297. doi: 10.1007/s10344-007-0097-z
Gren, I.-M., Häggmark-Svensson, T., Andersson, H., Jansson, G., & Jägerbrand, A. (2016).
Using traffic data to estimate wildlife populations. Journal of Bioeconomics, 18(1), 17–
31. doi: 10.1007/s10818-015-9209-0
Gulland, J. A. (1965). Estimation of mortality rates. Annex to Arctic Fisheries Working Group.
Report of Meeting in Ham-burg, 18-23 January 1965. ICES CM 1965/Gadoid Fish
Committee, 3.
Herrero, J., García-Serrano, A., & García-González, R. (2008). Reproductive and
demographic parameters in two Iberian wild boarSus scrofa populations. Mammal
Research, 53(4), 355–364. doi: 10.1007/BF03195196
Holland, E. P., Burrow, J. F., Dytham, C., & Aegerter, J. N. (2009). Modelling with uncertainty:
Introducing a probabilistic framework to predict animal population dynamics.
Ecological Modelling, 220(9), 1203–1217. doi: 10.1016/j.ecolmodel.2009.02.013
Imperio, S., Ferrante, M., Grignetti, A., Santini, G., & Focardi, S. (2010). Investigating
population dynamics in ungulates: Do hunting statistics make up a good index of
population abundance? Wildlife Biology, 16(2), 205–214. doi: 10.2981/08-051
Ives, A. R., & Dakos, V. (2012). Detecting dynamical changes in nonlinear time series using
locally linear state‐space models. Ecosphere, 3(6), 1–15.
Keuling, O., Baubet, E., Duscher, A., Ebert, C., Fischer, C., Monaco, A., … Thurfjell, H.
(2013). Mortality rates of wild boar Sus scrofa L. in central Europe. European Journal
of Wildlife Research, 59(6), 805–814. doi: 10.1007/s10344-013-0733-8
Kopij, G., & Panek, M. (2016). Effect of Winter Temperature and Maize Food Abundance on
Long-Term Population Dynamics of the Wild Boar Sus scrofa. Polish Journal of
Ecology, 64(3), 436–441. doi: 10.3161/15052249PJE2016.64.3.013
Lowe, S., Browne, M., Boudjelas, S., & De Poorter, M. (2000). 100 of the World’s Worst
Invasive Alien Species A selection from the Global Invasive Species Database. (p.
12pp.). Retrieved from https://www.eea.europa.eu/data-and-maps/indicators/invasive-
alien-species-in-europe/100-of-the-worlds-worst

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Massei, G., Kindberg, J., Licoppe, A., Gačić, D., Šprem, N., Kamler, J., … Náhlik, A. (2015).
Wild boar populations up, numbers of hunters down? A review of trends and
implications for Europe. Pest Management Science, 71(4), 492–500. doi:
10.1002/ps.3965
MathWorks (2015). MATLAB. Natick: The MathWorks Inc.
Melis, C., Szafrańska, P. A., Jędrzejewska, B., & Bartoń, K. (2006). Biogeographical variation
in the population density of wild boar (Sus scrofa) in western Eurasia. Journal of
Biogeography, 33(5), 803–811. doi: 10.1111/j.1365-2699.2006.01434.x
Melletti, M., & Meijaard, E. (Eds.). (2017). Ecology, Conservation and Management of Wild
Pigs and Peccaries. Cambridge: Cambridge University Press.
doi:10.1017/9781316941232
Newman, K., Buckland, S. T., Morgan, B., King, R., Borchers, D. L., Cole, D., … Thomas, L.
(2014). Modelling Population Dynamics: Model Formulation, Fitting and Assessment
using State-Space Methods. Retrieved from
www.springer.com/de/book/9781493909766
Övergaard, R., Gemmel, P., & Karlsson, M. (2007). Effects of weather conditions on mast
year frequency in beech (Fagus sylvatica L.) in Sweden. Forestry: An International
Journal of Forest Research, 80(5), 555–565. doi: 10.1093/forestry/cpm020
Servanty, S., Gaillard, J.-M., Ronchi, F., Focardi, S., Baubet, E., & Gimenez, O. (2011).
Influence of harvesting pressure on demographic tactics: implications for wildlife
management. Journal of Applied Ecology, 48(4), 835–843.
Servanty, S., Gaillard, J.-M., Toïgo, C., Brandt, S., & Baubet, E. (2009). Pulsed resources
and climate‐induced variation in the reproductive traits of wild boar under high hunting
pressure. Journal of Animal Ecology, 78(6), 1278–1290.
Skalski, J. R., Ryding, K. E., & Millspaugh, J. (2005). Wildlife Demography: Analysis of Sex,
Age, and Count Data. Elsevier.
Soulé, M. E. (1991). Conservation: Tactics for a Constant Crisis. Science, 253(5021), 744–
750. doi: 10.1126/science.253.5021.744
Vetter, S. G., Ruf, T., Bieber, C., & Arnold, W. (2015). What Is a Mild Winter? Regional
Differences in Within-Species Responses to Climate Change. PLOS ONE, 10(7),
e0132178. doi: 10.1371/journal.pone.0132178

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Supplementary Material
Supplementary Methods ……………………………………………………………………….
28
Supplementary Tables (S1 – S2) ……………………………………………………………..
30
Supplementary Figures (S1 – S3) …………………………………………………………….
32

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Supplementary Methods
Linearization state-space model equations
As introduced in the main text, we adapted the linearization process for the extended Kalman
filter as described in this section. Because linearizing all equations (measurement as well as
process equations; eq. 1a-d) in a standard fashion, i.e. with respect to the two state variables
!! and !!, rarely led to convergence, we linearized the process equations with respect to
these two state variables, whereas we linearized the measurement equations with respect to
!! and !!≔logit!!!!=!!!!. Following Durbin and Koopman (2012), the linearized
equations can be summarized in matrix notation as follows, where matrix !! contains the
linearized measurement equations and matrix !
! contains the linearized process equations;
note that for the analyzes we log-transformed the measurement equations (see Materials and
Methods in the main text):
!!=
!!"# !!
!!!!!
!!"# !!
!!!!!
!!"# !!
!!!!!
!!"# !!
!!!!!
, !
!=
!!!
!!!!!!!!!
!!!
!!!!!!!!!
!!!
!!!!!!!!!
!!!
!!!!!!!!!
.
In sum, following are the linearized equations used in this study:
!!=1
!!!!
!!!!!!
−1
10
(eq. S1a)
!
!=
!!!!!
!!!!!−!!!!
0
01
(eq. S1b)

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Age-at-harvest reconstruction
In this section we describe how we adapted Gulland’s (1965) age-at-harvest population
reconstruction to produce the total abundance time series depicted in Fig. 1. The data at our
disposal were structured according to stages and, for every hunting season, contained counts
of harvested piglets (0-12 months), yearlings (13-24 months), and adults (25+ months). To
reconstruct the population based on age classes, we used a minimization routine to find the
proportional distribution of adults as follows. We started, by using the life history data in Table
S1, with calculating the stable age distribution of an “average wild boar population”. We then
used the (properly scaled) proportional distribution of the six adult age classes to
proportionally distribute the hunting bag data for the most recent adult stage class (see
above), and we used these estimates as an initial condition to reconstruct the population. The
aim was to use the minimization routine to find the initial condition that was “identical” to the
distribution of the most recently reconstructed year. For the reconstruction, we used age-
dependent survival probabilities reported in Table S1.
Our second adaptation of Gulland’s method was the inclusion of mast-dependent harvesting
mortality. Because the data set spans 14 years, we first reconstructed the complete cohorts
and then calculated the average proportion harvested over the years with the same mast
intensity of the complete cohorts, assuming an otherwise unchanged harvesting effort; in light
of the results presented in the main text, the latter assumption is rather imprecise.
Finally, after having reconstructed all cohorts, we calculated yearly sums of all age classes to
give a total population abundance time series. By repeating the reconstruction with different
final years as an exploration, i.e. 12-14 years of data, it became evident that while early
abundances were stable (compared between reconstructions), the most recent abundances
showed variability between reconstructions, thus catching variability in mast conditions of the
most current year(s). For this reason, in the present study (Fig. 1) we use an averaged time
series based on three reconstructions with differing final year; note that the most current
abundance cannot be an average.

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Supplementary Tables
Table S1 | Parameter values used for the age-at-harvest reconstruction. Age-dependent survival probabilities
were used to reconstruct the cohorts, and all parameter values in Table S1 were used to calculate a stable age
distribution, using a post-breeding matrix, to initialize the reconstruction based on a minimization routine; see text
in this Supplementary material for additional details. The data in this table are based on Table 58 in Briedermann
(2009) and Table 6/14 from the previous edition (1990) and adapted by one of us (HG) to conditions prevailing in
the Canton of Thurgau (study area); note that the data were adapted prior to performing the state-space model
analyses.
Age class
Survival
Probability
Fertility
rate
Proportion females
reproducing
0-1
0.42
3.2
0.30
1-2
0.29
4.0
0.80
2-3
0.50
5.9
0.90
3-4
0.71
5.9
0.90
4-5
0.70
5.9
0.90
5-6
0.57
5.9
0.90
6-7
0.45
5.9
0.90
7-8
0.27
5.9
0.90
8-9
0
5.9
0.90
! !

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Table S2 | Parameter estimates from fitting the state-space model (eq. 1a-d) to data (Fig. S1a-b) using an
extended Kalman filter; see main text for additional details on the estimated parameters.
!!
0.2302
!!"#
!
60.7443
!!
0.0752
!!"#
!
~ 0
!!
0.0229
!!
!
0.0039
!!
172.7112
!!
!
0.0149
!!
~ 0
!!
1.4459
!!
0.1625
!
! !

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Supplementary Figures
!
!
Figure S1 | Raw data used in this study. (a) Hunting bag data, (b) compensation cost data (in Swiss Francs,
CHF). In both panels, the x-axes give the hunting season: e.g. 2010 is brief for the season 2010/11; the underlying
bars show the scaled mast index data, originally ranging from 0 (lack of mast, no bars) to 3 (full mast, highest
bars).
! !

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CH-8053 Zürich www.wildlifeanalysis.ch
33
!
Figure S2 | Variance approximation used for eq. 1a. The black line shows the variance of log-transformed
binomial random numbers (5000 for each population size) with a harvested proportion of p = 0.3 of the respective
population size, leading to the x-axis values. The red line shows the variance of the random variable
!!~!0,log !!
!!!!!+1 in eq. 1a, with !!
!=1−!!!!
!!.
! !

! ! ! ! !
Wildlife Analysis GmbH Telefon +41 44 550 40 30
Oetlisbergstrasse 38 hello@wildlifeanalysis.ch
CH-8053 Zürich www.wildlifeanalysis.ch
34
!
!
Figure S3 | Modeled and realized harvested proportions. Black lines show the modeled time- and mast-
dependent harvesting proportions (mean estimates as solid line, 95% confidence interval as dotted lines); the red
line shows the realized harvesting proportion, i.e. hunting bag data (Fig. S1a) divided by the mean reconstructed
abundances (Fig. 1).
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