1 2 3 4 5 kohji yamamura · hiroyuki matsuda · hiroyuki...
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Kohji Yamamura · Hiroyuki Matsuda · Hiroyuki Yokomizo · Koichi Kaji · Hiroyuki 5
Uno · Katsumi Tamada · Toshio Kurumada · Takashi Saitoh · Hirofumi Hirakawa 6
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Harvest-based Bayesian estimation of sika deer populations 10
using state-space models 11
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K. Yamamura 19
Laboratory of Population Ecology, National Institute for Agro-Environmental Sciences, 20
3-1-3 Kannondai, Tsukuba 305-8604, Japan 21
Tel. +81-29-838-8253, Fax. +81-29-838-8199 22
e-mail: [email protected] 23
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H. Matsuda 25
Department of Environmental Management, Yokohama National University, Yokohama 26
239-8501, Japan 27
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1
H. Yokomizo 2
The Ecology Centre, School of Integrative Biology, The University of Queensland, St. 3
Lucia, QLD 4072, Australia; CSIRO Sustainable Ecosystems, 306 Carmody Road, St 4
Lucia QLD 4067, Australia 5
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K. Kaji 7
Tokyo University of Agriculture and Technology, Fuchu 183-8509, Japan 8
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H. Uno · K. Tamada 10
Hokkaido Institute of Environmental Sciences, Sapporo 060-0819, Japan 11
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T. Kurumada 13
Eastern Hokkaido Wildlife Research Station, Hokkaido Institute of Environmental 14
Sciences, Kushiro 085-8588, Japan 15
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T. Saitoh 17
Field Science Center, Hokkaido University, Sapporo 060-0811, Japan 18
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H. Hirakawa 20
Forestry and Forest Products Research Institute, Sapporo 062-8516, Japan 21
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Abstract 1
We estimate the number of sika deer, Cervus nippon, in Hokkaido, Japan, in order to 2
construct the plan to reduce the agricultural damage caused by them. A population index 3
that is defined by the population divided by the population of 1993 is first estimated from 4
the data of a spotlight survey. A generalized linear mixed model (GLMM) with corner 5
point constraints is used in this estimation. Then, we estimate the population from the 6
index by evaluating the response of index to the known amount of harvest including 7
hunting. A stage-structured model is used in this harvest-based estimation. Estimates of 8
indices suffer large observation errors when the probability of observation fluctuates 9
widely. Hence, we apply state-space modeling to the harvest-based estimation to remove 10
the observation errors. We propose the use of Bayesian estimation with uniform 11
prior-distributions as an approximation of the maximum likelihood estimation, without 12
permitting an arbitrary assumption that the parameters fluctuate following 13
prior-distributions. It is shown that the harvest-based Bayesian estimation is effective in 14
reducing the observation errors in sika deer populations. However, the stage-structured 15
model requires many demographic parameters to be known beforehand of the analyses. 16
We cannot estimate these parameters from the observed time-series of index if the 17
amount of data is not sufficient. Then, we next construct a univariate model by 18
simplifying the stage-structured model. We show that the simplified model yields the 19
estimates that are nearly identical to those obtained from the stage-structured model. Such 20
a simplification of the model simultaneously clarifies which parameter is important in 21
estimating the population. 22
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Key words: Bayesian estimate; Cervus nippon; generalized linear mixed model; 24
observation error; population dynamics; state-space model 25
26
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Introduction 1
The estimation of the total number of wild sika deer, Cervus nippon Temminck, is one of 2
the major concerns in Hokkaido prefecture, Japan, because these deer cause severe 3
damage to agriculture, forestry and natural vegetation; the damage to agriculture and 4
forestry was 5,000 million yen in 1996 in Hokkaido prefecture. Sika deer populations are 5
characterized by their large intrinsic rate of population increase (rm = 0.15-0.19; Kaji et al. 6
2004). They have had no major predators since the extinction of gray wolves, Canis lupus, 7
that occurred in about 1890 (Inukai 1933). No significant density-dependent mortality 8
arises until the population crashes due to the complete exhaustion of food plants (Kaji et 9
al. 1988). Therefore, the Hokkaido Prefecture Government is artificially imposing 10
density-dependent mortality on sika deer populations to prevent agricultural damage and 11
to prevent the destruction of the ecosystem (Matsuda et al. 1999). Artificial mortality can 12
be divided into two categories: mortality caused by hunting, which is generally permitted 13
from late October to February, and mortality caused by ‘nuisance control’ to reduce the 14
agricultural damage. 15
The Hokkaido Government has been using indices that are defined by the 16
population relative to that in 1993, because the absolute population size of sika deer in 17
eastern Hokkaido was first estimated in 1993 (Hokkaido Institute of Environmental 18
Sciences 1995). Five indices are calculated based on the following: (1) spotlight survey, 19
(2) aerial survey, (3) catch per unit effort (CPUE), (4) sighting per unit effort (SPUE), and 20
(5) cost of damage to agriculture and forestry (Kaji et al. 1998). Among these indices, the 21
index based on spotlight survey seems to be the most reliable. It is less subject to artificial 22
biases (Uno et al. 2006). The index from aerial surveys is available only for a specific 23
range of areas and years, namely in Akan National Park and the Shiranuka Hills area in 24
February–March 1993, 1994, and 1997–2002. The first absolute estimate of population in 25
eastern Hokkaido in 1993, 12.0 ± 4.6 (×104), was based on the aerial survey of the 26
5
overwintering populations. This estimate may be underestimated; a part of populations 1
may have been hidden under trees, and the area of overwintering may have been also 2
underestimated. The indices from SPUE and CPUE were influenced by the change in the 3
hunting quota; the Hokkaido Government changed the permitted number of deer killed 4
per day per hunter as well as the permitted area of hunting in order to manage the deer in 5
an adaptive manner. The index from the cost of agricultural damage is influenced by the 6
agricultural practices; this index has been decreasing recently due to the construction of 7
deer fences that exclude deer from agricultural fields. 8
Although the population index is useful in evaluating the dynamics of animals, the 9
estimation of absolute population is necessary in managing the population; we must 10
determine the absolute number of deer that should be killed by hunting and nuisance 11
control in order to keep the population at an appropriate level. Anderson (2001, 2003) 12
stated that, without estimates of detection probabilities, the use of index values is without 13
a scientific basis. Matsuda et al. (2002) proposed a method for estimating the total 14
number of individuals from a population index. In this method, called harvest-based 15
estimation, the total number of individuals is estimated by examining the response of the 16
population index to the known number of individuals that have been artificially removed. 17
A stage-structured matrix model is used in this estimation. The population index derived 18
from a spotlight survey suffers large observation errors when the probability of 19
observation fluctuates widely, although it seems to be the most reliable among the 20
available indices. The harvest-based estimation of Matsuda et al. (2002) potentially 21
reduced the influence of observation errors. However, Matsuda et al. (2002) did not 22
explicitly incorporate the observation errors in estimating the population. 23
In this paper, we estimate the population by explicitly assuming the observation 24
errors in the population index. We use data from a spotlight survey because it seems to be 25
the most reliable method as we stated above. We divide the process of estimation of 26
populations into two parts. We first estimate the ‘observed population index’ that 27
6
contains observation errors. We next estimate the ‘true population’ that does not contain 1
observation errors. Then, we compare these estimates to show how the elimination of 2
observation error is effective. In estimating the observed population index, we use 3
generalized linear mixed model (GLMM) by considering spatial and temporal 4
heterogeneity. GLMM is recently becoming a popular tool (Martínez-Abraín et al. 2003, 5
McCulloch 2003, Kubo and Kasuya 2006, Littell et al. 2006). In estimating the true 6
population, we use the estimated index instead of using the raw data for simplicity and for 7
generality. We combine the state-space modeling and the harvest-based estimation that 8
uses a stage-structure model; state-space modeling is becoming a popular approach in 9
estimating populations that suffer observation errors (de Valpine and Hastings 2002, 10
Calder et al. 2003, de Valpine 2003, Punt 2003, Saitoh et al. 2003, Stenseth et al. 2003, 11
Williams et al. 2003, Buckland et al. 2004, Clark and Bjørnstad 2004, de Valpine and 12
Hilborn 2005, Viljugrein et al. 2005, Newman et al. 2006, Yamamura et al. 2006, Clark 13
2007). 14
In estimating the state-space model, we propose the use of Bayesian estimation as 15
an approximation of the maximum likelihood estimation. Bayesian estimation is widely 16
used recently due to the development of algorithm of Markov chain Monte Carlo that is 17
abbreviated by MCMC (Meyer and Millar 1999, Millar and Meyer 2000, Gross et al. 18
2002, Clark 2005, Clark and Gelfand 2006, Clark 2007, McCarthy 2007). However, 19
Bayesian estimation suffers the criticism from non-Bayesian scientists, because the 20
assumption of prior-distribution lacks sufficient scientific basis in many cases. In our 21
paper, we do not permit an arbitrary assumption that the parameters fluctuate following 22
prior-distributions. Then, we obtain the maximum likelihood estimates approximately by 23
utilizing the procedure of Bayesian estimation. 24
The stage-structured model contains many demographic parameters that are 25
treated as being known beforehand. We next simplify the stage-structured model, so that 26
we can estimate all parameters of the model. We further show that such simplification of 27
7
the model clarifies which parameter is important in estimating the population. 1
2
3
Estimation of population index 4
Spotlight survey 5
Spotlight survey was conducted annually for 14 years, from 1992 to 2005, at 156 survey 6
routes that are distributed over 12 management units of sika deer populations on 7
Hokkaido Island (Fig. 1, Uno et al. 2006). The survey was performed between late 8
October and early November, before the start of the hunting season. Each survey route 9
was about 10 km in length. Two observers on a vehicle (at a speed ranging from 20 to 40 10
km/h) search for deer on both sides of the survey route by using hand-held spotlights 11
(Q-Beam 160,000 candle-power, Brinkmann, Tx.) from about 19 to 22 o’clock. The total 12
number of deer detected along the routes was recorded. 13
There are at least three populations of sika deer (the Akan population, Hidaka 14
population, and Taisetsu population) on Hokkaido Island, all of which have different 15
haplotype compositions (Nagata et al. 1998). The Akan population ranges over units 16
9–12 in eastern Hokkaido, while the Hidaka and Taisetsu populations range over units 17
1–8 in western Hokkaido (Fig. 1). Therefore, we estimate the population index for two 18
areas: eastern Hokkaido (units 9–12) and western Hokkaido (units 1–8). Among the 156 19
survey routes on Hokkaido Island, 46 survey routes were distributed over units 9–12, 20
while 110 routes were distributed over units 1–8. The survey routes did not always remain 21
fixed over the 14-year survey. The number of survey routes in western Hokkaido 22
increased gradually during the 14 years. In recent years, several survey routes in eastern 23
Hokkaido were enclosed with deer fences that excluded deer from agricultural fields. In 24
order to avoid underestimation of the population index, we exclude data that were 25
obtained after the enclosure. 26
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1
Structure of population index 2
We can efficiently estimate the population index by considering the spatial and temporal 3
structure of the index. We first define the population index in year t by 4
5
1993 1993
( )( )
t itt
i
EIE
λλ
= , (1) 6
7
where λit indicates the expected number of observed individuals per 1 km of the survey 8
route in a spotlight survey that was conducted in October on the ith survey route in year t 9
(i = 1, 2, ... ; t = 1992, 1993, 1994, ..., 2005), Et indicates the spatial expectation over all 10
survey routes in year t (see Table 1 for a list of symbols). Thus, Et(λit) indicates the 11
expected number of individuals per 1 km over all survey routes in year t. We will call the 12
quantity of Eq. 1 as ‘observed population index’ if necessary to discriminate it from the 13
‘true population index’ that will be defined later. Matsuda et al. (2002) and Uno et al. 14
(2006) estimated the index using a moment method; they estimated the index by 15
substituting the observed average number of individuals for Et(λit) in Eq. 1. This simple 16
method of estimation has a shortcoming, however. If several survey routes were changed 17
between 1993 and year t, the index may be biased because the abundance of deer varies 18
depending on the survey routes. To solve this problem, we should eliminate the influence 19
of road by using a model that explicitly contains the influence of road. We use the 20
following model: 21
22
λit = AiEt(λit), (2) 23
24
where Ai is the factor (road factor) that determines the abundance at the ith survey route. 25
Ai is the average abundance of deer at the ith route across all years; we have Et(Ai) = 1. 26
9
Et(λit) is the factor (temporal factor) that determines the abundance in year t, as we 1
defined previously. The actual λit is not given exactly by the multiplication of Ai and 2
Et(λit); there are some random deviations. In describing random factors in such a 3
multiplicative system, a logarithmic model is generally preferable. If we use a 4
multiplicative form such as Eq. 2, the error involved in Et(λit) is amplified by the factor Ai. 5
In contrast, if we use a logarithmic form of Eq. 2, the error involved in loge[Et(λit)] 6
becomes additive to loge(Ai). Hence the error involved in loge[Et(λit)] is not amplified by 7
the factor loge(Ai). This argument also applies to the error involved in loge(Ai). If we use a 8
logarithmic form, therefore, the variance of the resultant error becomes nearly constant, 9
that is, homoscedasticity arises in the resultant error in loge(λit) if loge[Et(λit)] and loge(Ai) 10
have homoscedasticity. Simultaneously, the distribution of the resultant error approaches 11
a normal distribution with increasing number of error factors, irrespective of the original 12
distribution of each error, due to the central limit theorem. We then use the following 13
model as an approximation. 14
15
log ( ) log ( ) log [ ( )]e it e i e t it itc A E eλ λ= + + + , ite ~ 20Normal(0, )σ , 16
(3) 17
18
where eit is a random variable that follows a normal distribution with a mean of zero and a 19
constant variance σ02. We add a constant, c (= −σ0
2/2), to adjust for the shift in the 20
expectation that emerges in the process of logarithmic transformation (McCulloch and 21
Searle 2001, Diggle et al. 2002, §7.4). Equation 3 is alternatively expressed as λit = exp(c) 22
Ai Et(λit) exp(eit). The expectation of λit in year t is given by 23
24
( ) exp( ) ( ) ( ) [exp( )]t it t i t it t itE c E A E E eλ λ= . (4) 25
26
The random variable exp(eit) follows a lognormal distribution with mean exp(σ02/2) 27
10
(Minotani 2003). We have Et(Ai) = 1 by the definition. Thus, we can confirm the equality 1
in Eq. 4, and hence we can confirm the validity of Eq. 3. By substituting Eq. 1 for Eq. 3 2
we obtain 3
4
1993 1993log ( ) log ( ) log ( ) log [ ( )]e it e i e t e i itc A I E eλ λ= + + + + . (5) 5
6
Let Lit be the survey length (km) at the ith survey route in year t, and mit be the observed 7
number of individuals at the ith survey route in year t. Then, the expectation of mit is given 8
by Litλit. We assume that the variance of mit for a given Litλit is approximately proportional 9
to Litλit, that is, we assume that the distribution of mit is given by an over-dispersed 10
Poisson distribution having a fixed overdispersion parameter. 11
12
GLMM estimation 13
The model belongs to a generalized linear mixed model (GLMM) (McCulloch and Searle 14
2001, McCulloch 2003), and hence we can estimate the parameters, Ai and It, by using 15
statistical software that enables the estimation of GLMM. The survey length (Lit) is 16
known, and hence loge(Lit) is an ‘offset variable’ (McCullagh and Nelder 1989). The 17
quantity of loge(It) is zero when t = 1993 by definition. Therefore, we can directly 18
estimate loge(It) (t = 1994, 1995, ..., 2005) by using the ‘corner point constraints’ where 19
the corner point is set at 1993. We use the glmmPQL function of R (Venables and Ripley 20
2002). The summary of R program is listed in Electronic Supplementary Material S1. 21
22
Results of estimation 23
Figure 2 indicates the estimated indices in eastern Hokkaido (units 9–12) and western 24
Hokkaido (units 1–8). The 95% confidence intervals were calculated from their Wald 25
standard errors. The index in eastern Hokkaido was larger than 1 from 1995 to 1998 but it 26
11
became smaller than 1 from 1999 to 2002. In contrast, the index in western areas seems to 1
have increased monotonically. 2
The estimated indices fluctuated widely in both eastern and western Hokkaido. 3
The index abruptly decreased in 1994 and 2004, but we cannot find potential factors that 4
would have reduced the population in 1994 and 2004. It is known that the survival rate of 5
sika deer becomes lower if they suffer severe snow in the winter (Takatsuki et al. 1994, 6
Uno et al. 1998), but the amount of snow was not high in 1994 and 2004, and the artificial 7
mortality was also not high in these years. The population abruptly increased in 2003; the 8
population became 1.31 times larger from 2002 to 2003. The observed rate of increase 9
was larger than the intrinsic rate of population increase of 1.21 that is achieved in an 10
optimal environment. Thus, we consider that the large fluctuation seen in tI in Fig. 2 was 11
likely to have been partly caused by the fluctuation of the probability of observation. We 12
can reduce the influence of such fluctuation in estimating populations by using a 13
population model. We describe such a model in the next section. 14
15
16
Population model 17
Stage-structured model 18
Matsuda et al. (2002) constructed a stage-structured model for sika deer populations by 19
incorporating the estimates of their demographic parameters. We summarize their model 20
in this section. Let NCt, NFt, and NMt be the number of fawns, adult females (over 1.5 years 21
old), and adult males (over 1.5 years old), respectively, at the end of October in the year t. 22
About 90% of adult females yield one fawn in the spring (Hokkaido Institute of 23
Environmental Sciences 1997). These fawns are 0.5 years old at the end of October. 24
Artificial mortality can be divided into two categories as stated above: mortality caused 25
by hunting, which is permitted from late October to February, and mortality caused by 26
12
nuisance control to reduce agricultural damage. The amount of mortality resulting from 1
nuisance control from late October to February is relatively small, because the nuisance 2
control is not generally performed in locations where hunting is permitted. The natural 3
mortality of adults in spring, summer or autumn is also relatively low compared to the 4
winter mortality. Hence, we can consider that the following three types of mortality act 5
sequentially through the course of a year: (1) hunting mortality from late October to 6
February; (2) natural mortality from January to April; and (3) mortality due to nuisance 7
control from March to late October. Birth occurs approximately between the second and 8
third mortality seasons. 9
Let HFt and HMt be the number of adult females and males, respectively, killed by 10
hunting from late October in year t to February in year t + 1. About 19% of the killed 11
individuals are fawns (K. Kaji and H. Uno, Hokkaido Institute of Environmental 12
Sciences, unpubl.). Hence, the number of killed female fawns, male fawns, adult females, 13
and adult males are given by 0.19 HFt, 0.19 HMt, 0.81HFt, and 0.81HMt, respectively. Let 14
SC, SF, and SM be the proportion of survival of fawns, adult females, and adult males, 15
respectively, under natural conditions from January to April. Let CFt and CMt be the 16
number of females and males, respectively, killed by nuisance control from March to late 17
October in the year t + 1. About 13% of killed individuals are fawns (Kaji 2001). Hence, 18
the number of killed adult females and adult males are given by 0.87CFt and 0.87CMt, 19
respectively. Let r be the expected number of female fawns per female adult. The sex 20
ratio is approximately 1:1 at birth. Fawns are small from spring to October, and hence 21
they are killed with their mothers by the nuisance control activities that are undertaken in 22
this season. If their mothers are killed via nuisance control, the fawns without their 23
mothers are likely to die even if the fawns are not killed directly. Hence, the number of 24
fawns killed by nuisance control in year t is given by 2r ·0.87 CFt. Fawns are not the direct 25
target of nuisance control because they are small in this season. It is rare that a fawn is 26
killed directly without killing its mother. Hence, 2r 0.87 CFt includes the directly killed 27
13
fawns 0.13(CFt + CMt). As we stated previously, the sika deer populations suffer 1
density-dependent mortality only under extremely high densities where almost all food is 2
consumed. Such extreme consumption of food is not observed in Hokkaido except for the 3
small area of Nakanoshima Island and Cape Shiretoko. Hence, we do not incorporate the 4
density-dependence in the model. Thus, we can use the following model: 5
6
NCt = 2r SF (NFt−1 – 0.81HFt−1) – 2r · 0.87CFt−1, (6) 7
NFt = SC (0.5NCt−1 – 0.19 HFt−1) + SF (NFt−1 – 0.81HFt−1) – 0.87CFt−1, (7) 8
NMt = SC (0.5NCt−1 – 0.19 HMt−1) + SM (NMt−1 – 0.81 HMt−1) – 0.87CMt−1. (8) 9
10
We can express these equations by a matrix form: 11
12
= −t t-1 t-1N AN BH , (9) 13
14
where 15
16
[ ]C F M, ,t t tN N N ′=tN , (10) 17
[ ]F M F M, , ,t t t tH H C C ′=tH , (11) 18
F
C F
C M
0 2 00.5 00.5 0
rSS SS S
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
A , (12) 19
F
C F
C M
1.62 0 1.74 00.19 0.81 0 0.87 0
0 0.19 0.81 0 0.87
rS rS S
S S
⎡ ⎤⎢ ⎥= +⎢ ⎥⎢ ⎥+⎣ ⎦
B . (13) 20
21
Matsuda et al. (2002) estimated the expectation of the parameters as follows: ˆ 0.45r = , 22
Cˆ 0.73S = , F
ˆ 0.89S = , and Mˆ 0.80S = . Their estimates were based on the increase rate 23
of 1.15 that was observed in a closed population in Nakanoshima Island that may have 24
14
been subject to weak density-dependent mortality (Kaji et al. 1988). We instead adopt the 1
increase rate of 1.21 that was observed in a population in Cape Shiretoko (Kaji et al. 2
2004). We modified the quantities of SC, SF, and SM so that the eigenvalue of the matrix A 3
in Eq. 12 becomes 1.21 without changing the relative amount of these three. Then, we 4 obtained the following estimates of parameters: ˆ 0.45r = , C
ˆ 0.77S = , Fˆ 0.94S = , and 5
Mˆ 0.85S = . Let Nt be the total number of individuals at the end of October in year t: 6
7
Nt = NC t + NF t + NM t. (14) 8
9
We define the ‘true population index’, Ut, by 10
11
1993/t tU N N= . (15) 12
13
The total number of artificially killed deer including those killed by hunting and nuisance 14
control is shown in Fig. 3. 15
16
Fluctuation of demographic parameters 17
The demographic parameters, r, SC, SF, and SM, will fluctuate year by year due to the 18
fluctuation of meteorological factors in the fields. As we stated above, it is known that the 19
survival rate of sika deer becomes lower if they suffer severe snow in the winter 20
(Takatsuki et al. 1994, Uno et al. 1998). We underestimate the possibility of population 21
decline or population outbreak if we do not consider the yearly fluctuation of parameters. 22
Let us denote the demographic parameters in year t by rt, SCt, SFt, and SMt. In describing 23
the fluctuation of SF, Matsuda et al. (2002) used a uniform distribution of the range 24
F Fˆ ˆ[0.9 ,1.1 ]S S that does not exceed 1; the mean and variance are given by FS and 25
2F
ˆ / 300S , respectively. However, actual SF will follow a more smooth distributions 26
15
defined in the range [0, 1]. Hence, we use the beta-distribution with the same mean and 1
variance, which seems more reasonable because a beta distribution can express a smooth 2
distribution defined in the range [0, 1]. This beta-distribution is a distribution to generate 3
process errors; it is not a prior-distribution of parameters. The quantity of SF is rather 4
close to 1, and hence the corresponding beta-distribution is skewed toward right (i.e., 5
negatively skewed). The parameter of the beta distribution was determined by the 6
moment method as follows. The mean and variance of a beta distribution (with parameter 7
α and β) is given by /( )α α β+ and 2/[( ) ( 1)]αβ α β α β+ + + , respectively. By equating 8
the mean and variance to FS and 2F
ˆ / 300S , respectively, we obtained the estimates: α = 9
17.1 and β = 1.1. The fluctuation of meteorological factors influences fawns, adult 10
females and adult males simultaneously. Therefore, as a simple approximation, we 11
assume that the ratio of survival rate is kept approximately at a constant when SF t 12 fluctuates. Thus, we assume SCt = F C F
ˆ ˆ( / )tS S S = 0.77 SFt and SMt = F M Fˆ ˆ( / )tS S S = 0.85 13
SFt. 14
In describing the fluctuation of 2rt, Matsuda et al. (2002) used a uniform 15
distribution of the range ˆ ˆ[1.8 , 2.2 ]r r that does not exceed 1, because the number of new 16
fawns per female in year t, that is given by 2rt, does not exceed 1. The mean and variance 17
are given by ˆ2r and 2ˆ / 75r , respectively. Hence, we use a beta distribution with the 18
same mean and variance. This beta-distribution is a distribution to generate process 19
errors; it is not a prior-distribution of parameters. We determined the parameter of the 20
beta distribution by the same method as stated above: α = 31.5 and β = 3.5. 21
22
23
State-space model 24
State-space model 25
We estimate the number of wild individuals (Nt) from the estimates of the population 26
16
index ( tI ) by using the stage-structured model described above. In this estimation, we use 1
the estimated index instead of using the raw data for simplicity and for generality. Two 2
equations appear in state-space modeling: a state equation and an observation equation. A 3
state equation describes the dynamics of the true number of individuals, while an 4
observation equation describes the relation between observed number of individuals and 5
the true number of individuals. The observation errors in the population index are not 6
mutually independent. By the definition in Eq. 1, the population index is expressed by 7
8
1993 1993log ( ) log [ ( )] log [ ( )]e t e t it e iI E Eλ λ= − . (16) 9
10
We assume that the estimates of log [ ( )]e t itE λ are subject to observation errors that 11
independently follows a normal distribution with the same variance. Then the estimates 12
of log ( )e tI are subject to observation errors that follows a multivariate normal 13
distribution with a correlation coefficient of 0.5. Let ˆlog ( )e tI be the vector of logarithm 14
of observed population index from 1994 to 2005, log ( )e tU be the vector of the logarithm 15
of true population index. Then, we have 16
17 ˆlog ( ) log ( )e e= +t t 1tI U e , Normal( , )1t 1e 0 Σ∼ (17) 18
19
where 1te is the vector of errors, and 20
21 2 2 2
1 1 12 2 2
1 1 1
2 2 21 1 1
2
2
2
σ σ σ
σ σ σ
σ σ σ
⎡ ⎤⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1Σ . (18) 22
23
Equations 6, 7, 8, 14, and 15 are state equations, while Eqs. 17 and 18 are observation 24
17
equations. 1
2
Maximum likelihood estimation 3
The probability that a given set of estimated indices tI (t = 1994, 1995, ..., 2005) is 4
obtained is expressed by 5
6
1994 1995 2005ˆ ˆ ˆPr( , ,..., )I I I 7
1994 1995 2005
1994 1995 2005 1994 1995 2005ˆ ˆ ˆ... Pr( , ,..., | , ,..., )
N N NI I I N N N= ∫ ∫ ∫8
1994 1995 2005 1994 1995 2005Pr( , ,..., )d d ...dN N N N N N× , (19) 9
10
where 1994 1995 2005Pr( , ,..., )N N N is the probability that the set of Nt is generated. This 11
probability is calculated by using state equations. 1994 1995 2005ˆ ˆ ˆPr( , ,..., |I I I 12
1994 1995 2005, ,..., )N N N is the conditional probability that the set of tI is obtained under 13
a given set of Nt. This probability is calculated using observation equations. The 14
integration is performed over the entire range of Nt. 15
We must determine the stage-distribution in 1993 as an initial state before 16
calculating the likelihood. The sex ratio of adults (M/F) was about 0.5 in 1993 (Kaji et al. 17
1998). Hence, we have the following relation. 18
19
NM1993 = 0.5 NF1993. (20) 20
21
We assume that NF1993 = NF1992 for convenience. If there is no artificial mortality, we have 22
the following relation from Eq. 6. 23
24
C1993 F19932 0.45 0.94N N= × × × . (21) 25
26
18
By substituting Eqs. 20 and 21 for Eq. 14, we obtain the initial stage-distribution in 1993. 1
Then, we can perform the integration in Eq. 19 numerically by using simulations. We can 2
numerically search the maximum likelihood estimates of N1993 and σ1 that maximize the 3
quantity of Eq. 19. However, we cannot estimate the SE of maximum likelihood 4
estimates by the standard methods such as that using Hessian matrix. Numerical 5
evaluation of the second derivatives in Hessian matrix is not straightforward in our case. 6
Hence, we use Bayesian estimation as an approximation of the maximum likelihood 7
estimation. 8
9
10
Bayesian estimation 11
Approximation of maximum likelihood estimates by Bayesian estimates 12
Bayesian estimation contains a logical problem that is a subject of debate between 13
Bayesian and non-Bayesian scientists. Recent articles on Bayesian inference sometimes 14
portray the debate as ‘Bayesian versus classical approach’ (e.g., Ellison 2004, Clark 2005, 15
Clark 2007). It should be noted that Bayesian inference that originates from Bayes (1763) 16
is historically older than the classical approach. Fisher (1922) criticized the logical 17
problem of Bayesian approach, and developed the new theory of the maximum likelihood 18
estimation that is currently called classical approach. Nayman (1934) later developed the 19
concept of confidence intervals. The approach of Neyman-Pearson, which is called 20
‘frequentist’, is much different from the approach of Fisher, which is called ‘Fisherian’ 21
(Efron 1998, Salsburg 2001, Hubbard and Bayarri 2003, Shibamura 2004). However, we 22
can use the term ‘non-Bayesian’ to refer to both Fisherian and frequentist. 23
We find no problems if we assume prior-distributions for the parameters that 24
actually fluctuate. Bayesian inference will be especially preferable if the 25
prior-distribution is completely known. Problems arise when we assume 26
19
prior-distributions for the parameters that do not actually fluctuate, that is, being fixed. 1
The discrimination will be related to the ‘repeatability’. If the parameters can be 2
repeatedly yielded actually, the parameters are repeatable and hence the assumption of 3
prior-distribution will be appropriate. If the parameters cannot be actually yielded 4
repeatedly, we cannot assume prior-distributions for the parameters. It is not always clear 5
whether a phenomenon is repeatable or not; it sometimes changes depending on the 6
purpose. In our estimation of deer populations, two parameters, N1993 and σ1, are 7
unknown. These parameters seem not logically repeatable. Only the measurement, that is, 8
the procedure of estimation, is potentially repeatable. Therefore, we do not adopt an 9
arbitrary assumption that N1993 and σ1 fluctuate following prior-distributions. The 10
non-Bayesian scientists invite us to admit that, in many cases, the ‘parameters’ are not 11
fluctuating but the ‘estimates of parameters’ are instead fluctuating. 12
We can nonetheless use the Bayesian estimate as an approximation of maximum 13
likelihood estimate, even if the parameters do not actually fluctuate following 14
prior-distributions. If the likelihood surface of parameters is given by a multivariate 15
normal distribution, the maximum likelihood estimates approximately coincide to the 16
Bayesian estimates using uniform prior-distributions (See Electronic Supplementary 17
Material S2). The confidence intervals approximately coincide to the Bayesian credible 18
intervals that are given by the quantiles of the posterior-distributions. Under the regular 19
conditions, the likelihood in the parameters approaches a normal likelihood with 20
increasing sample size (Aitkin et al. 2005). Relying upon the asymptotic normality, we 21
can use Bayesian estimates using uniform prior-distributions as approximations of 22
maximum likelihood estimates. In estimating N1993 and σ1, therefore, we use Bayesian 23
estimates using uniform prior-distributions. 24
We should calculate the likelihood for all combinations of N1993 and σ1 over 25
uniform distributions that cover the entire range of parameters, such as (−∞, +∞), after an 26
appropriate transformation. We should transform the variable beforehand so that the 27
20
likelihood surface becomes close to a multivariate normal distribution that is required for 1
using Bayesian estimates as approximations of maximum likelihood estimates. Thus, the 2
prior-distribution is determined without being influenced by other information. For the 3
sake of convenience, however, we use uniform distributions in an arithmetic scale for the 4
prior-distributions of N1993 and σ1 in this paper. We perform simulations for σ1 = 0.01, 5
0.02, ..., 0.5. For each quantity of σ1, we perform simulations for N1993 = 1000, 2000, ..., 6
500000. We later confirmed that the parameter values outside these ranges do not 7
contribute to the likelihood in our data. For each combination of N1993 and σ1, we repeat 8
the simulations 1000 times. Then, the posterior-distributions are obtained by 9
accumulating the likelihood of the time-series according to Bayes’ theorem. 10
In calculating the approximate confidence intervals or SE of the estimates of Nt (t 11
= 1994, 1995, ..., 2005), that are endogenous variables, we can use the Bayesian 12
posterior-distributions of Nt due to the following reasons. Let us imagine the 13
posterior-distributions of Nt (t = 1994, 1995, ..., 2005) under the maximum likelihood 14
estimates of N1993 and σ1. These posterior-distributions correspond to the point estimates 15
of the distributions of Nt. The estimates of distributions fluctuate due to the fluctuation of 16
the maximum likelihood estimates of N1993 and σ1 under the fixed amount of the true N1993 17
and σ1. We can obtain the ‘distribution of distribution’ of Nt (t = 1994, 1995, ..., 2005) by 18
calculating the weighted sum of distributions with a weight proportional to the likelihood 19
of N1993 and σ1. These weighted sums of distributions are identical to the Bayesian 20
posterior-distributions of Nt if we are using uniform prior-distributions for N1993 and σ1. 21
Thus, relying on the asymptotic normality, we can use the Bayesian 22
posterior-distributions of Nt (t = 1994, 1995, ..., 2005) to calculate the approximate 23
confidence intervals or SE of ˆtN (t = 1994, 1995, ..., 2005). To obtain the Bayesian 24
posterior-distributions for Nt (t = 1994, 1995, ..., 2005), we accumulate the likelihood for 25
Nt (t = 1994, 1995, ..., 2005) for every 10000 intervals. For example, the total likelihood 26
for N1999 = 155000 is given by accumulating the likelihood of time-series that passed 27
21
between N1999 = 150000 and 160000. We do not use other algorithms for numerical 1
Kalman-filtering such as the Monte Carlo smoothing of Kitagawa (1996ab), because the 2
population indices have covariance in our case. 3
4
Results of estimation 5
The estimate of N1993 ± SE was 17.1 ± 3.2 (×104) in eastern Hokkaido and 11.1 ± 6.3 6
(×104) in western Hokkaido. The estimate of σ1 ± SE was 0.23 ± 0.06 in eastern Hokkaido 7
and 0.19 ± 0.06 in western Hokkaido. Figure 4 indicates the estimated total population. 8
The population changed gradually year to year. In eastern Hokkaido, the population 9
estimate decreased from 1998 to 2001, but it remained around 19 (×104) after 2003. In 10
contrast, the population estimate in western Hokkaido consistently increased after 1994. 11
The estimated proportion of deer artificially killed in eastern Hokkaido was larger than 12
that in western Hokkaido (Fig. 5). The proportion changed in the period between 1994 13
and 2005 in both eastern and western Hokkaido. However, the ratio of the proportion of 14
killed deer, which is given by (proportion in western area)/(proportion in eastern area), 15
was kept around 0.5, as indicated by the dotted line in Fig. 5. 16
Most of the recent articles on Bayesian estimation use Markov chain Monte Carlo 17
(MCMC) in their calculations. The free software for Bayesian estimation, WinBUGS, is 18
frequently used. However, MCMC may not always generate appropriate estimates; the 19
user’s manual of WinBUGS cautions us that we should be extremely careful if using 20
WinBUGS for serious statistical analysis. The manual tells us that ‘MCMC sampling can 21
be dangerous!’ (Spiegelhalter et al. 2003). We used WinBUGS to perform the same 22
estimation that is given by Eq. 19. The program for WinBUGS is listed in Electronic 23
Supplementary Material S3. This program yielded the results that are identical to Fig. 4. 24
Therefore, we could confirm that the MCMC yields valid results in our case. We will 25
therefore use MCMC in the following section; MCMC enables us to estimate a larger 26
22
number of parameters. 1
2
3
Simplified model 4
Influence of other parameters 5
The stage-structured model described by Eqs. 6, 7, and 8 contains the following 11 6
parameters when we consider the fluctuation of rt, SFt, SMt, and SCt: two parameters to 7
describe the distribution of rt; two parameters to describe the distribution of SFt; the ratio 8
between SFt and SMt; the ratio between SFt and SCt; the sex ratio of fawns; the proportion of 9
fawns in the female individuals killed by hunting; the proportion of fawns in the male 10
individuals killed by hunting; the proportion of fawns in the female individuals killed by 11
nuisance control, and the proportion of fawns in the male individuals killed by nuisance 12
control. In performing the above Bayesian estimation, we assumed that these parameters 13
have already been established based on other data by Matsuda et al. (2002) and Kaji et al. 14
(2004). However, we should consider the uncertainty in these parameters. We performed 15
estimation by changing the natural increase rate per year from 1.21 to 1.15 to evaluate the 16
influence of the change in parameters. The estimates of observation errors were nearly the 17
same: the estimates of σ1 ± SE were 0.23 ± 0.06 in eastern Hokkaido and 0.19 ± 0.05 in 18
western Hokkaido. However, the estimates of population became much different when 19
we used the increase rate of 1.15; the estimates of N1993 ± SE were 26.0 ± 5.3 (×104) in 20
eastern Hokkaido and 27.3 ± 10.8 (×104) in western Hokkaido. Thus, the natural increase 21
rate greatly influences the estimates of population, although it does not much influence 22
the estimation of relative abundance such as N2005/N1993. Thus, it is preferable to estimate 23
the natural increase rate as well as N1993 and σ1. Then, we tried the estimation of the 24
female survival rate (SFt) by MCMC by using the software WinBUGS (Spiegelhalter et al. 25
2003), but we failed; the posterior-distribution of population became almost the same as 26
23
the uniform prior-distribution, probably because of the insufficient amount of data. 1
2
Univariate model 3
It is unlikely that we can estimate all of the 11 parameters with sufficient precision form 4
current data, because we have only 13 years data without density dependent regulation. 5
As we stated above, the estimation actually failed even when we tried to estimate only 6
three parameters, N1993, σ1, and SFt. In order to estimate all parameters in the model, we 7
construct a univariate model by simplifying the stage-structured model described by Eqs 8
6, 7, and 8. Let eCt, eFt, and eMt be the random deviation in the number of fawn, female, 9
and male, respectively. We describe the influence of fluctuation of rt, SFt, SMt, and SCt in 10
Eq. 9 by a simpler form approximately. 11
12
= − +t t-1 t 2tN AN BH e , Normal( , )2t 2e 0 Σ∼ (22) 13
14
where A is a matrix that the elements rt, SFt, SMt, and SCt in the matrix A were replaced 15
by their expectation. 2te is defined by C F M( )t t te e e ′=2te . We assume that 2te 16
follows a multivariate normal distribution with variance-covariance matrix 2Σ as an 17
approximation. Let tρ be the average rate of increase per individual in year t; the 18
parameters tρ changes depending on the relative amount of NCt, NFt, and NMt. Let tλ be 19
an average influence of harvesting one individual in year t; the parameters tλ changes 20
depending on the relative amount of HFt, HMt, CFt, and CMt. Let Kt be the total number of 21
harvest, i.e., Kt = HMt + HFt + CMt + CFt. Then, we have 22
23
1 1 1 1 3t t t t t tN N K eρ λ− − − −= − + . 23 3Normal(0, )te σ∼ (23) 24
25
We assume that tρ is approximately constant and denote it by ρ . We further assume that 26
24
the influence of harvesting one individual is approximately 1. Then, we have 1
2
1 1 3t t t tN N K eρ − −= − + . 23 3Normal(0, )te σ∼ (24) 3
4
We again use Eqs. 15, 17, and 18 to describe the observation error. 5
We estimate the parameters of Eq. 24 by using the software WinBUGS 6
(Spiegelhalter et al. 2003). We use the prior-distributions that are similar to those in the 7
stage-structured model; we use the uniform prior-distributions defined in [0.01, 0.5] for 8
the errors in state and observation equations. However, we use uniform distributions with 9
restricted ranges for the prior-distribution of N1993 to avoid the problem of calculating 10
negative populations; we use a uniform distribution defined in [10, 50] (×104) in eastern 11
Hokkaido and that defined in [2, 50] (×104) in western Hokkaido. We should evaluate the 12
convergence of estimates when we use MCMC in estimating parameters. Various 13
methods have been proposed to judge the convergence (Cowles and Carlin 1996, 14
Gamerman 1997, Carlin and Louis 2000). We use a multiple-chain convergence statistic 15
called Corrected Scale Reduction Factor (CSRF) (Brooks and Gelman 1998) that is 16
provided by the program BOA for R/S-PLUS software (Smith 2005). We calculate CSRF 17
for the final estimation by using 5 parallel chains for 15,000 iterations each. 18
We are using the Bayesian inference as an approximation of the maximum 19
likelihood inference in this paper. Hence, we can select the optimal model by using AIC 20
(the abbreviation of ‘An Information Criterion’ according to Akaike 2007) that is 21
popularly called Akaike’s Information Criterion. AIC selects the model that has the 22
largest expected power of prediction that is evaluated by the measure of Kullback-Leibler 23
information (Akaike 1973). We use the deviance at the Bayesian estimate as an 24
approximation of the deviance at the maximum likelihood estimate. Then, we can 25
calculate the relative quantity of AIC by: AIC = deviance + 2×(number of parameters). 26
We do not use Deviance Information Criterion (DIC) that is frequently used in comparing 27
25
models in Bayesian inference. DIC is calculated by a similar basis as AIC except that the 1
information at the unknown true parameters is replaced by the expected information 2
under their posterior-distribution (Spiegelhalter et al. 2002). AIC and DIC will be 3
approximately the same under the regularity conditions if we use uniform 4
prior-distributions. However, DIC cannot detect the redundancy of parameters; DIC is 5
insensitive to the incorporation of parameters that do not contribute to the likelihood, that 6
is, it is insensitive to the failure in a regularity condition. 7
8
Results of estimation 9
We could not obtain appropriate parameters even for this simplified model; the 10
posterior-distribution of population became almost the same as the uniform 11
prior-distribution, probably because of the insufficient amount of data. Hence, we fixed 12
the quantity of ρ at 1.21. However, the shape of the posterior-distributions of 3σ was 13
again close to the uniform prior-distribution in both eastern and western Hokkaido. 14
Approximate AIC of the model without 3te is smaller than that of the model with 3te in 15
both eastern and western Hokkaido. Hence, we adopted the model without error of the 16
state equation in both eastern Hokkaido and western Hokkaido. The estimates of 1σ ± SE 17
in Eq. 18 were 0.23 ± 0.06 in eastern Hokkaido and 0.20 ± 0.05 in western Hokkaido, 18
which are close to those obtained by the stage-structured model. The Bayesian estimates 19
of tN are shown in Fig. 6. The estimates are also close to those obtained by the 20
stage-structured model (Fig. 4). The SE of estimates is smaller in Fig. 6 mainly because 21
the error terms of the state equations were dropped by the AIC in the simplified model. 22
The CSRF statistic was smaller than 1.01 for both N1993 and 1σ in both eastern and 23
western Hokkaido; it indicates that the convergence is sufficiently good. 24
25
26
26
Discussion 1
We showed that the combination of the state-space modeling and the harvest-based 2
estimation is effective in estimating the total number of sika deer where the probability of 3
observation fluctuates widely. If we use the observed population index given in Fig. 2 to 4
manage the population of sika deer, we must frequently change the amount of harvest 5
including the hunting and nuisance control. However, the improved estimate of the total 6
number of individuals (Fig. 4) indicates that we should not change the amount of harvest 7
so frequently. The current amount of harvest in Eastern Hokkaido is near the threshold 8
level (upper panel of Fig. 4); the mortality caused by the harvesting is almost balancing 9
out to the natural increment. Figure 5 indicates that the proportion of killed deer in 10
Eastern Hokkaido in 2002, 2003, 2004, and 2005 was around 0.21 that is the natural 11
increment of sika deer. If we wish to keep the population at a lower level in Eastern 12
Hokkaido, therefore, we should temporarily increase the amount of harvest. In contrast, 13
the lower panel of Fig. 4 indicates that the population in Western Hokkaido is increasing 14
consistently. Figure 5 indicates that the amount of harvest in Western Hokkaido is around 15
the half of the natural increment (0.21). Therefore, the amount of harvest should be 16
doubled if we wish to stop the increase of population in Western Hokkaido. 17
We used Bayesian estimation as an approximation of the maximum likelihood 18
estimation. If we want to avoid subjectivity in the analyses, we should use the maximum 19
likelihood estimation without using arbitrary prior-distributions unless the parameters 20
actually fluctuate following distributions. However, the maximum likelihood estimation 21
is numerically difficult in a complicated model such as a nonlinear, non-normal 22
state-space model. In such a case, we should first transform the space of parameters so 23
that the likelihood surface becomes close to the normal distribution. Then, we can use the 24
Bayesian estimation with uniform prior-distributions as an approximation of the 25
maximum likelihood estimation. The confidence intervals are approximately given by the 26
27
credible intervals. For convenience, however, we did not use transformations in the 1
current analyses. The posterior-distributions that are given in Electronic Supplementary 2
Material S4 indicate that the likelihood surface for N1993 and σ1 is not symmetric 3
especially in Western Hokkaido. Thus, some bias in the estimation should be suspected in 4
Western Hokkaido. Logarithmic transformation will be generally preferable in the 5
analyses of the populations as we discussed above in performing GLMM. The problem of 6
transformation should be further explored. 7
We estimated the dynamics of sika deer populations by using two models: the 8
stage-structured model and the simplified univariate model. In constructing the 9
stage-structured model, we assumed that the demographic parameters (11 parameters) 10
have already been established based on other data by Matsuda et al. (2002) and Kaji et al. 11
(2004). Thus, we obtained conditional estimates of populations under given quantities of 12
these 11 parameters. We tried to estimate these parameters as well as N1993 and σ1, but we 13
failed probably due to the limited amount of data. The posterior-distribution was nearly 14
the same as the uniform prior-distribution. Hence, we constructed a simplified model so 15
that we can estimate all parameters in the model. However, we again failed to obtain 16
appropriate estimate even for this simplified model. The posterior-distribution was again 17
nearly the same as the uniform prior-distribution. When we set the quantity of population 18
increase at 1.21 in the simplified model as well as the stage-structured model, we could 19
successfully obtain estimates. These estimates are conditional estimates under the 20
increase rate of 1.21. The estimates were similar to those obtained by stage-structured 21
model. Thus, the simplified model indicated that the parameter of natural increase rate is 22
quite important while other parameters are relatively less important in estimating the 23
populations. 24
The similarity in the estimates of populations between stage-structured model and 25
simplified univariate model gives us an insight into the purpose of constructing a model. 26
Actual systems in the fields will be influenced by an infinite number of parameters. 27
28
Therefore, several scientists are considering that there is no ‘true model’ that completely 1
describes the system (Stone 1979, Konishi and Kitagawa 2004, Kitagawa 2007). AIC 2
provides a practical solution for this problem; AIC does not select a ‘true model’ but 3
selects a ‘good model’ in a sense that it has a highest power of prediction. The ‘true’ and 4
‘good’ are quite different characteristics (Amari 2007). However, AIC causes a different 5
problem that the selected model changes depending on the amount of data; the model 6
selected by AIC usually becomes more complicated as the amount of data increases. The 7
actual system is influenced by many parameters, but it is unlikely that many parameters 8
have equivalent level of influence on the system. Hence, Yamamura et al. (2006) defined 9
a ‘true model’ as ‘a model that appropriately summarizes the actual system by ignoring 10
the minor components of the system’. The similarity between Figs. 4 and 6 suggests the 11
existence of such a ‘true model’ in the fields. 12
13
Acknowledgments: We thank Dr. Y. Iwasa for reviewing the manuscript. We also thank 14
anonymous reviewers for their comments that improved the manuscript greatly. This 15
work has been supported by a Grant-in-Aid for Scientific Research of JSPS to H.Y. 16
17
18
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6
7
35
Figure legends 1
Fig. 1. The study area to estimate the total number of individuals. Numbers inside the map 2
indicate the sika deer management unit (Hokkaido Institute of Environmental Sciences 3
1994). Hokkaido Island is divided into two areas, eastern area (units 9–12) and western 4
area (units 1–8), that have different populations of sika deer 5
6
Fig. 2. Population indices that were estimated by the function glmmPQL in R. Dotted 7
lines indicate the 95% Wald confidence intervals. 8
9
Fig. 3. Time series of the total number of artificially killed deer. The quantity of HFt−1 + 10
HMt−1 + CFt−1 + CMt−1 is shown for year t 11
12
Fig. 4. Bayesian estimates of populations that were obtained from the state-space model 13
based on the stage-structured model (Eqs. 6, 7, 8, 14, 15, 17and 18). Circles indicate the 14
estimates of Nt. Dotted lines indicate the estimates of Nt ± SE 15
16
Fig. 5. Change in the artificial mortality. Circle indicates the estimated proportion of 17
killed deer that was calculated by (HFt + HMt + CFt + CMt) / 1ˆ
tN − , that is, the ratio of the 18
quantity in Fig.3 to that in Fig. 4. Dotted line indicates the ratio of the proportion of killed 19
deer between western and eastern Hokkaido 20
21
Fig. 6. Bayesian estimates of populations that were obtained from the state-space model 22
based on the simplified univariate model (Eqs. 24, 15, 17and 18). Circles indicate the 23
estimates of Nt. Dotted lines indicate the estimates of Nt ± SE 24
Figure 1Yamamura et al.
0 100
44°N
144°E142°E42°N
200km
N
8
912
1110
5
6
7
3
4
2
1
0
0.5
1
1.5
2
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
inde
x
Year
Eastern Hokkaido
0
0.5
1
1.5
2
2.5
3
3.5
4
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
inde
x
Year
Western Hokkaido
Figure 2Yamamura et al.
0
1
2
3
4
5
6
1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Eastern HokkaidoWestern Hokkaido
Num
ber o
f kille
d de
er (t
en th
ousa
nd)
Year
Figure 3Yamamura et al.
0
5
10
15
20
25
30
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
estim
ates
(ten
thou
sand
)
Year
Eastern Hokkaido
0
5
10
15
20
25
30
35
40
45
50
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
estim
ates
(ten
thou
sand
)
Year
Western Hokkaido
Figure 4Yamamura et al.
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.2
0.4
0.6
0.8
1
1993 1995 1997 1999 2001 2003 2005
Year
Est
imat
ed p
ropo
rtion
of k
illed
deer
Eastern Hokkaido
Western Hokkaido
Ratio of proportion (W
estern / Eastern)
Ratio of W/E
Figure 5Yamamura et al.
0
5
10
15
20
25
30
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
estim
ates
(ten
thou
sand
)
Year
Eastern Hokkaido
0
5
10
15
20
25
30
35
40
45
50
1993 1995 1997 1999 2001 2003 2005
Pop
ulat
ion
estim
ates
(ten
thou
sand
)
Year
Western Hokkaido
Figure 6Yamamura et al.
1
Tables 1
Table 1. List of principal mathematical symbols that are used in the model (in an 2
alphabetical order) 3
4
Symbol Meanings
Ai Factor that determines the average abundance at the ith survey route
CFt Number of adult females that were killed by nuisance control from
March to late October in year t + 1
CMt Number of adult males that were killed by nuisance control from March
to late October in year t + 1
c constant to adjust the shift in the expectation that emerged in the process
of logarithmic transformation
Et Operator to indicate the spatial expectation over survey routes in year t
HFt Number of adult females that were killed by hunting from late October in
year t to February in year t + 1
HMt Number of adult males that were killed by hunting from late October in
year t to February in year t + 1
It Population index in year t. The definition is given by Eq. 1
Kt Total number of harvested individuals (= CMt + CFt + HMt + HFt)
Lit Survey length at the ith survey route in year t
mit Number of observed individuals of the ith survey route in year t
NCt Number of fawns in October in year t
NFt Number of adult females in October in year t
NMt Number of adult males in October in year t
Nt Total number of individuals in October in year t (= NCt + NFt + NMt)
Oit Expected number of observed individuals per 1 km of the ith survey route
2
in year t
pH Proportion of adult deer killed by hunting
pC Proportion of adult deer killed by nuisance control
rt Expected number of newly born female fawns per adult female in year t
SCt Proportion of survived fawns under the natural condition from January to
April in year t
SFt Proportion of survived adult females under the natural condition from
January to April in year t
SMt Proportion of survived adult males under the natural condition from
January to April in year t
Ut True population index in year t. The definition is given by Eq. 16
α, β Parameters of beta-distribution.
tλ Average influence of harvesting one individuals in year t in the
simplified model
tρ Average rate of increase per individual in year t in the simplified model
1
1
Electronic Supplementary Material
Kohji Yamamura, Hiroyuki Matsuda, Hiroyuki Yokomizo, Koichi Kaji, Hiroyuki Uno, Katsumi Tamada,
Toshio Kurumada, Takashi Saitoh, and Hirofumi Hirakawa (2007) Harvest-based Bayesian estimation of
sika deer populations using state-space models. Population Ecology 49:
Supplement S1: Summary of R-program for estimating population index in eastern Hokkaido using generalized linear mixed model (GLMM)
# Format of the data file (data.txt).
# Four from 592 observations in eastern Hokkaido are shown.
# OBS PLACE YEAR SUM CODE LEN
# 1 Kita1 1993 8 0 10
# 2 Kita2 1994 0 1994 10
# 3 Abash 1995 9 1995 10
# 4 Higas 1996 56 1996 10
# PLACE is the name of place. LEN is the length of road (km).
# SUM is the number of observed individuals.
# CODE is the same as YEAR except that 1993 is replaced by 0.
data <- read.table("C:¥¥data.txt",header=T)
data$PLACE <- factor(data$PLACE)
data$CODE <- factor(data$CODE)
library(MASS)
data.result <- glmmPQL(SUM ~ -1 + offset(log(LEN)) + PLACE
+ CODE, random = ~ 1 | OBS, family = poisson, data = data)
summary(data.result)
Supplement S2: Relation between maximum likelihood estimate and Bayesian estimate under uniform prior-distributions
Let us consider the estimation of a parameter b. Let P(b) be the prior-distribution of b, P(data|b) be the
probability of obtaining the data under a given amount of b, that is, the likelihood function of b. The
maximum likelihood estimate of b is given by the quantity of b that maximizes the quantity of P(data|b).
2
Bayes’ theorem indicates that the posterior-distribution of b is given by
(data ) ( )
( data)(data ) ( )d
P b P bP b
P b P b b=∫
(A1)
We first consider a situation where P(data|b) have a finite range of b. Equation A1 indicates that P(b|data)
is proportional to P(data|b) if P(b) is given by an uniform distribution that covers the entire range of b
satisfying P(data|b) > 0. In this case, the maximum likelihood estimate of b is identical to the quantity of b
that gives the mode of the posterior-distribution. When we adopt the mean of posterior-distribution as the
Bayesian estimate, the maximum likelihood estimate is identical to the Bayesian estimate if P(data|b) is
given by a symmetrical unimodal distribution. Under regularity conditions, P(data|b) generally
approaches a normal distribution with increasing size of data. Then, P(data|b) is approximately given by a
symmetrical unimodal distribution having a finite range of b. Therefore, the maximum likelihood estimate
is approximately given by the Bayesian estimate if we use a uniform distribution having a sufficient range
of b as the prior-distribution.
Supplement S3: WinBUGS program for estimating the population (in ten thousands) in eastern Hokkaido using stage-structured model
# Data for eastern Hokkaido
list(
Hf=c(0,0.2272,0.3187,0.3311,0.3726,1.4659,1.1383,1.4237,0.9907,1.0092,1.0239,1.2415,0),
Hm=c(1.2449,1.1506,1.6871,1.3928,1.3514,1.9850,1.4256,1.4538,1.1051,1.1601,1.1151,1.1065,0),
Cf=c(0.4025,0.5303,0.9093,1.25,1.7149,1.6029,1.2581,1.1448,1.0228,0.8647,0.9634,1.0548,0),
Cm=c(0.3949,0.5249,0.8164,0.9844,0.8605,0.7842,0.6489,0.6613,0.5778,0.5045,0.5088,0.5613,0),
I=c(1,0.7953,1.0766,1.0506,1.1017,1.2206,0.8805,0.6771,0.7836,0.742,1.2306,0.6791,1.0433),
P=13)
# Initial parameters
list(N=c(20,20,20,20,20,20,20,20,20,20,20,20,20),sigma=0.2)
# Start of model
model{
# Calculation of the logarithm of population index.
# The logarithm of index in 1994 is defined as IL(1).
for (t in 1:P-1){
3
IL[t] <- log(I[t+1])
}
# Generating the population of 1993.
N1 ~ dunif(0.1,50)
sratio <- 0.5
birth <- 0.45*0.94
Nf1 <- N1/(1.0 + sratio + 2.0*birth)
Nm1 <- sratio*Nf1
Nc1 <- 2*birth*Nf1
Nc[1] <- Nc1
Nf[1] <- Nf1
Nm[1] <- Nm1
N[1] <- Nc[1]+Nf[1]+Nm[1]
# Generating the variance covariance matrix of observation errors.
sigma ~ dunif(0.01,0.5)
sigma2 <- sigma*sigma
for (t1 in 1:P-1){
for (t2 in 1:P-1){
var[t1, t2] <- sigma2 + equals(t1,t2)*sigma2
}
}
ivar[1:P-1,1:P-1] <- inverse(var[,])
# Simulation of the time series of population.
for (t in 1:P-1) {
Nc[t+1] <- max(r2[t+1]*Sf[t+1]*(Nf[t]-0.81*Hf[t])-r2[t+1]*0.87*Cf[t], 1.0E-6)
Nf[t+1] <- max(Sc[t+1]*(0.5*Nc[t]-0.19*Hf[t]) + Sf[t+1]*(Nf[t]-0.81*Hf[t])-0.87*Cf[t], 1.0E-6)
Nm[t+1] <- max(Sc[t+1]*(0.5*Nc[t]-0.19*Hm[t]) + Sm[t+1]*(Nm[t]-0.81*Hm[t])-0.87*Cm[t], 1.0E-6)
N[t+1] <- Nc[t+1]+Nf[t+1]+Nm[t+1]
Sc[t+1] <- Sf[t+1]*(0.73/0.89)
Sm[t+1] <- Sf[t+1]*(0.80/0.89)
Sf[t+1] ~ dbeta(17.0606,1.08898)
r2[t+1] ~ dbeta(31.5,3.5)
}
4
# Calculating the index from the simulated time-series of population.
for (t in 1:P-1) {
U[t] <- log(N[t+1]/N[1])
}
# Logarithm of index is assumed to follow a multivariate normal distribution.
IL[1:P-1] ~ dmnorm(U[1:P-1], ivar[1:P-1,1:P-1])
}
# End of model.
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
1994
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1995
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1996
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
1997
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1998
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1999
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
2000
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2001
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2002
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
2003
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2004
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2005
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
Population in 1993 (N1993)
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5
Pro
babi
lity
dens
ity
Standard deviation
Standard deviation (σ1)
Supplement S4-1: Posterior-distributions for Eastern Hokkaido
Posterior-distributions of parameters
Posterior-distributions of populatins from 1994 to 2005
5
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
1994
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1995
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1996
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
1997
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1998
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
1999
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
2000
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2001
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2002
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
2003
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2004
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Population (ten thousand)
2005
0
0.05
0.1
0.15
0.2
5 10 15 20 25 30 35
Pro
babi
lity
dens
ity
Population (ten thousand)
Population in 1993 (N1993)
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5
Pro
babi
lity
dens
ity
Standard deviation
Standard deviation (σ1)
Supplement S4-2: Posterior-distributions for Western Hokkaido
Posterior-distributions of parameters
Posterior-distributions of populatins from 1994 to 2005
6