8 September 2015
Author approved edits
Supplemental Material
Record Books Do Not Capture Population Trends in Bovid Trophy Size
MARCO FESTA-BIANCHET,1 Département de Biologie, Université de Sherbrooke,
Sherbrooke, QC J1K 2R1, Canada
SUSANNE SCHINDLER,² Department of Zoology, University of Oxford, Oxford OX1 3PS,
England, United Kingdom
FANIE PELLETIER,¹ Département de Biologie, Université de Sherbrooke, Sherbrooke, QC
J1K 2R1, Canada
MODEL DESCRIPTION
We constructed an individual-based model of a population of male bighorn sheep (Ovis
canadensis) and coded it in JAVA. Below we describe the data we use and the statistical
analyses performed to estimate survival and horn growth. We list the estimates in Tables 1–3.
DATA
We use data from male bighorn sheep at Ram Mountain, Alberta, Canada. Although data are
available from 1975 to 2013, we only use records from 1975 to 1992, because a combination of
density-dependent and selective effects from 1992 to 2013 caused slower horn growth, so that
the majority of male sheep died without reaching the legal definition at four-fifths-curl (more
details on the curl-regulation are given below). For the survival analysis we use data from 1975
to 2013. Available measurements represent survivors from the hunt, so data from males older
than 4 years are biased.
SURVIVAL
We use survival estimates of Gaillard et al. (2004), adding a survival rate of 30% for 13-yearolds and 0% for 14-year-olds.
HORN LENGTH
Fitted parameters describing horns growing in length are listed in Table 3. We have sufficient
data of males aged between 4 and 9 years to estimate annuli increments of 4–8-year-olds. For
older males we initially assumed that horns grow on average 1 cm/year with a standard deviation
(SD) set to the mean value of SDs. With these measures we obtained unrealistic long horns, so
we reduced the mean of increments for older ages (≥6 yr) to account for wear (Table 3). To
reduce the bias in horn length through early death of hunted males, we used annuli increments to
model growth instead of changes in absolute horn lengths, but we had to account for tear and
wear at the horn tips accordingly.
LEGALITY
We model bighorn sheep harvested under the four-fifths-curl hunting regulation. The probability
of a male with given horn length to be legal under the four-fifths-curl regulation has been
determined in Festa-Bianchet et al. (2014) and parameters of the fit are reproduced in Table 2.
THE MODEL
We implemented an individual-based model in the JAVA programming language and compiled
it with the javac-compiler of version 1.6.0_24. The program was run on a Dell Sandybridgemachine with 64-bit processor and the Red Hat Enterprise Linux operating system.
In our model, a male recruits to the population at age 4 and lives for maximally 10 further
years, so that all males are aged between 4 and 14 years. In addition to its age, a male has 3
further properties: the length of its horn (without loss of generality we focus on one horn only),
the male’s legal status, and whether it is eligible for the record book if it is shot.
One time step corresponds to 1 year, and at each time step a number of 4-year-old males
recruit to the population. This number is Gaussian distributed (see Table 1 for parameter
settings). The horn lengths of recruits are Gaussian distributed (Table 1) and the probability of
being legal is based on horn length (Table 2). These probabilities were derived from field
observations of males of known age and horn length (Festa-Bianchet et al. 2014).
According to the age-specific survival rates, we determine which males survive (Table 3).
Those that survive become older (i.e., their age increases by 1) and their horns grow. That means
horns lengthen by increments that are Gaussian distributed (Table 3). Negative increments are set
to zero, which corresponds to no growth. We assumed that after 9 years the average increment in
horn size was equal to the average amount of wear (Table 3).
After 10 initial time steps all age classes are potentially represented and hunting with an intensity
of 40% is introduced after 5 additional time steps. That means the probability of a legal male to
be shot during the hunt is 40% (see Figs. 1–2 for a simulation with a 10% harvest rate of legal
males).
The sequence of operators within a time step (year) in the model is as follows:
recruitment and update of legal status of each male (May); hunting season (Aug to Oct);
followed by natural mortality, horn growth, and ageing (Nov to May).
Table 1: Parameters for 4-year old bighorn male recruits.
Mean cohort size of 4-year olds
10000
SD of cohort size of 4-year olds
1000
Mean horn length at age 4
61 cm
SD of horn length at age 4
5.94
Table 2: Parameters for probability function that a bighorn male would be legal. A male with
horn length x is legal with probability p(x) = 1 / (1 + e − (a + b x)) under the 4/5-curl regulation. We
use the estimates in Festa-Bianchet et al., (2014).
Intercept a
-19.01
Slope b
0.25
Table 3: Age-specific natural survival rates and annuli increments (in cm) for bighorn males.
Increments include a reduction to account for wear. We assumed no net increase in length from 9
years of age onward.
Age
4
5
6
7
8
9
10
11
12
13
Probability to survive
0.860
0.841
0.826
0.805
0.788
0.77
0.74
0.72
0.69
0.3
14
0
Annuli increments
Mean (wear) in cm
SD
10.59
1.477
8.15
1.501
4.45 (2)
1.162
2.73 (2)
1.367
1.49 (2)
1.071
0 (1)
1.316
0 (1)
1.316
0 (1)
1.316
0 (1)
1.316
0 (1)
1.316
0 (1)
1.316
Figure 1: Simulated changes in average horn length of bighorn sheep males under a harvest
regime of 10%. Plots show average horn length of harvested males with horns of at least 97 cm
(record), all harvested males with horns describing at least 4/5 of a curl, and all males aged 4
years and older (Total). We compared three populations with a starting average length of 61 cm
for 4-year-old males, then simulated (A) an increase of horn length of 4-year-olds by 1% a year,
(B) a decrease by 1% a year and (C) no temporal change in horn length of 4-year-olds.
Figure 2: Simulated changes in the standard deviation of horn length under a harvest regime of
10%. Plots show the standard deviation of horn length of harvested males with horns at least 97
cm (record), all harvested males with horns describing at least 4/5 of a curl, and all males aged 4
years and older (total) in a bighorn population with a starting standard deviation of 5.94 and
average length of 61 cm for 4-year-old males, then (a) average horn length of 4-year-olds
increases by 1% a year, (b) declines by 1% a year and (c) is stable.
LITERATURE CITED
Festa-Bianchet, M., F. Pelletier, J. T. Jorgenson, C. Feder, and A. Hubbs. 2014. Decrease in horn
size and increase in age of trophy sheep in Alberta over 37 years. Journal of Wildlife
Management 78:133–141.
Gaillard, J. M., A. Viallefont, A. Loison, and M. Festa-Bianchet. 2004. Assessing senescence
patterns in populations of large mammals. Animal Biodiversity and Conservation 27:47–58.
Table 1. Parameters for 4-year-old male bighorn sheep recruits.
Mean cohort size of 4-yr-olds
10,000
SD of cohort size of 4-yr-olds
1,000
Mean horn length at age 4
61 cm
SD of horn length at age 4
5.94
Table 2. Parameters for probability function that a male bighorn sheep would be legal. A male
with horn length x is legal with probability p(x) = 1 / (1 + e − (a
+ b x)
) under the four-fifths-curl
regulation. We use the estimates produced by (Festa-Bianchet et al. 2014).
Intercept a
−19.01
Slope b
0.25
Table 3. Age-specific natural survival rates and annuli increments (in cm) for male bighorn
sheep. Increments include the reduction to account for wear. We assumed no net increase in
length from 9 years of age onward.
Annuli increments
Age
Probability to survive
Mean (wear) in cm
SD
4
0.860
10.59
1.477
5
0.841
8.15
1.501
6
0.826
4.45 (2)
1.162
7
0.805
2.73 (2)
1.367
8
0.788
1.49 (2)
1.071
9
0.77
0 (1)
1.316
10
0.74
0 (1)
1.316
11
0.72
0 (1)
1.316
12
0.69
0 (1)
1.316
13
0.3
0 (1)
1.316
14
0
0 (1)
1.316
Figure 1. Simulated changes in the average horn length under a restrictive harvest regime of
10%. Plots show average horn length of harvested male bighorn sheep with horns of ≥97 cm
(record), all harvested males with horns describing at least four-fifths of a curl, and all males
aged ≥4 years (Total). We compared 3 bighorn populations with a starting average length of 61
cm for 4-year-old males, then simulated (A) an increase of horn length of 4-year-olds by
1%/year, (B) a decrease by 1%/year, and (C) no temporal change in horn length of 4-year-olds.
Figure 2. Simulated changes in the standard deviation of horn length under a restrictive harvest
regime of 10%. Plots show the standard deviation of horn length of harvested male bighorn
sheep with horns ≥97 cm (record), all harvested males with horns describing at least four-fifths
of a curl, and all males aged ≥4 years (total) in a bighorn population with a starting standard
deviation of 5.94 and average length of 61 cm for 4-year-old males, then (a) average horn length
of 4-year-olds increasing by 1%/year, (b) declining by 1%/year, and (c) stable.