Appendix 1. Mathematical formulation and instructions for implementing the model in
program E-SURGE.
At a given sampling occasion, an animal may be prebreeder (state PB), breeder (state B) with
no or some experience (i=0, 1 or 2), nonbreeder confined (state NBc) or not (state NBnc) to a
cliff with some experience (i=1 or 2) and dead bird (state D). The following observations may
be made: ‘1’ (if detected PB), ‘2’ (if detected B), ‘3’ (if detected NB confined), ‘4’ (if
detected NB not confined) and ‘0’ (if no detected). We define the initial state vector Π, the
transition matrix (which is decomposed in 3 matrix: survival matrix S, breeding matrix B and
confinement matrix C) and the event matrix E.
Π=
S=
B=
,
,
,
C=
,
E=
.
Columns of the matrix Π, S and C correspond respectively to state PB0, B0, NB1C, NB1NC, B1,
NB2C, NB2NC, B2 and D. However, columns of the matrix B correspond respectively to 11
intermediate states (left exponents is for the state at the previous breeding season): PB0, B0,
B
NB1, CNB1, NCNB1, B1, BNB2, CNB2, NCNB2, B2 and D. Columns of the matrix E correspond
to the observations ‘no detected’ , ‘detected PB’, ‘detected B’, ‘detected NB confined’ and
‘detected NB not confined’.
Rows of the matrix S, B and E correspond respectively to state PB0, B0, NB1C, NB1NC, B1,
NB2C, NB2NC, B2 and D while rows of the matrix C correspond to the 11 intermediate states.
To simplify notation, we did not distinguish probabilities in matrices. However, transitions are
not all equal. According to effects that we wanted to test, we differentiated some probabilities
from the others. For example, if we considered that survival of prebreeders and adults were
different, we differentiated probability of the first raw from the other raws. This comment
is valid for all matrices.
These models can be implemented in program E-SURGE (Choquet et al., 2009), which is
freely downloadable at http://www.cefe.cnrs.fr/biom/En/softwares.htm. The first step is to
load the data into the program and specify the number of groups (1 here), states (9 here),
events (5 here), age classes (22 here) and covariate (0 here).
Then the model specification procedure is decomposed into i) implementing the basic
structural form of the matrices using the GEPAT interface, ii) setting linear model of each
parameter using the GEMACO interface and iii) fixing initial parameters using IVFV
interface.
In the GEPAT module in E-SURGE, ‘*’ entries denote the complement of the sum of positive
rows entries, and ‘-’ entries denote zero. So, matrices introduced above are specified as
follows:
Π=
,
S=
,
B=
,
C=
,
E=
.
In the GEMACO interface, predefined shortcuts are used to specify which parameters are
time-constant, time-specific or state-specific (e.g. ‘i’ denotes constancy, ‘t’ means time effect,
‘a’ denotes an age effect, and ‘from’ means that parameters are not equal in each matrix row).
For the best model, we used ‘i’ for initial parameters, ‘from(1).[a+t]+from(2:8)’ for survival
probabilities, ‘from(1).a+from(2,5,8)+from(3 4 6 7)+t’for breeding probabilities and ‘t’ for
confinement probabilities. For the events, we distinguished the first from the following
encounter occasions. All individuals are first captured as PB. So, we used,
‘firste.from+nexte[from(1).a+f(2:11)] for the detection probabilities.
In the IVFV interface, when specifying initial parameter values, one should then constrain the
first beta value associated with the first detection for PB to 1 and the following beta value
associated with the first detection of adult to 0.
REFERENCES
Choquet, R., Rouan, L. & Pradel, R. 2009 Program E-SURGE : A software application for
fitting multievents models. In Modeling demographic processes in marked populations,
pp. 845-865, Ed. Cooch E. & Conroy M. Thomson D., Berlin [et al.]: Springer.