Appendix C. Description of the multi-state mark-recapture model in matrix notation.
To estimate age-specific survival probabilities, age-specific probabilities to start reproduction and recapture probabilities a multi-state model with the two states “metamorph” and “breeder” was used. From one occasion to the next, individuals may move between these two states, they may survive and the may be recaptured. These events can be described by a matrix of state transition probabilities, by a vector of state-specific survival probabilities and a vector of state-specific recapture probabilities, respectively. In the following, the states at time t are in rows and the states at time t + 1 in columns. Because the parameters change with age, we show the model for each age class:
From age 0 to 1 year: | |||||||||
1 - α_{1} | α_{1} | φ_{1y} | 0 | ||||||
0 | 1 | φ_{ad} | p | ||||||
From age 1 to x - 1 years: | |||||||||
1 - α_{z} | α_{z} | φ_{ad} | 0 | ||||||
0 | 1 | φ_{ad} | p | ||||||
where α_{z} is the probability to start to reproduce at age z, and where z {2, …, x - 1}
From age x - 1 to x years (full reproduction is assumed at age x [in fact a_{x} = 1]): | |||||||||
0 | 1 | φ_{ad} | 0 | ||||||
0 | 1 | φ_{ad} | p | ||||||
From age x onwards: | |||||||||
1 | 0 | φ_{ad} | 0 | ||||||
0 | 1 | φ_{ad} | p | ||||||
The parameters in this model are:
α_{z}: probability to start to reproduce at age z
φ_{1y}: survival probability in the first year
φ_{ad}: survival probability after the first year
p: recapture probability