Appendix D. Description of the projection matrix model.
To estimate the asymptotic population growth rate we formulated a female based matrix projection model with 5 age classes (toadlet, 1 year old, 2, years old, 3 years old, more than 3 years old) using a post-breeding census and a projection interval of one year (Caswell 2001). The dominant eigenvalue of the projection matrix is an estimate of the asymptotic population growth rate, and thus of fitness. The projection matrix for the spadefoot toads is:
A = | |||||||||||
φ_{1y}A_{1}F_{1}φ_{tadpole}0.5 | φ_{ad}A_{2}F_{2}φ_{tadpole}0.5 | φ_{ad}A_{3}F_{3}φ_{tadpole}0.5 | φ_{ad}A_{4}F_{4}φ_{tadpole}0.5 | φ_{ad}A_{5}F_{5}φ_{tadpole}0.5 | |||||||
φ_{1y} | 0 | 0 | 0 | 0 | |||||||
0 | φ_{ad} | 0 | 0 | 0 | |||||||
0 | 0 | φ_{ad} | 0 | 0 | |||||||
0 | 0 | 0 | φ_{ad} | φ_{ad} | |||||||
where:
A_{1} = α_{1}
A_{2} = α_{1} + (1 - α_{1})α_{2}
A_{3} = α_{1} + (1 - α_{1})α_{2} + (1 - α_{1})(1 - α_{2})α_{3}
A_{4} = α_{1} + (1 - α_{1})α_{2} + (1 - α_{1})(1 - α_{2})α_{3} + (1 - α_{1})(1 - α_{2})(1 - α_{3})
The parameters in the model are:
φ_{tadpole}: tadpole survival using the relationship logit(φ_{tadpole}) = -5.24 + 8.09 × 10^{-5} x - 8.49 × 10^{-10} x², where x is the number of eggs in the pond (we used x = 90’000; regression model from Hels and Nachman 2002).
φ_{1y}: first year survival (own model averaged estimates)
φ_{ad}: adult year survival (based on our own data, we used an estimate of 0.43)
α_{1}: probability to start to reproduce when 1 year old (own model averaged estimates)
α_{2}: probability to start to reproduce when 2 year old (own model averaged estimates)
α_{3}: probability to start to reproduce when 3 year old (own model averaged estimates)
F_{1}: number of eggs a female of age 1 is producing, using the relationship F_{1} = 80.4BM_{1} (Hels 2002), where BM_{1} is body mass of 1 year old toads (12.8g)
F_{2}: number of eggs a female of age 1 is producing, using the relationship F_{2} = 80.4BM_{2} (Hels 2002), where BM_{2} is body mass of 1 year old toads (15.6g)
F_{3}: number of eggs a female of age 1 is producing, using the relationship F_{3} = 80.4BM_{3} (Hels 2002), where BM_{3} is body mass of 1 year old toads (21.2g)
F_{4}: number of eggs a female of age 1 is producing, using the relationship F_{4} = 80.4BM_{4} (Hels 2002), where BM_{4} is body mass of 1 year old toads (25.1g)
F_{5}: number of eggs a female of age 1 is producing, using the relationship F_{5} = 80.4BM_{5} (Hels 2002), where BM_{5} is body mass of 1 year old toads (30.2g)
First year survival and the age-specific probabilities to reproduce depended on body condition (PC1) and timing of metamorphosis, and we therefore calculated the population growth rate for different conditions. This population model assumes that females which have reproduced once, reproduce every year and that the males are an unlimited resource. To account for uncertainty in parameter estimates, we estimated confidence intervals for the population growth rate using simulation (see Schaub et al. 2009 for details). In each of the 1000 iterations we sampled φ_{1y}, φ_{ad}, α_{1}, α_{2}, and α_{3} from beta distributions that were given by the estimated means and sampling variances of the corresponding parameters. For the other parameters of the matrix model no measures of uncertainty were available.