Back-calculated length at age and length increment
The back-calculated length increment was estimated for a sub-sample of perch from Lake Vaggatem and Lake Skrukkebukta. On the opercular bones of individual perch, we measured the width of annual growth increments as the distance between the opaque zones. We used these measurements in addition to the total opercular length and the body length of the individual perch to estimate length-at-age with the nonlinear body-proportional hypothesis method. This method is commonly used for perch (Thoresson 1996) and assumes that the deviation in body size from the expected body size given by the operculum size does not change through life (Thoresson 1996),
\(L_{a}=L_{A}({\frac{O_{a}}{O_{A}})}^{\beta_{1}}\),
where \(L_{a}\) is the back-calculated length-at-age a, \(O_{a}\) the measured operculum radius at age a, \(O_{A}\) the observed operculum size at time of capture, and \(L_{A}\) the observed fish length at time of capture. \(\beta_{1}\) is the linear regression slope coefficient estimated from the log-log relationship between the body length and operculum length at capture. A total of 1646 perch were used in the back-calculation procedure. We back-calculated length increment (mm·year-1) for juvenile fish in the age group of 1-4 years with individual perch ranging from age 2 to 10 years, giving us length increment data from year 1995 to year 2018. We did not asses growth during the sampling year, because those estimates would be dependent on sampling time within the year, which was not exactly the same every year. A comparison between back-calculated length at final winter before capture and length at capture revealed a good fit of the back-calculation model (p<0.001, F= 2501 on 1 and 1644 df, adj-R2=0.94) (Appendix Fig. S6 & Table S9). Back-calculated length increment (mm·year-1) for 1, 2, 3 and 4 year old perch were related to summer water temperature and relative density in a Linear Mixed Effect model (LME) with sampling year and age at capture as random effects, with the nlme-package in R. In addition, we estimated cohort-mean (year class) length increment from age 1 to 4, which was related to mean summer water temperature (°C) and mean relative density (CPUE) for the same time period (three-year moving-average) with linear regression.