Statistical analysis
Posterior medians were used as appropriate point estimates in the following statistical analysis. To examine how the strengths of intra- and interspecific density dependence of C. dalli and G. furcata change depending on intrinsic growth rates, the posterior median of each parameter obtained from the state-space model was used for regression analysis. We used the intrinsic growth rates of the two species as explanatory variables, and the parameters of the four strengths of density dependence were treated as response variables. Without knowing whether the strength of density dependence changes linearly or nonlinearly with environmental suitability, we used a generalized linear model (GLM) and a generalized additive model (GAM) simultaneously and selected the optimal model based on the Akaike information criterion (AIC).
Model selection. To determine how the environmental suitability of C. dalli (rsC ) and G. furcata(rsG ) affect the strength of density dependence and whether the effect is linear or nonlinear, we started by modelling all possible covariate combinations and specifications for the relationship between strength of density dependence and environmental suitability. For each of αii , αjj , αij , and αji , a total of four candidate GLMs and six candidate GAMs were defined (Table 1). Then, we screened for optimal models based on AIC.
All GLMs were fitted with the R function lm (R Core Team 2013), and all GAMs were fitted with the function gam in the R packagemgcv v1.8 (Wood 2017) in R v3.6.2. Model selection was conducted using the R function AIC in R v3.6.2 (R Core Team 2013).