Figure 1. CCM results of mean shoot length vs. growing degree days (measured May – July) and insect herbivory. Prediction skill is measured as Pearson’s correlation coefficient. Significant causal forcing is indicated in cases where prediction skill increases significantly with library length. P-values indicate significant increase in prediction skill for the longest library length tested relative to the shortest library length tested at lag = 0 and lag = 1, respectively, as well as a comparison of the predictive skills for lag = 0 vs. lag = 1.
Using this information, we found that the best embeddings (i.e. predictive models) of system dynamics were achieved using four time-lagged dimensions, and a tuning parameter that indicated moderately nonlinear dynamics (θ = 0.75). To construct a predictive model of willow shoot growth dynamics as a function of the full set of causally related variables, we therefore used two lagged dimensions of willow shoot lengths, one lagged dimension of climate, and one non-lagged dimension of insect herbivory. These relationships yielded a model of the form:
mean_shoot (t +1) =β 0+β 1Shoot (t )+β 2Shoot (t –1)+β 3Climate (t –1)+β 4Herbivory (t )
where βi indicates fitted values, which are allowed to vary through state-space based on historical dynamics. For example, β 1 might be high for low values ofmean_shoot (t ), indicating that increases in shoot biomass also lead to increases in growth, whereas it might be low for larger values of mean_shoot (t ), indicating self-limitation.
In general, the fitted parameters β 3 andβ 4 can be interpreted as partial derivatives describing the effect of thermal climate and insect herbivory on shoot growth (i.e. ∂Shoot /∂Climate and ∂Shoot /∂Herbivory , respectively). Positive values indicate that shoot growth increases as a function of warmer climate or higher insect herbivory. Negative values indicate that growth declines with warming, or declines with increased insect herbivory. Note that these can be interpreted identically to slopes in a standard linear regression (e.g. where the slope term describes the partial derivative of response variable y relative to explanatory variable x , ∂y /∂x ).