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 ).