Theoretical models
To better understand the mechanisms underlying trophic cascades in our
experiment and extend them to more general settings, we also used food
chain models to investigate the relationship between trophic cascade,
predator efficiency and primary production. The model is described as
follows:
\begin{equation}
\left\{\begin{matrix}\frac{\text{dC}}{\text{dt}}=e_{C}F_{C}-d_{C}C\\
\frac{\text{dH}}{\text{dt}}=e_{H}F_{H}-F_{C}-d_{H}H\\
\frac{\text{dR}}{\text{dt}}=rR\left(1-\frac{R}{K}\right)-F_{H}\\
\end{matrix}\right.\ \nonumber \\
\end{equation}where R , H , and C are the biomasses of the basal,
herbivore, and predator species, respectively (as in Fig. 1), \(r\) and
K are the intrinsic growth rate and carrying capacity of the basal
species, respectively. \(F_{H}\) and \(F_{C}\) are the foraging rates of
the herbivore and predator species, respectively, for which we consider
three types of functional responses. For the type I functional response,\(F_{H}=a_{H}\text{RH}\) and \(F_{C}=a_{C}\text{HC}\). For the type
II functional response,\(F_{H}=\frac{a_{H}\text{RH}}{1+a_{H}h_{H}R}\) and\(F_{C}=\frac{a_{C}\text{HC}}{1+a_{C}h_{C}H}\). For the type III
functional response, \(F_{H}=\frac{a_{H}R^{2}H}{1+a_{H}h_{H}R^{2}}\)and \(F_{C}=\frac{a_{C}H^{2}C}{1+a_{C}h_{C}H^{2}}\). In these
formulas, \(a_{H}\) and \(a_{C}\) are attack rates, \(h_{H}\) and\(h_{C}\) are handling times, \(e_{H}\) and \(e_{C}\) are assimilation
efficiencies, and \(d_{H}\) and \(d_{C}\) are mortality rates of the
herbivore and predator species, respectively.
Let \((R_{3}^{*},\ H_{3}^{*},\ C_{3}^{*})\ \)denote the equilibrium
biomasses of the three species in the presence of the top predator and\((R_{2}^{*},\ H_{2}^{*})\) the equilibrium biomasses of the basal and
herbivore species in the absence of top predator. Similar to our
empirical analyses, we quantified the strength of trophic cascade as:
\begin{equation}
STC=\ln\left(R_{3}^{*}/R_{2}^{*}\right)\nonumber \\
\end{equation}and the predator efficiency as:
\begin{equation}
E_{\text{pre}}=\frac{e_{C}F_{3C}^{*}}{e_{H}F_{3H}^{*}}\nonumber \\
\end{equation}where \(F_{3C}^{*}\) and \(F_{3H}^{*}\) represent the equilibrium
foraging rates of the predator and herbivore, respectively, in
three-species food chain models. In addition, primary productivity is
given by (Loreau 2010):
\begin{equation}
P_{\text{pro}}=rR_{3}^{*}\nonumber \\
\end{equation}We then performed analytical investigations and simulations to
understand the relationships between STC , \(E_{\text{pre}}\) and\(P_{\text{pro}}\). For the type I functional response model, we derived
analytic solutions for the relationship between trophic cascade strengthSTC and predator efficiency \(E_{\text{pre}}\) (see SI
Appendix 2 ). For all types of functional responses, we also performed
numerical simulations to investigate the relationships betweenSTC, \(E_{\text{pre}}\), and \(P_{\text{pro}}\). In our
simulations, we fixed two intrinsic parameters for the plant or predator
(both fixed in our experiment), i.e., \(r=0.8\) and \(d_{C}=0.5\),
but changed other parameters to capture variation in the herbivore
species \(H\) and the nutrient supply level, as in our experiment.
Specifically, we randomly draw parameters\(K\sim U\left[2,5\right]\),\(a_{H},a_{C}\sim U\left[0.2,0.8\right]\),\(h_{H},h_{C}\sim U\left[0.01,0.3\right]\),\(d_{H}\sim U\left[0.1,0.5\right]\), where\(U\left[x,y\right]\) denotes the uniform distribution
over the interval \(\left[a,b\right]\). For assimilation
efficiencies, we used both fixed values (Yodzis & Innes 1992), i.e.,\(e_{H}\)= 0.45 and \(e_{C}\)= 0.85, and varying values, i.e.,\(e_{H}\sim U\left[0.3,0.6\right]\ \text{and\ }e_{C}\sim U\left[0.5,0.9\right]\).
For each type of functional response, we retained 1000 replicates of
simulated food chains where all three species can stably persist, for
which we recorded the strength of trophic cascades, predator efficiency
and primary productivity (see details in SI Appendix 2 ).
To test the robustness of our results to model complexity, we further
considered a four-species model that includes one plant, two herbivores,
and one predator (McCann 2011). Because our simulations of three-species
food chains showed highly consistent results across different types of
functional responses, we considered only the type II functional response
for the four-species model (see details in SI Appendix 2 ).
Overall, all models (with different functional responses or herbivore
richness) showed similar patterns, so in the main text we presented only
results based on the three-species food chain model with type II
functional response. Results from other models were presented inSI Appendix 2 .