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 .