## 3 Results
### Model Simulation and Comparison  
Table 1 summarizes the baseline parameter values used for the simulation results. While most parameters have been taken from previous studies [10; 12]. \(a\) and \(m\) were chosen so that \(\alpha(\cdot)\) gives a similar value to \(b_{1}\). Note that the maximum conversion efficiency for energy (as indexed by C) must be less than 1 due to the second law of thermodynamics. However, in theory, the maximum conversion efficiency for a limiting resource such as \(P\) or \(N\) can be 1, if the organism can physiologically sequester all resources it obtains from its food or environment. [PERSON] and colleagues demonstrated that the LKE model contains \"a paradox of energy enrichment\", where energy enrichment (increasing the light-limited carrying capacity \(K\)) decreases the grazer density (and, indeed, can cause grazer extinction) by lowering food quality, an important contribution to the theory of predator-prey by highlighting the impacts of food quality. Later models built on the same framework also show similar qualitative properties, which led to the impression that the paradox of energy is unique to the threshold (minimum) formulations in the LKE model [12; 22]. Here we show that the MRL model, with its distinct formulation without minimum terms, retains this key feature of the LKE model. Specifically, increasing \(K\) results in the deterministic extinction of the grazer in the MRL model, see Figure 4. Furthermore, as expected from the model formulation, the MRL model gives a lower bound on the growth dynamics. For example, when \(K=0.25\) mg C/L, the MRL model shows transient oscillations that persist for some time before dampening toward a steady-state with lower values than those seen in the LKE model. When \(K=2\) mg C/L, the MRL model also shows a clear distinction in the speed of convergence to the steady-state value. For moderate \(K\), the MRL model shows periodic solutions that are typical of predator-prey models [19; 34; 35; 36]. However, the periodic behaviors in the MRL model seem to act on a slower time scale, creating long phases of slow increase/decrease in the density of the producer and grazer as compared to the LKE model. For example, when \(K=0.75\), the oscillation period of the grazer population is about 40 days in the MRL model vs. 20 days in the LKE model. The simulation results in Figure 4 suggest that, due to slower growth dynamics, the MRL model is more susceptible to oscillations, including transient ones, compared to the LKE model. This hints at the possibility that the MRL model may be suitable for the study of periodic behaviors on long time-scales, and thus is relevant for certain biological systems, such as snowshoe hares [37]. \begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Value** & **Unit** \\ \hline P & Total phosphorus & 0.025 & mg P L\({}^{-1}\) \\ \(\mu\) & Overall maximal growth rate & 1.2 & day\({}^{-1}\) \\ \(K\) & Producer carrying capacity (light limited) & 0.5 & mg C L\({}^{-1}\) \\ \(q\) & Producer minimal P/C & 0.0038 & (mg P)/(mg C) \\ \(\theta\) & grazer constant P/C & 0.03 & (mg P)/(mg C) \\ \(b_{1}\) & Theoretical maximum efficiency of biomass production & 0.8 & \\ \(a\) & Maximum efficiency of biomass production (energy) & 8 & \\ \(m\) & Half-saturation of biomass production efficiency & 200 & (mg C)/(mg P) \\ \(c\) & Maximum ingestion rate of the grazer & 0.81 & day\({}^{-1}\) \\ \(k\) & Half-saturation of grazer ingestion response & 0.25 & mg C L\({}^{-1}\) \\ \(d\) & Grazer loss rate & 0.25 & day\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Baseline parameter values taken from [10; 12]. \(a\) and \(m\) were chosen so that \(\alpha(\cdot)\) gives values in the proximity of \(b_{1}\) (note that \(\alpha(\cdot)\) varies with the model dynamics). Next, we turn our attention to the effect of resource limitation on model dynamics. Figure 5 shows the asymptotic behaviors of the MRL (A) and LKE (B) models with respect to resource limitation parameters \(K\) and \(P\).