## 2 Methods
### Producer-Grazer Model Formulation  
To formulate the producer-grazer model based on the MRL approach introduced here, we first state three main assumptions, which were used in the formulation of the LKE model [10]:  
1. The producer-grazer system is closed. This means the total amounts of phosphorus and carbon are conserved at all times. Furthermore, P and C are present either in the grazer or in the producer (i.e., there are no free pools of P or C). 2. The minimum P:C ratio for the producer is \(q\). 3. The P:C ratio of the grazer is fixed at \(\theta\). 4. The grazer dies at a constant rate, \(d\). The model also has a set of assumptions distinct from the LKE model formulation:  
Figure 3: Comparison of [PERSON]’s growth law and the MRL growth based on the Droop cell quota model. (**A**) Red dots are [PERSON]’s growth, and green dots are MRL growth. Increasing the number of limiting resources decreases the growth rate in the MRL growth, but has no effect on the growth rate under [PERSON]’s growth. The green shaded region should contain the true growth. In this case, all resources were assumed to have the same limiting effect. (**B**) Effect of one resource being the most dominant limitation. Blue, magenta, and black represent the scenarios where one resource is slightly, moderately, and significantly more limited than all other resources, respectively. As a resource becomes more limiting, the distance between the two growth formulations decreases. However, this effect quickly saturates as more than three resources are considered. In this case, except for the most limiting resource, all other resources were assumed to be equally limiting. \(P_{IN}\) can be partitioned into multiple parts for different purposes [17]. * The density of the producer is limited by light at a maximum \(K\) due to self-shading [10]. * The energy conversion efficiency for the grazer is an increasing function of the C:P ratio in the producer. The densities of the producer and grazer are denoted by \(x\) and \(y\), respectively, and are measured in terms of carbon in units of [mg C] [L\({}^{-1}\)]. The dynamical system can be expressed in words as:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\] \[(\text{grazer rate of change}) =(\text{Conversion to biomass})\times(\text{Predation rate})-(\text{ Death rate}).\]  
By Assumption (4), we have  
\[(\text{Death rate})=dy. \tag{8}\]  
Next, we represent the predation rate as a function of the producer \(x\), specifically:  
\[(\text{Predation rate})=f(x)y. \tag{9}\]  
Predation supplies both phosphorus and carbon for the grazer. In order to determine the amount of phosphorus intake per \(f(x)y\), we first need to find the amount of phosphorus per \(x\). We note that P: C is fixed at \(\theta\) for the grazer, whose mass/density is given in terms of carbon, so \(\theta y\) is the total amount of phosphorus in the grazer population at any given time. Since the total amount of phosphorus (\(P\)) is conserved in the system, the total amount of phosphorus in the producer population is given by \((P-\theta y)\). Then, the P:C ratio for the producer is \(Q=(P-\theta y)/x\). Thus, the total amount of phosphorus acquired from predation is \(Q\times f(x)y\). However, not all of the acquired phosphorus is used for biomass production. Some are used to maintain and build cellular machinery that produces new biomass [32; 33]. Assumption (5) allows us to partition the intake phosphorus \(P_{IN}\) into two parts: \(P_{R}\) and \(P_{P}\). \(P_{R}\) is the amount of phosphorus that goes into maintaining the existing machinery. On the other hand, \(P_{P}\) is the phosphorus reserved to be used for biomass synthesized by \(P_{R}\) (i.e., growth). We consider an optimal proportional function \(\alpha(\cdot)\), which optimizes the growth process by optimizing the fraction of phosphorus that goes into \(P_{P}\) and \(P_{R}\). In other words, the most efficient division would have each \(P_{R}\) process the \(\alpha(\cdot)\) unit of \(P_{P}\) per unit of time to build biomass.