## 2 Methods
### Producer-Grazer Model Formulation
The \(\alpha(\cdot)\) relates to the biomass production efficiency; for example, if \(\alpha(\cdot)\) is much less than 1, then there is very low efficiency--almost all the P intake goes into maintaining the machinery. On the other hand, if \(\alpha(\cdot)\) is large, then most of the P intake goes into making new biomass. Together, we have  
\[P_{IN}=P_{R}+P_{P}=P_{R}+\alpha(\cdot)P_{R}=\big{(}1+\alpha(\cdot)\big{)}P_{R}. \tag{10}\]  
Solving for \(P_{R}\) and \(P_{P}\), we get:  
\[P_{R}=\frac{1}{1+\alpha(\cdot)}P_{IN}\qquad\text{and}\qquad P_{P}=\frac{\alpha (\cdot)}{1+\alpha(\cdot)}P_{IN}. \tag{11}\]  
Since the only source of phosphorus acquisition for the grazer in this model is through predation, \(P_{IN}=Q\times f(x)y\). Thus, the rate of phosphorus being used for the synthesis of biomass is:  
\[P_{P}=\frac{\alpha(\cdot)}{1+\alpha(\cdot)}P_{IN}=\frac{\alpha(\cdot)}{1+ \alpha(\cdot)}Qf(x)y. \tag{12}\]  
However, note that the biomass synthesis must be in terms of carbon (since units of \(x\) and \(y\) are given in terms of carbon). This means we need to convert P biomass production to carbon. Since we know that P:C is fixed at \(\theta\) for the grazer, the biomass acquired from hunting in terms of carbon is:  
\[(\text{Conversion to biomass})\times(\text{Predation rate})=\left(\frac{1}{\theta} \frac{\alpha(\cdot)}{1+\alpha(\cdot)}\right)\left(Qf(x)y\right)=\frac{\alpha( \cdot)}{1+\alpha(\cdot)}\frac{Qf(x)y}{\theta}. \tag{13}\]  
Together, the rate of change equation for the grazer is:  
\[\frac{dy}{dt}=\frac{\alpha(\cdot)}{1+\alpha(\cdot)}\frac{Qf(x)y}{\theta}-dy. \tag{14}\]  
Next, we look at the growth rate of \(x\). By Assumption (6), we can use the Droop formulation for the growth rate of \(x\), so its growth assumes the carbon limitation:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]0  
Additionally, producer growth can be limited by phosphorus supply. Since the minimum P:C ratio for the producer is \(q\), we have a second upper limit on producer density (in terms of carbon) of \((P-\theta y)/q\). This gives an expression for growth limited by phosphorus as:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]1  
Using the proposed co-limitation form of growth in Equation (6) in Section 2.1, the growth of the producer is given by:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]2  
In the overall growth under MRL, the net growth is assumed to be modulated by the Droop formula relative to each resource. At infinite resource (\(K\rightarrow\infty,Q\rightarrow\infty\)), the net growth under the proposed MRL formulation converges to half of the absolute maximum growth \(\mu\). Together with Equation (14), we obtain the governing equations for the producer-grazer dynamics:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]3 \[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]4  
We shall use the abbreviation MRL model to denote this new proposed model, which, in contrast to earlier stoichiometric formulation such as the LKE model [10; 12], contains no minimum function. In the MRL model, the function \(\alpha(\cdot)\) incorporates the effects of energy conversion efficiency using C as a surrogate, as in the previous formulation [10]. By Assumption (7), we take \(\alpha(\cdot)\) to be an increasing function of \(1/Q\) (or the C:P ratio). We use a saturation function to represent the functional form of \(\alpha(\cdot)\) :  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]5  
Here, \(a\) is the maximum efficiency of biomass production (which is based on the amount of phosphorus partitioned into biomass production). The \(m\) is the half-saturation of biomass production efficiency. This saturation form of \(\alpha(\cdot)\) gives:  
\[(\text{Producer rate of change}) =(\text{Growth rate})-(\text{Predation rate})\]6Finally, for the predation functional response \(f(x)\), we choose a Monod-type function [10].