###### Abstract  
Organism growth is often determined by multiple resources interdependently. However, growth models based on the Droop cell quota framework have historically been built using threshold formulations, which means they intrinsically involve single-resource limitations. In addition, it is a daunting task to study the global dynamics of these models mathematically, since they employ minimum functions that are non-smooth (not differentiable). To provide an approach to encompass interactions of multiple resources, we propose a multiple-resource limitation growth function based on the Droop cell quota concept and incorporate it into an existing producer-grazer model. The formulation of the producer's growth rate is based on cell growth process time-tracking, while the grazer's growth rate is constructed based on optimal limiting nutrient allocation in cell transcription and translation phases. We show that the proposed model captures a wide range of experimental observations, such as the paradox of enrichment, the paradox of energy enrichment, and the paradox of nutrient enrichment. Together, our proposed formulation and the existing threshold formulation provide bounds on the expected growth of an organism. Moreover, the proposed model is mathematically more tractable, since it does not use the minimum functions as in other stoichiometric models.  
multiple resource limitation; Liebig's law of the minimum; colimitation; producer-grazer; predator-prey; dynamics; growth law; ecological stoichiometry +
Footnote †: journal: Applied sciences  
strategies to maintain the same growth rate under various resource limitations [15; 16; 17]. Others have also shown that plant-virus dynamics are altered under different resource treatments [18; 19; 20; 21]. Growth functions that depend on biochemical constraints often ignore most aspects of the molecular mechanisms for growth, for example, they only consider the available supplies of each resource in proportion to the others. Thus, such functions would not be able to explain the mechanisms behind various growth strategies that arise in response to changes in growth conditions. On the other hand, it is straightforward to incorporate the constraints of growth machinery into threshold growth models [10; 12]. However, while multiple resources are modeled in threshold models, growth is explicitly limited by a single resource at any given time. Thus, a gap exists between these two types of model formulations for multiple resource limitation (MRL).  
In this study, we formulate a multiple resource limitation version of a previous threshold-type stoichiometric extension of the Lotka-Volterra model [10; 11; 12; 13]. By considering the process of protein translation (an essential process driving growth), we also build a resource-explicit biomass conversion term. The new growth formulation retains key dynamical characteristics, especially at very high or low resource supplies, but also exhibits some unique characteristics. This model provides a description of growth based on multiple resource limitation and accounts for important aspects of the machinery that drives growth. As the threshold formulations of growth limitation often hinder global analysis, we hope that the smoothness of the new model may facilitate global analysis to uncover underlying biological properties and implications [22; 23; 24; 25].