Rich Dynamics of a General Producer–Grazer Interaction Model under Shared Multiple Resource Limitations

Tin Phan, James J. Elser, Yang Kuang

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Article number4150
JournalApplied Sciences (Switzerland)
Issue number7
StatePublished - Apr 2023


  • Liebig’s law of the minimum
  • colimitation
  • dynamics
  • ecological stoichiometry
  • growth law
  • multiple resource limitation
  • predator–prey
  • producer–grazer


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