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

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.

Within‐plant coexistence of viruses across nitrogen and phosphorus supply rates

“…These results depend on the assumption of single resource limitation (Equations and ), and we constrained the invader's growth rate to depend on one resource throughout a simulation by setting its minimum nutrient concentration for the other nutrient to a very low number (e.g., Figure 2c). However, increasing evidence suggests primary producers and microorganisms are co‐limited by N and P (Bratt et al, 2020; Frenken et al, 2021; Harpole et al, 2011), which, if applied to the viruses, would likely make invasion even more difficult under this mechanism (Phan et al, 2023). Because invasion occurred even with low nutrient supplies in our experiment, we can assume the virus's minimum nutrient concentrations required for replication are lower than the nutrient supplies used in the experiment.…”