Temperature- and Nutrient-dependent Droop model for population dynamics of phytoplankton
Abstract Global change is rapidly and fundamentally altering many of the processes regulating the flux of energy throughout ecosystems and although researchers now understand the effect of temperature on key rates (such as aquatic primary productivity), the theoretical foundation needed to generate forecasts of biomass dynamics and extinction risk remains underdeveloped. We develop new theory that describes the interconnected effects of nutrients and temperature on phytoplankton populations and show that the thermal response of equilibrium biomass (i.e., carrying capacity) always peaks at a lower temperature than for productivity (i.e., growth rate). This difference results from trade-offs between the thermal responses of growth, death, and per-capita impact on the nutrient pool, making this thermal mismatch highly general and applicable to widely used population models. We further show that non-equilibrium dynamics depend on the pace of environmental change relative to underlying vital rates, and that populations respond to variable environments differently at high vs. low temperatures due to thermal asymmetries.
The Model To explore the interaction between nutrient availability and temperature on population growth, biomass and dynamics, we use a 3-dimensional system of ordinary differential equations to describe the coupled dynamics of nutrient (N) availability, intracellular nutrient flux modeled using a dynamic quota (Q) and population biomass of phytoplankton (B). We build off the framework first described by Droop (1974; 1977), which was recently amended by Anderson et al. (unpublished) to incorporate the role of temperature (T). The dynamic nutrient quota component in this model separates nutrient uptake and assimilation into two separate, temperature-dependent processes. In this model, nutrient drawdown and biomass accrual is largely regulated by nutrient accessibility – that is, the availability and uptake rate of nutrients. Nutrient assimilation, which determines the rate at which stored nutrients (via Q) are converted into biomass, regulates the magnitude of quota build-up, and together uptake and assimilation rates (both of which are temperature dependent) create a trade-off determining the accumulation (rate and magnitude) of an internal nutrient pool via the quota. More broadly, the quota regulates the total flux from resource (nutrients) to biomass and is determined by the balance between density- (and temperature-) dependent uptake and assimilation rates, as well as the available external nutrient pool.
Here, nutrient availability (N) is modeled as a chemostat, with nutrient uptake by phytoplankton (B) following a type-II functional response (Monod function), thus allowing for saturation in per-capita uptake. Within the cell, the nutrient quota, Q, determines the flux of nutrients based on differences between temperature-dependent uptake and assimilation rates.
All analyses were done in Wolfram Mathematica v13.1