Recent observations have shown that phytoplankton biomass increases in the North Atlantic during winter, even when the mixed layer is deepening and light is limited. Current theories suggest that this is due to a release from grazing pressure. Here we demonstrate that the often-used grazing models that are linear at low phytoplankton concentration do not allow for a wintertime increase in phytoplankton biomass. However, mathematical formulations of grazing as a function of phytoplankton concentration that are quadratic at low concentrations (or more generally decrease faster than linearly as phytoplankton concentration decreases) can reproduce the fall to spring transition in phytoplankton, including wintertime biomass accumulation. We illustrate this point with a minimal model for the annual cycle of North Atlantic phytoplankton designed to simulate phytoplankton concentration as observed by BioGeoChemical-Argo (BGC-Argo) floats in the North Atlantic. This analysis provides a mathematical framework for assessing hypotheses of phytoplankton bloom formation.

One of the most prominent biological events in the surface ocean is the North Atlantic spring bloom

The traditional theory of phytoplankton population dynamics in the North Atlantic attributes the spring bloom to the release of phytoplankton from
light limitation, which causes phytoplankton growth rates to increase. This has become known as the “critical depth hypothesis”

An alternative hypothesis proposed by

The critical depth hypothesis and the disturbance-recovery hypothesis differ in their predictions of the evolution of winter loss rates. Process-level
understanding and quantification of phytoplankton population loss rates is challenging, because it is very difficult to directly measure the factors
that contribute to loss for the whole population. Phytoplankton are thought to be tightly controlled by grazing and loss processes

The interactions between phytoplankton and zooplankton can be modeled through mathematical relationships that express the rate of phytoplankton
consumption by zooplankton as a function of phytoplankton concentration

During the spring bloom, phytoplankton accumulation is exponential due to the rapid increase in growth rates that makes loss processes relatively much smaller. In the wintertime, the observed phytoplankton accumulation is slower and leading hypotheses of phytoplankton bloom formation differ in their predictions both of phytoplankton population dynamics and of phytoplankton loss rates. Comparing phytoplankton–zooplankton models with different representations of grazing against the observations of biomass accumulation during sub-optimal growth conditions, such as during the wintertime, may constrain the range of appropriate grazing functions for winter conditions or even the winter–spring transition. Here, we demonstrate that the disturbance-recovery hypothesis requires a grazing function that decreases more rapidly than linearly at low prey concentrations. We show that a model with a quadratic grazing function at low winter phytoplankton concentrations can capture the full annual cycle of phytoplankton biomass in the North Atlantic, i.e., both weak wintertime biomass accumulation and an explosive springtime bloom. Our aim is to provide empirically motivated guidance for the formulation and testing of grazing models.

Grazing rate

In this section we formulate a simple ecosystem model and examine different grazing functions to clarify the relationship between grazing rates and
mixed layer depth during winter conditions (Fig.

The vertical coordinate,

To illustrate the importance of the form the grazing term, we will examine the model in Eq. (

We formulate a bulk mixed layer model by employing these simplifying assumptions and taking the vertical average of the equations in
Eq. (

The term

We can derive an equation for the standing stock of biomass in the mixed layer by taking a vertical integral of the equations in
Eq. (

In contrast to the average concentration, the total biomass does not change due to the physical effects of dilution. However, when the mixed layer
shoals, biomass is lost below the mixed layer through detrainment and the total biomass decreases at a rate given by

In the following subsections we will analyze the phenology of phytoplankton for different choices of grazing functions
(Fig.

The exponent

Phytoplankton are known to respond faster than zooplankton to environmental changes

While the critical depth hypothesis has become the most widely accepted framework to interpret the onset of spring blooms – but there are growing
objections

Ratio of grazing to growth as a function of phytoplankton biomass and mixed layer depth for mixed layer integrated models, e.g., Eq. (

Consider first the saturating (type II) grazing function. In winter, prey concentrations are very low and this function is approximately linear

Assuming the natural mortality of phytoplankton is negligibly small, the growth and grazing terms in the

It could be rebutted that winter accumulation is possible if zooplankton mortality is represented as a linear, rather than quadratic, loss term. In
that case, as the mixed layer deepens, zooplankton biomass loss rates would not decrease as quickly as the rate of zooplankton grazing on
phytoplankton, eventually reaching a crossing-over point at which there would be a net loss of zooplankton biomass and consequently an increase in
phytoplankton biomass. This is the case of Lotka–Volterra predatory–prey dynamics in a variable environment

The situation is different if we prescribe a phytoplankton grazing function that decreases more rapidly than linearly as

In this case, the grazing rate decreases faster than the phytoplankton growth rate as the mixed layer deepens due to the additional

We aim to demonstrate that when implemented in a full nutrient–phytoplankton–zooplankton (NPZ) model, a grazing function with a quadratic (or higher than linear) dependence on phytoplankton
concentration at low

This growth rate has temporal dependence through the mixed layer depth and through the surface irradiance. It also depends on nutrient concentration
through the function

We calibrate the NPZ model using the averaged annual cycle of phytoplankton biomass as observed by BGC-Argo floats in the high-latitude North Atlantic

We estimated phytoplankton concentration

In contrast to

Parameters used in Figs.

The NPZ model equations (Eq.

The focus of this
article is on the functional formulation of the model. If the model cannot reproduce the key features of the observations for any values of the
parameters, then the model must be rejected. If we can find parameter values for which the model reproduces key features of the observations, we then
assess if those values are consistent with observational estimates. The parameters related to grazing and mortality are therefore calibrated by
fitting each model accumulation rate and concentrations to observations over the full annual cycle. We use a trust-region-reflective least-squares
algorithm

The temporal rescaling used to average the observational time series creates a spurious peak in net population growth rate at the beginning of winter
(days 315 to 4). Throughout the winter there is variability in accumulation rates among individual time series, including some negative values, even
when the average is positive. Our choice to define the start of winter as the period when all time series have positive accumulation rates creates the
spurious maximum in the observations at that time. We remove this artifact before parameter fitting by interpolating linearly from day 315 to day 4
(Fig.

Using either Holling type II or III grazing functions, the model with the optimal fit parameters generates a spring bloom with a rapid increase in
phytoplankton concentration and biomass that coincides with the spring shoaling of the mixed layer (Fig.

During the winter (day 320–365 and continuing 1–75), the phytoplankton concentration is larger when using the type III grazing function than when
using the type II grazing function (Fig.

During the summertime, the mixed layer depth is fairly constant, and phytoplankton and zooplankton populations are close to equilibrium. This model does not include export from the mixed layer through sinking or migrating particles. Instead, any carbon export from the mixed layer only occurs when the mixed layer is shoaling due to biomass being left in the stratified layer below the new mixed layer.

Surface phytoplankton versus zooplankton concentrations through the annual cycle for both models. Green curve shows concentrations that result from the type III model. Orange curve shows concentrations that result from the type II model. The labeled dots indicate day of year for particular locations in the phase portrait. Over the annual cycle, the phytoplankton and zooplankton populations transit these curves counter-clockwise. The grey curves show the steady state concentrations throughout the annual cycle.

The modeled relationship between phytoplankton and zooplankton shows notable differences between the two grazing functions. This is best illustrated
by plotting the temporal evolution of the two communities in a

The simple

Our work suggests that the winter accumulation of biomass recently documented from float observations in the North Atlantic

Relatively little is known about phytoplankton loss through grazing

The wintertime growth is not only important to sustain phytoplankton populations in winter, but also it is believed to play a crucial role in the development of the subsequent spring bloom. We showed that the reduction in grazing rate results in larger populations of both zooplankton and phytoplankton at the end of winter than would occur with a linear coupling. Furthermore, the larger zooplankton concentrations result in a faster acceleration in zooplankton grazing once phytoplankton concentrations increase during a bloom. The combination of more abundant wintertime populations and stronger/more rapid coupling between phytoplankton and zooplankton populations curb explosive phytoplankton growth. The magnitude and timing of the spring bloom and interactions between zooplankton and phytoplankton populations in the springtime may be affected by factors not considered here, such as a non-linearities in phytoplankton photophysiology, but our goal was to illustrate in as simple a framework as possible the impact of the functional form of grazing on winter growth and not to derive the most comprehensive model of phytoplankton phenology.

There are other possible explanations for wintertime biomass accumulation beyond the dilution of phytoplankton. The biological functions encapsulated
in the NPZ model parameters may vary over time. For example, the zooplankton assimilation rate,

The sensitive dependence of phytoplankton phenology on the rate of grazing by higher trophic levels at low concentrations provides a powerful
quantitative framework in which to evaluate theories of plankton phenology. Observations of wintertime phytoplankton biomass accumulation have been
interpreted as evidence of a release from grazing pressure in deep mixed layers, but little attention has been given to the key role played by the
choice of grazing functions in these theories. Some studies have used a Holling type III grazing function

It is worth commenting on the ecological underpinnings for the different models of grazing. The grazing functions used in our model represent the
coupling between all species of each trophic level of phytoplankton and zooplankton; the phytoplankton class includes all autotrophs, while the
zooplankton class includes all grazers that consume phytoplankton. A superlinear decrease in grazing rates at low prey concentration has been observed
in the lab studies of aquatic vertebrates and invertebrates and in theoretical studies

A reduction in the grazing rate at low phytoplankton concentration has been proposed as the mechanism to explain the emerging observation that biomass
often increases, albeit weakly, during the wintertime when mixed layers deepen

While our analysis focused on wintertime conditions, we believe that more attention on the functional form of grazing functions may shed light on
other phases of phytoplankton phenology as well. Observations show a tight coupling between the evolution of phytoplankton and zooplankton populations
in all seasons

Observational validation of the functional forms of grazing functions is key to build confidence in predictions based on biogeochemical
models. Different models can be tuned to provide reasonable estimates of the annual cycle of phytoplankton biomass, like our NPZ model with a
saturating grazing function. However, in order to make predictions that are robust to changing conditions, it is important that models have the
correct functional dependencies. Tuning of model parameters is no guarantee of model performance in an evolving environment that has not been observed
yet. Climate change may reshape North Atlantic phytoplankton populations and primary production

All code and data are available at

All authors conceptualized the research. AM curated the BGC Argo data. MF wrote the model code and performed the simulations and model analysis. MF, GF, and RF performed the mathematical analysis. MF wrote the manuscript with input from all co-authors.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Biogeochemistry in the BGC-Argo era: from process studies to ecosystem forecasts (BG/OS inter-journal SI)”. It is not associated with a conference.

The authors would like to thank Amala Mahadevan, Stephanie Dutkiewicz, Emmanuel Boss, and three anonymous reviewers for feedback on earlier drafts of this article.

This research has been supported by the NDSEG fellowship and Martin fellowship.

This paper was edited by Giorgio Dall'Olmo and reviewed by three anonymous referees.