TempSizeMod - Temperature-size Model

Background

Over half of the global lakes are recording increases in lake temperature and reporting rising bloom events (O'Reilley et al. 2015; Ho et al. 2019; Dokulil et al. 2021). Climate warming is increasingly threatening freshwater ecosystem biodiversity (Paerl et al. 2016; Dudgeon 2019; Reid et al. 2019). Temperature change leads to alterations in the temporal dynamics (also known as phenological shifts) and community compositions of phytoplankton (Winder and Sommer 2012; Zohary et al. 2021). It is crucial to understand how temperature drives changes to the assembly and the dynamics of the phytoplankton community (Petchey et al. 1999; Yvon-Durocher et al. 2011; Shurin et al. 2012; Striebel et al. 2016).

In the current realm of global environmental change, understanding the effects of temperature on the phytoplankton community is highly important for ecosystem management (Paerl et al. 2016). Nevertheless, in a multi-stressor system, disentangling the effects of temperature on phytoplankton is not easy (Dudgeon 2019). Temperature often interacts with other environmental factors like lake mixing and stratification, lights, and zooplankton grazer (Pomati et al. 2020), thus affecting phytoplankton communities both directly and indirectly (Zohary et al. 2021). This makes predicting plankton dynamics difficult. For instance, the occurrence of phytoplankton blooms is becoming more unpredictable as blooms are found in both cold and warm waters (e.g. Sterner et al. 2020; Reinl et al. 2021, 2023).

Trait-based models are valuable tools to disentangle possible effects and test the possible mechanisms that water temperature has on phytoplankton (Litchman 2023). Phytoplankton cell size is a master trait and has been widely used in modelling work (Litchman and Klausmeier 2008). Size-based models allow explorations of aggregated community properties such as total biomass and community mean cell size of phytoplankton (e.g. Ward et al. 2012; Acevedo-Trejos et al. 2018; To et al. 2024). They are useful for understanding changes in the macroecological patterns of the phytoplankton community size compositions in response to changes in environmental conditions.

Model description

The model is adapted from the well-established Nutrient-Phytoplankton-Zooplankton-Detritus (NPZD) model (sensu Fasham et al., 1990; Post et al. 2024) incorporated into a size-based framework (e.g., Armstrong 1994; To et al. 2024). The model is differential equation-based and includes one nutrient source, phosphorus $PO_4^{3-}$, available for uptake by different phytoplankton size classes ($P_i$). The phytoplankton are subject to grazing by two zooplankton of different size groups ($Z_1$, $Z_2$). The growth of phytoplankton is limited by light and nutrients, and is scaled by temperature dependence. The detritus pool, $D$, collects the dead and ungrazed matters, followed by recharging the nutrient pool through remineralization processes.

The model focuses on capturing size-dependent bottom-up and top-down interactions through data-driven allometric relationships between phytoplankton growth and zooplankton grazing. In this study, the allometric parameters are calibrated against lake-specific data within the reported ranges (Hansen et al. 1994, 1997; Edwards et al. 2012). For this study, the model aims at capturing nutrient and plankton dynamics in a Swiss lake, Greifensee, followed by projecting changes to these dynamics based on IPCC reported climate change scenarios (RCPs). Alternatively, the model can be used to study changes in the size compositions of lake phytoplankton communities under varying temperature and nutrient conditions.

## Temperature dependence in the model The temperature dependence for phytoplankton growth follows a bell-shaped thermal tolerance curve, given by,

$$E(T) = e^{0.063T} \left[1- \left(\frac{T-T_{opt}}{\sigma_T}\right)^2 \right] $$,

where $T$ is the ambient lake water surface temperature (LWST), $T_{opt}$ is the thermal optima that determines the median of the curve, and $\sigma_T$ is the thermal tolerance that determines the width of the curve. In this study, we assume a community mean thermal tolerance curve to all phytoplankton size classes.

The maximum ingestion rates of zooplankton follows a Q10 model such that, the maximum grazing increases with temperature. The equation for the dependence is,

$$I_{max}(S_j^Z) \cdot Q_{10}^{\frac{T-T_{ref}}{10}} $$.

The Q10 temperature coefficient here specifies the amount of maximum ingestion rate increases with a 10 $^{\circ}$ C temperature increase. It describes the sensitivity of zooplankton response to a higher temperature. $T$ refers to the lake temperature, while $T_{ref}$ refers to the reference temperature when the rate is equal to the baseline rate (i.e. no effects from temperature).

Allometric relationships in the model

The model comprises of three allometric equations. These allometries allow an ecological trade-off to arise in the model based on water temperature throughout the year. The small phytoplantkon can grow faster than the large phytoplankton, but are subject to stronger grazing from the smaller zooplankton, who will selectively graze on the small cells.

The allometric relationships considered in the model are:

$$\mu_{max}(S_i^P) = \beta_{\mu_{max}}\cdot (S_i^P)^{\alpha_{\mu_{max}}}$$

$$I_{max}(S_j^Z) = \beta_{I_{max}}\cdot (S_j^Z)^{\alpha_{I_{max}}}$$

$$P_{opt}(S_i^P, S_j^Z) = \beta_{P_{opt}}\cdot (S_j^Z)^{\alpha_{P_{opt}}}$$
representing, respectively, maximum growth rate, $\mu_{max}(S_i^P)$, for phytoplankton size class $i$, and maximum ingestion rate, $I_{max}(S_j^Z)$, and optimal prey size, $P_{opt}(S_i^P, S_j^Z)$, for zooplankton size class $j$.

Model calibration

We fine-tuned selected parameters against the time-averaged plankton data collected from lake Greifensee, consisting plankton size (bio-area) and abundance (ROI/sec). For details, please refer to the related publication of this model (under review). After calibrating the parameter values, we obtained a best-fit model that reflect the current average state of the lake.

Experiments

Using the standard model, we conduct two numerical experiments:

a projection based on Representative Climate Pathways (RCPs) issued by IPCC
a sensitivity test for the two thermal traits, $T_{opt}$ and $\sigma_T$ in different RCP scenarios

For details, please refer to the related publication of this model (under review).

Suggestions for future works

The current model are structured flexibly and can be extended for future studies. Below are some options.

Experiments on model hypothesis - How different assumptions on the temperature dependence of phytoplankton growth and zooplankton grazing would change the predictions on communities compositions of phytoplankton community? e.g. Comparisons between:
- Eppley curve or Q10 model (Ho et al. 2013) versus thermal tolerance curve (Chen, 2022)
- Size-dependent Eppley curves (e.g. To et al. 2024)
- Community-averaged unimodal curve (Cagle and Roelke 2021; the present study)
- Size-dependent unimodal curve (due to be examined)
Experiments on model hypothesis - mechanisms between temperature dependence and resource levels
- optimal growth temperature, $T_{opt}$, with nutrient levels (Bestion et al. 2018)
- optimal growth temperature, $T_{opt}$, with irradiance levels (Edwards et al. 2016)
Numerical explorations on the relationships between phytoplankton size and thermal traits, e.g. $T_{opt}$, $\sigma_{T}$
- What are the probable/observable range of parameters for such allometric relationships?
Ecological explorations of altered stratification phenology in lakes (Woolway et al. 2021), for example,

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