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 $N$, available for uptake by different phytoplankton size classes ($A_i$). 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. Phytoplankton growth and zooplankton grazing are size-dependent. The depth of the epilimnion is known as the mixed layer depth (MLD) and is seasonally varying. The blue arrows indicate ecological and biogeochemical processes (solid lines) and physical processes (dashed lines). Environmental forcing to this model is provided by in-situ observations of photosynthetically active radiations (PAR), mixed layer depth (MLD – an indicator of lake mixing), and water temperature (LWST) from Greifensee, Switzerland. To approximate real-world grazing, the grazed material is split into (1) a portion that is not ingested (due to sloppy feeding) and is immediately lost into detritus, (2) a portion that is ingested and assimilated by zooplankton, and (3) an additional portion that is not assimilated after being ingested and quickly excreted into nutrient. The loss of zooplankton to higher order predation is density-dependent and formulated as a quadratic function. We assume that the loss of nutrients from the epilimnion is recharged through changes in MLD by a constant nutrient source, $N_0$, from the hypolimnion. The value of $N_0$ is determined by adjusting the modelled nutrient dynamics with observations and is also prescribed as forcing to the model.

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:

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

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

$$P_{opt}(S_i^P, S_j^Z) = \beta_{P_{opt}}\cdot (S_j^Z)^{\alpha_{P_{opt}}}$$

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 conducted two numerical experiments:

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

Supplementary tests

• a sensitivity test for the model parameters for the standard model run
• a sensitivity run for varying lake mixing parameters for the worst warming scenario

Getting Started

1. Clone the repository or extract the Zenodo zip file.
2. Create a virtual environment: python -m venv .venv
3. Activate it: source .venv/bin/activate (or .venv\Scripts\activate on Windows)
4. Install packages: pip install -r requirements.txt
5. Run the analysis: python Models/FinalMod/FinalMod_run.py

Model run

To run the model, one needs the model script FinalMod_Sbm.py and the model run script FinalMod_run.py.

• FinalMod_Sbm.py

This is the file containing the size-based NPZD model. The default parameter values are coded in the model.

• FinalMod_run.py

In the run script, one could adjust the number of year and of phytoplankton size classes according to their own preferences.

Pnum = 10      # specify the nos of phytoplankton size groups
Ynum = 10       # no. of modelling year

To change model parameter values from the default values, one enters their preferred values in below lines.

sol = SizeModLogspace(mld=mld_run, par=PAR_run, sst=LWST_run, dmdt=dmdt_run, N0=N0, numP=Pnum, numYears=Ynum).solution

For example, if one wants to change remineralization rate to 0.5, please input it as

sol = SizeModLogspace(mld=mld_run, par=PAR_run, sst=LWST_run, dmdt=dmdt_run, N0=N0, numP=Pnum, numYears=Ynum, Rem=0.5).solution

or if one wants to change the parameters that determine the allometric relationships for $\mu_{max}$ i.e. $\alpha_{\mu_{max}}$ to 0.25,

sol = SizeModLogspace(mld=mld_run, par=PAR_run, sst=LWST_run, dmdt=dmdt_run, N0=N0, numP=Pnum, numYears=Ynum, mu_alpha=0.25).solution

The coding name for the parameters can be found in FinalMod_Sbm.py.

• FinalMod_proj.py

In this script one can run a number of projection sceanrios, i.e. linear (or, non-linear depending on the input forcing) increase in temperature. If one wants to configure the increase in temperature by their own preferred values, or adjust the number of scenarios, please change at the below line:

temp_incr = [0, 1.4, 2.5, 4.2]

Data repository

To, S. W., Chakraborty, S., Acevedo-Trejos, E., Pomati, F., & Merico, A. (2025). Future inorganic nutrient and plankton dynamics in a temperate lake. https://doi.org/10.5281/zenodo.14802352

License

The data files in this directory are distributed under the Creative Commons Attribution 4.0 International License (CC-BY-4.0).

You are free to share and adapt this data for any purpose, provided you give appropriate credit to the original authors and provide a link to the license.

For the full legal code, visit: https://creativecommons.org

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

Reference:

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