insidephy: Inter- and intra-specific Size Diversity of Phytoplankton

insidephy is a package for modelling inter- and intra-specific size variability of phytoplankton. The package is designed to capture the growth dynamics of phytoplankton species under common laboratory conditions, for example using batch cultures or chemostats. insidephy is an open-source project written in Python, with routines to quantify phytoplankton community size composition based on size classes (SBMc) or individuals (SBMi) and with parallelization routines to improve the computation capacity of simulation experiments, particularly, those using the individual-based model type.

Modelling approach

We follow a trait-based modelling approach to capture the cell size variability and to untangle the effects of intra- and interspecific size variability on the assembly and functioning of phytoplankton communities. An empirically derived trade-off is constructed based on the allometric relationships between cell size and maximum growth rate, maximum uptake rate, maximum resource quota, minimum resource quota and nutrient half-saturation. These allometries allow phytoplankton cell sizes with different nutrient competitive abilities to be favoured as resource conditions change.

Size-based model using size classes (SBMc)

In the package insidephy, the module SMBc contains a class of the same name with a single method to quantify the size-based model resolving size classes. The SBMc model type consists of a system of ordinary differential equations describing Droop's growth dynamics, with state variables for the resource, the internal quota of each size class and the abundance or density of individuals on each size class. The model is solved numerically using scipy.odeint library.

Size-based model using individuals (SBMi)

In the package insidephy, the module SMBi contains two classes one that describes the agent and another with the routines to solve numerically the individual-based model.

The agent class is named PhytoCell, for which the cell size is assigned from a log-scaled size distribution within the provided size range (between a minimum and maximum cell size provided by the user). Using the assigned sizes to each cell, the other eco-physiological traits are determined using the above mentioned allometric relationships. The PhytoCell agent also has a method to update its growth based on the available resources following Droop's formulation similarly as for the SBMc model.

The second class, named SBMi, contains the routines to quantify the size-based model. The model has four main methods, named initialize, update, run and split-combine. The initialize method creates a given number of phytoplankton cell agents from the class PhytoCell all agents with a randomly selected size within the specified size range. The update method contains the main rules that dictate how an agent reproduces and dies. The run method executes the model and saves the result into arrays (via method save_to_array). Last, the split-combine method is the algorithm used to make sure the number of (super) individuals stays within a computationally tractable range. Below there is a flow diagram showing the steps taken during the execution of a single time step of the SBMi model.

Installation details

To install and utilize the insidephy package a running distribution of Python (preferably 3.7 or above) is required. To install insidephy simply download the tarball for the latest version of the package from the GitHub repository and install from the source as:

python install insidephy-0.0.1.tar.gz

Basic usage and minimal running example

Once the package has been successfully installed on Python, then specific modules of the insidephy package can be imported, for example, the SBMi model type as:

from insidephy.size_based_models.SBMi import SBMi

To make a simulation first we need to define the initial conditions. Here notice that some initial values have to be provided as a list or tuple:

ini_resource = 0.0002
ini_density = [1e4, 1e4]
min_size = [1.5e1, 1.5e4]
max_size = [2.5e1, 2.5e4]
spp_names = ["Aa", "Bb"]
dilution_rate = 0.0
volume = 1.0
nsi_spp = [500, 500]
nsi_min = 100
nsi_max = 1900
time_end = 30
time_step = 1 / 24

Then to execute the simulation simply type:

sbmi = SBMi(ini_resource=ini_resource, ini_density=ini_density, minsize=min_size, maxsize=max_size, spp_names=spp_names, dilution_rate=dilution_rate, volume=volume, nsi_spp=nsi_spp, nsi_min=nsi_min, nsi_max=nsi_max, time_step=time_step, time_end=time_end)

The result of the execution will be stored as multidimensional arrays in the object sbmi. The aggregate variables with only time dimension are resource, biomass, abundance and quota. The results of the model in dimensions per time and agent are agents_size, agensts_biomass and agents_abundance. These results can be accessed as instances of SBMi object using the dot operator, like:

sbmi.resource
sbmi.biomass
sbmi.abundance
sbmi.quota
sbmi.agents_size
sbmi.agents_biomass
sbmi.agents_abundance

Similarly, the SBMc model using the same parametrization can be calculated and compared with the results of SBMi model. The full code to compute this example is included in the folder examples as part of the minimal running example. The results of the comparison between SBMc and SBMi size-based models can be found below. In the leftmost column are shown the temporal dynamics of the main variables and by aggregating the results of the idealized species Aa and Bb. The middle and rightmost column summarizes the results for each of the idealized species. As can be observed both models captures similarly the temporal dynamics of the variables, with the exception of the mean cell size, which is here calculated as a weighted mean.

Other examples

Other cases to calculate and store single and multiple species experiments using parallel computations are also included in the same folder. For example, simulations using the 22 species reported by Marañón et al. (2013 Eco. Lett.) show an agreement between observations and both size-model types.

Relevant references

Acevedo-Trejos, E. Marañón, E., Hellweger F. L., and Merico A. (submitted) insidephy: A package for modelling inter- and intra-specific size variability of phytoplankton.

Hellweger, F. L., & Kianirad, E. (2007). Individual-based modeling of phytoplankton: Evaluating approaches for applying the cell quota model. Journal of Theoretical Biology, 249(3), 554–565. https://doi.org/10.1016/j.jtbi.2007.08.020

Marañón, E., Cermeño, P., López-Sandoval, D. C., Rodríguez-Ramos, T., Sobrino, C., Huete-Ortega, M., Blanco, J. M., & Rodríguez, J. (2013). Unimodal size scaling of phytoplankton growth and the size dependence of nutrient uptake and use. Ecology Letters, 16(3), 371–379. https://doi.org/10.1111/ele.12052