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pyReefCore

pyReefCore is a 1D model which simulates evolution of mixed carbonate-siliciclastic system under environmental forcing conditions (e.g. sea-level, water flow, siliciclastic input). The carbonate production model simulates the logistic growth and interaction among species based on Generalized Lotka-Volterra equations. The environmental forces are converted to factors and combined into one single environmental value to model the evolution of species. The interaction among species is quantified using a community matrix that captures the beneficial or detrimental effects of the presence of each species on the other.

Generalized Lotka-Volterra model

The most common models of species evolution in ecological modeling are the predator-prey Lotka-Volterra (LV) equation and its modifications.

From LV equations, one can formulate the Generalized Lotka-Volterra (GLV) equation that allows unlimited number of species and different types of interactions among species. The GLV equation is mainly formed by two parts, the logistic growth/decay of a species and its interaction with the other species,

$$\frac{dx_i}{dt} = \epsilon_i x_i + \sum_{j=1}^{N_s} \alpha_{ij}x_ix_j$$

where $x_i$ is the population density of species i; $\epsilon_i$ is the intrinsic rate of increase/decrease of a population of species i (also called Malthusian parameter); $\alpha_{ij}$ is the interaction coefficient among the species association i and j, (a particular case is $\alpha_{ii}$, the interaction of one species association with itself); and t is time. Equation 1 can be written in matrix formulation as:

$$\frac{dx_i}{dt} = diag[X](\epsilon + AX)$$

where $X$ is the vector of population densities of each species i, $\epsilon$ is the vector of all Mathusian parameters, $A$ is the matrix of interaction coefficients, also known as community matrix, and $diag[X]$ is a square matrix with diagonal elements equal to $X$, and zeros outside the diagonal.

Definition of species rate and community matrix

To solve the ODEs, the user needs to define several initial conditions:

  • the initial species population number $X0$
  • the intrinsic rate of a population species $\epsilon$
  • the interaction coefficients among the species association $\alpha$

Several other input are required and will need to be set in a XmL inputfile. An example of such file is provided in here.

Solving the ODEs system

The mathematical model for the species population evolution results in a set of differential equations (ODEs), one for each species associations modeled. The Runge-Kutta-Fehlberg method (RKF45 or Fehlberg as defined in the odespy library) is used to solve the GLV ODE system.

The Fehlberg method requires 5 parameters :

  • the step-size (or time step)
  • an initial population of the species association
  • the relative tolerance for solution
  • the absolute tolerance for solution
  • the minimum step size for an adaptive algorithm.

Carbonate production

Once a species association population is computed, carbonate production is calculated using a carbonate production factor. Production factors are specified for the maximum population, and linearly scaled to the actual population following the relation $$ \frac{dP}{dt} = R_{max}\frac{x_i}{K_i}$$

where $P$ is the carbonate production, $t$ is time, $R_{max}$ is the carbonate production factor when population is at its maximum, and $K_i$ is the maximum population of species i, computed as

$$K_i=\frac{\epsilon_i}{\alpha_{ii}}$$

which gives:

$$ \frac{dP}{dt} = R_{max}\frac{\alpha_{ii}, x_i}{\epsilon_i}$$

We define the maximum carbonate production rate (m/y) for each species in the XmL input file.

Installation

Local install

The code is available from our github page and can be obtained either from this page or using git

git clone https://github.com/pyReef-model/pyReefCore.git

Once donwloaded, navigate to the pyReefCore folder and run the following command:

pip install -e /workspace/volume/pyReefCore/

Docker container

The code is available from Docker Hub at pyreefmodel/pyreef-Docker and can be downloaded using Kitematic. An example of model is provided in the Tests folder using IPython Notebook.

Usage

pyReefCore can be use from an IPython notebook or a python script directly. An example of functions available is provided below:

%matplotlib inline
import matplotlib.pyplot as plt

import numpy

%config InlineBackend.figure_format = 'svg'
from pyReefCore.model import Model

# initialise model
reef = Model()

# Define the XmL input file
reef.load_xml('input.xml')

# Run to a given time (for example 500 years)
reef.run_to_time(500.,showtime=100.)

# Define a colorscale to display the core
# Some colormaps are available from the following link:
# http://matplotlib.org/examples/color/colormaps_reference.html
from matplotlib.cm import terrain
nbcolors = len(reef.core.coralH)+3
colors = terrain(numpy.linspace(0, 1, nbcolors))

# Plot evolution of species population with time
reef.plot.speciesTime(pop=reef.coral.population, time=reef.coral.iterationTime, colors=colors,
                      size=(8,4), font=8, dpi=80,fname='pop.pdf')

# Plot evolution of species population with depth
reef.plot.speciesDepth(pop=reef.coral.population, depth=reef.core.thickness, colors=colors,
                       size=(8,4), font=8, dpi=80)

# Plot coral facies distribution core and assemblages
reef.plot.drawCore(pop=reef.core.coralH, depth=reef.core.thickness, surf=reef.core.topH,
                   colors=colors, size=(8,10), font=8, dpi=380, fname='out.pdf')