# PZ package

Version 1.0.0 by Lawrence Murray

Lotka-Volterra-like model of the interaction of phytoplankton (prey) and zooplankton (predator).

## Synopsis

./run.sh


This samples from the prior and posterior distributions. The oct/ directory contains a few functions for plotting these results (GNU Octave and OctBi required).

A synthetic data set is provided, but a new one one may be generated with init.sh (GNU Octave and OctBi required).

## Description

This package is based on a Lotka-Volterra model of the interaction between phytoplankton $P$ (prey) and zooplankton $Z$ (predator). It differs from the classic Lotka-Volterra by having a stochastic growth term for phytoplankton, and quadratic mortality term for zooplankton.

The process model is given by the equations: \begin{eqnarray} \frac{dP}{dt} &=& \alpha_t P - cPZ \
\frac{dZ}{dt} &=& ecPZ - m_lZ - m_q Z^2, \end{eqnarray} where $t$ is time in days, and the stochastic growth term $a_t$ is drawn daily as $\alpha_t \sim \mathcal{N}(\mu,\sigma)$, with $\mu$ and $\sigma$ being the two parameters of the model.

This version of the model was originally used in Jones, Parslow & Murray (2010). Its behaviour under sampling with the particle marginal Metropolis-Hastings (PMMH) sampler is also studied in Murray, Jones & Parslow (2013).

## References

Jones, E.; Parslow, J. & Murray, L. M. A Bayesian approach to state and parameter estimation in a Phytoplankton-Zooplankton model. Australian Meteorological and Oceanographic Journal, 2010, 59, 7-16.

Murray, L. M.; Jones, E. M. & Parslow, J. On collapsed state-space models and the particle marginal Metropolis-Hastings sampler, 2013.