Unstable oscillations and spatial structure: The Nicholson-Bailey model of host-parasitoid dynamics

Basic Problem

Nicholson and Bailey proposed a model for host-parasitoid dynamics in the 1930s. This model describes the population of hosts and parasitoids in discrete time steps. A central characteristic of the Nicholson-Bailey model is that both populations undergo oscillations with increasing amplitude until first the host and then the parasitoid population dies out. This module first investigates the population dynamics of the simple Nicholson-Bailey model, and then extends the model to consider spatial distribution of both host and parasitoid to investigate the effect of spatial structure on stabilizing the population dynamics.

General approach

We will first implement the difference equations of the spatially homogeneous Nicholson-Bailey model in an R script. Then we will extend the Nicholson-Bailey model to incorporate spatial structure, by simulating the dynamics on a two-dimensional lattice (i.e., a square grid of cells).

What can be learned?

Concepts:

Unstable oscillations
Effects of spatial structure on population dynamics
Emergence of spatial structure in populations

Methods:

Simulation of simple two-species difference equations
Simulating spatial structure

Starting point

Download the Downloadhandout (PDF, 589 KB) describing the background of the problem and the equations for both the simple Nicholson-Bailey model and for its spatial extension.
Develop R script for both models according to the instructions in the handout.

Interesting questions that you can investigate

Why does the Nicholson-Bailey model show unstable oscillations and how could it be stabilized?
How does the spatial dispersal rate of hosts and parasitoids affect the population dynamics and why?
How do lattice size and boundary conditions affect the dynamics?
What would happen if the hosts or the parasites spread out equally over the entire lattice?

Some advanced questions:

Can parasitoids facilitate the coexistence of competing hosts?
Measure the correlation between the dynamics at two sites as a function of their distance. How strong is the spatial correlation and how does that depend on the parameters controlling parasitoid and host dispersal?
(Find a more extensive list of questions in the Downloadhandout (PDF, 589 KB).)

Glossary

Difference equations: Also referred to as maps, difference equations describe the dynamics (change) of variables (e.g., the population size) in discrete time, whereas differential equations describe the dynamics in continuous time.
Unstable oscillations: Oscillations with increasing amplitudes.
Parasitoid: A lifestyle that mixes the features of true parasites and predators. Similar to parasites, parasitoids develop within their host, but the completion of their lifecycle always results in the death of the host.

Literature & Weblinks

Hassell, M.P. et al. (1991). external pageSpatial structure and chaos in insect population dynamics. Nature, 353: 255-258.
Comins, H. N., Hassell, M. P. and May, R. M. (1992). external pageThe spatial dynamics of host-parasitoid systems. J. Animal Ecology, 61:735-48.
Kerr, B. et al. (2006). external pageLocal migration promotes competitive restraints in a host-pathogen 'tragedy of the commons'. Nature, 442: 75-78.
Preedy, KF et al. (2007). external pageDisease induced dynamics in host–parasitoid systems: chaos and coexistence. J. R. Soc. Interface, 4: 463-471.