Jonathan A. Sherratt, Department of Mathematics, Heriot-Watt University

Spatiotemporal Patterns of Cyclic Field Voles

Field voles in Kielder Forest

Field vole (microtus agrestis) Kielder Forest is a large area of managed woodland on the border between Scotland and England. The forest is a good habitat for field voles (Microtus agrestis), and this population cycles with a period of about 4 years. Detailed spatiotemporal field studies show that the vole cycles are not synchronous in space; rather, they are out of phase in different parts of the forest, and are organised into a periodic travelling wave. Thus bands of high and low vole density move across the forest; the speed of movement is 15-20 km per year.

Wave generation by Kielder Water

A key question arising from this field data is: what causes the waves to form? One potential explanation is suggested by the geography of the forest. At its centre is a large (10 square km) reservoir, Kielder Water - one of the largest man-made bodies of water in Europe. It is long and thin in shape, and the wave fronts found in the vole data lie approximately parallel to the reservoir. This suggests that the reservoir may explain the formation of the periodic waves. I have investigated this in collaboration with Tom Sherratt, Xavier Lambin and Chris Thomas. We assumed that the vole population cycles are driven by the interaction with their main terrestrial predator, weasels (Mustela nivalis). We show that in this type of cyclic predator-prey system, large-scale landscape features such as Kielder Water can generate periodic travelling waves. The figure shows the periodic travelling wave that is generated in a simulation of weasel-vole dynamics in Kielder Forest. The colours indicate vole density (red=high, blue=low) and the "hole" in the middle is a representation of Kielder Water. Move the mouse over the image to see the effects of removing the reservoir: the vole density gradually becomes uniform in space. This simple mathematical experiment shows that it is the reservoir that is causing the periodic wave in vole density.

Periodic Wave Selection

The variation in wavelength with obstacle radius Equation systems such as our predator-prey model have a family of periodic travelling wave solutions. Therefore it is natural to ask what determines the particular wave that forms around a large scale landscape feature such as Kielder Water. I have investigated this question in collaboration with Tom Sherratt and Xavier Lambin.We showed that travelling wave selection depends on the size and shape of the landscape feature. The figure opposite shows the variation in wavelength with the radius of a cicular obstacle for parameter values corresponding to the weasel-field vole interaction. Smaller obstacles generate waves with longer wavelength. For non-circular obstacles, our numerical studies show that it is the largest linear dimension that is important. Thus for Kielder Water, which long and thin, it is the length that controls periodic wave selection.

The work described on this page is discussed in the following papers:

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