Generation of periodic waves by landscape features in cyclic predator–prey systems

J. A. Sherratt, X. Lambin, C. J. Thomas, T. N. Sherratt

Abstract

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator–prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular ‘target patterns’ or spirals. This is, to our knowledge, a new mechanism for periodic–wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator–prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland–England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large–scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real–world environments, our results suggest that complex spatio–temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.