Bulletin of Mathematical Biology 67 (2005) 33–55 www.elsevier.com/locate/ybulm Plant–herbivore models, where more grass means fewer grazers Belinda Barnes a,∗ , Harvinder Sidhu b a Ecosystem Dynamics Group, Research School of Biological Sciences, Australian National University, Canberra 0200, Australia b School of Physical, Environmental and Mathematical Sciences, UNSW at ADFA, Canberra 2600, Australia Received 24 February 2004; accepted 3 June 2004 Abstract Classical theory has led us to believe that where more grazing is available herbivores will inflict heavier pressure on the grass, thus keeping its height low. This approach is hotly debated, although still widely accepted. Based on field data collected, van der Koppel et al. [van der Koppel, J., Huisman, J., van der Wal, R., Olff, H., 1996. Patterns of herbivory along a productivity gradient: an empirical and theoretical investigation. Ecology 77, 736–745] contest the standard plant–herbivore models, arguing that herbivores do not ‘control’ the plant growth entirely, and propose two differential equation models. In this paper we describe briefly how van der Koppel et al. (1996) derive their uncontrolled plant–herbivore interaction models, and then expand on the specific mathematical results cited in their paper to provide a global overview of the dynamics of such systems, for a broad range of parameter values. © 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved. 1. Introduction van der Koppel et al. (1996) noted from an island research site, that when the vegetation became particularly dense, the number of grazing animals decreased, rather than increased. This observation was in stark contrast to the predictions of earlier researchers ∗ Corresponding author. E-mail address: belinda.barnes@anu.edu.au (B. Barnes). 0092-8240/$30 © 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.bulm.2004.06.001