Theoretical Population Biology 67 (2005) 273–284 Threshold policies control for predator–prey systems using a control Liapunov function approach $ Magno Enrique Mendoza Meza a,Ã , Amit Bhaya a , Eugenius Kaszkurewicz a , Michel Iskin da Silveira Costa b a Department of Electrical Engineering, COPPE, Federal University of Rio de Janeiro, P.O. Box 68504, RJ 21945-970, Brazil b Laboratory Nacional de Computac - a˜o Cientı´fica, Av. Getu´lio Vargas, 333 - Quitandinha Petro´polis-RJ 25651-070, Brazil Received 10 September 2004 Available online 8 April 2005 Abstract Thestabilityofpredator–preymodels,inthecontextofexploitationofrenewableresources,subjecttothresholdpolicies(TP)is studiedinthispaperusingtheideaofbacksteppingandcontrolLiapunovfunctions(CLF)wellknownincontroltheory,aswellas theconceptofvirtualequilibria.TPsaredefinedandanalysedfordifferenttypesofoneandtwospeciespredator–preymodels.The models studied are the single species Noy-Meir herbivore-vegetation model, in a grazing management context, as well as the Rosenzweig–MacArthur two species predator–prey model, in a fishery management context. TPs are shown to be versatile and useful in managing renewable resources, being simple to design and implement, and also yielding advantages in situations of overexploitation. r 2005 Elsevier Inc. All rights reserved. Keywords: Predator–prey systems; Control Liapunov function; Sustainable yield; Variable structure system; Threshold policy; Global stability; Virtual equilibrium 1. Introduction Grazing management refers to the manipulation of livestock to systematically control periods of grazing and no grazing (usually termed deferment or rest). The primaryobjectivesaretocontroltheeffectsofgrazingat the individual plant level in order to protect soil watershed and improve livestock production (Heitsch- midt and Stuth, 1991). In grazing management, it is possible to control the consumption of the herbivore (predator) by allowing or not allowing grazing. A mathematical model that is much used in the study of herbivore grazing was proposed by Noy-Meir (1975) andwillbeexaminedinthispaper.TheNoy-Meirmodel describes vegetation growth under the assumption that it is subject to the action of a constant herbivore population. In common with most other single species models in the literature, it has a logistic growth term, and a consumption term that models the action of the herbivore. In the grazing management context, when a scheme such as short duration or deferred rotation is used, it means that the consumption term is being switched on (when grazing of a particular paddock is allowed) and off (when the livestock is fenced out of the paddock) (Heitschmidt and Stuth, 1991). Another possibility arises in grazing models of coral reefs which can flip between coral- and algae-dominated states. It has been postulated that the interplay between herbivorous fish and algae is an important factor in determining the flippingdynamics,sinceremovalofthefishmightinduce ARTICLE IN PRESS www.elsevier.com/locate/ytpbi 0040-5809/$-see front matter r 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.tpb.2005.01.005 $ ThisresearchwaspartiallyfinancedbyProjectNos.140811/2002- 8, 551863/2002-1, 471262/03-0 of CNPq, and also by the agencies CAPES and FAPERJ. Ã Corresponding author. E-mail addresses: magno@vishnu.coep.ufrj.br (M.E.M. Meza), amit@nacad.ufrj.br (A. Bhaya), eugenius@nacad.ufrj.br (E. Kaszkurewicz), michel@lncc.br (M.I. da Silveira Costa).