A simple model of optimal territory size for drift-feeding fish JAMES W. A. GRANT AND DAVID L. G. NOAKES Department of Zoology, University of Guelph, Guelph, Ont., Canada NlG 2W1 Received February 17, 1986 GRANT, J. W. A., and D. L. G. NOAKES. 1987. A simple model of optimal territory size for drift-feeding fish. Can. J. Zool. 65: 270-276. A simple model of optimal feeding-territory size is developed for drift-feeding fish by modifying Hixon's (M. A. Hixon. 1980. Am. Nat. 115: 5 10-530) model for energy maximizers. Our model predicts that territory size should vary directly with food density when the fish is subject to time constraints, and directly with intruder pressure when the fish is subject to processing constraints. These unique predictions arise because any food not immediately eaten is carried downstream out of the territory. Our model also predicts that territory size should vary inversely with intruder pressure when the fish is subject to time constraints, and inversely with food density when the fish is subject to processing constraints. The qualitative predictions of the model are not affected by changes in the shape of the cost or benefit curves, unlike other simple models. A review of studies of salmonid temtory size indicates that most are inadequate tests of the model because (i) food was not presented in a natural manner, (ii) ration levels were not controlled at levels that were clearly above or below a maximum daily ration, and (iii) the confounding effects of intruder pressure were not controlled. Future experiments will have to incorporate these factors to distinguish between the competing predictions posed by this and previous models. GRANT, J. W. A., et D. L. G. NOAKES. 1987. A simple model of optimal territory size for drift-feeding fish. Can. J. Zool. 65 : 270-276. Ce modele simple, qui permet de prCdire la taille optimale des territoires rCservCs a I'alimentation chez les poissons qui se nourrissent dans la dCrive, est une modification du modele de maximisation de 1'Cnergie mis au point par Hixon (M. A. Hixon. 1980. Am. Nat. 115 : 510-530). Le modkle indique que les dimensions du territoire doivent varier directement en fonction de la densit6 de nourriture lorsque les poissons sont soumis a des contraintes temporelles, et directement en fonction de la pression des intrus lorsque les poissons sont soumis a des contraintes reliCes a la digestion. Ces prddictions plut6t Ctonnantes sont dues au fait que toute noumture qui n'est pas consommCe immkdiatement est entrainee en aval du territoire par le courant. Le modele indique Cgalement que le territoire est fonction inverse de la pression des intrus lorsque les poissons sont soumis a des contraintes temporelles et fonction inverse de la densit6 de nourriture lorsque les poissons sont soumis a des contraintes relikes i la digestion. Les prkdictions qualitatives du modele ne sont pas affectkes par des changements dans la forme de la courbe de coiit ou de bCnCfice, contrairement a ce qui se produit dans le cas d'autres modkles simples. Une rCvision des Ctudes sur les dimensions des temtoires chez les salmonidCs indique que la plupart de ces travaux constituent de mauvais tests du modele puisque (i) la nourriture n'a pas CtC prCsentCe dans des conditions naturelles, (ii) les quantitks prCsentCes n'Ctaient pas clairement plus grandes ou plus faibles qu'une ration quotidienne maximale et (iii) les effets de la pression des intrus n'ktaient pas contr6lCs, ce qui peut entrainer de la confusion. A I'avenir, les expdriences devront tenir compte de ces facteurs pour permettre de distinguer les predictions de ce modkle de celles des modtiles antkrieurs. [Traduit par la revue] Introduction Juvenile salmonids living in streams feed primarily on invertebrate drift (Dill et al. 1981 ; Bachman 1984; McNicol et al. 1985), often within a defended territory (e.g . , Kalleberg 1958; Puckett and Dill 1985). However, none of the simple models of optimal feeding-territory size (reviewed by Schoener 1983) seems to be entirely applicable for drift-feeding fish. Hixon's (1980) model, probably the most general for energy maximizers whose food intake is limited by the time available to feed (Schoener 1983), assumes that food produced on the territory is continuously renewing, immobile, and accumulates until harvested by the resident. Therefore, the time required to harvest the sustainable yield of food on the territory (Trf) is directly proportional to territory area (Fig. 1). Briefly, Hixon (1980) also assumes the following: the time required to chase all intruders (T,) is directly proportional to territory area; and the time available for feeding (Taf) is equal to the total hours of daylight (T,) minus T,, and hence is a negative linear function of area. Therefore, net energy gain, which is proportional to feeding time (Tf), is limited by the lesser of Taf or Trf and is maximized at the territory area where Tti = Taf (A*, Fig. 1). However, the assumption that food accumulates on the territory is not appropriate for drift-feeding fish because any food that is not immediately eaten is carried downstream out of the territory. To harvest all of the food drifting through its territory a fish would have to spend all of its time foraging, either scanning its territory for food or feeding, so Tti of Fig. 1 would have zero TERRITORY AREA FIG. 1. Hixon's (1980) model of optimal territory size for an energy maximizer (T,, total hours of daylight; Td, time required to harvest all the food on the territory; T,, time spent chasing intruders; Taf, time available to feed; Tf, time spent feeding (bold line); A*, territory size that maximizes time spent feeding and net energy gain). Modified from Schoener ( 1983). slope and y-intercept of T,. Clearly, Hixon's model is inappro- priate for drift-feeding fish defending territories. Dill (1978~) developed a complex simulation model of optimal territory size for juvenile salmonids feeding on terrestri- al insects that fall onto the surface of the water. His model makes quantitative predictions of territory size, provided that a number of input parameters can be estimated. The purpose of this paper is to develop a simple model for drift-feeding fish that qualitatively predicts the effects of food density and intruder pressure on territory size. We have attempted to modify Hixon's Can. J. 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