Geometry for the eccentric territory

Abstract

Basic theory predicts that two-dimensional territories should be circular, but results from stream-dwelling juvenile salmonid fishes appear to deviate consistently from such a shape. We present a model for cardioidal territories based on data from juvenile brook charr (Salvelinus fontinalis) and show that circles can be considered as a limiting case for cardioids. Since they are regular geometric shapes directly related to circles, but with one concave side, cardioids may be useful for some analyses of territorial data.

Environmental influences on territoriality of juvenile brook charr, Salvelinus fontinalis, in a stream environment

“…The shape of their territories (Noakes & McNicol 1982) and their defense (McNicol & Noakes 1981) support the hypothesis that territorial defense is determined by energetic costs and benefits (Davies 1978, Dill 1978a). How they respond to changes in costs and benefits of defense should therefore test this hypothesis.…”

“…Typical cardioid territories (McNicol & Noakes 198 1, Noakes & McNicol 1982) were defended at all current velocities tested, though no apparent relationship between territory shape and current velocity was evident from territory plots. Ratios of CHG to CHS interaction distances to the front and side were calculated and analyzed to determine if the outer region of a territory (where CHG predominates, see McNicol & Noakes 198 1) was compressed more or less than the inner region (where CHS Table S.…”

Quantitative field investigations of feeding and territorial behaviour of young-of-the-year brook charr, Salvelinus fontinalis

“…1) to the model y = a(1-kcosO) + ncos20 + mcos30, (1) using multiple regression, where y is the distance between the resident and the territory perimeter at angle 0 . This model assumes that these territories are basically circular (Noakes & McNicol 1982), and measures the degree of deviation from this basic shape as well as the eccentricity of the territorial station . The latter is tested by the term a(1-kcosO), which measures how much the shape is distorted toward a cardioid ; at k = 0, the territory is Y=26 .84 (I-0.66 COS 0) CHG Scale 1 :4 Fig .…”

“…as indicated a perfect circle and at k = 1, a perfect cardioid . Previous laboratory studies indicated that juvenile brook charr defend cordate territories (McNicol & Noakes 1981, Noakes & McNicol 1982 . Any further distortion is indicated by the magnitude of the other two terms of the equation .…”

“…In such studies they defend territories of a specific shape and structure in an apparently adaptive manner (McNicol & Noakes 1981, Noakes & McNicol 1982 . However, reported field observations on this behaviour too, are only qualitative (Griffith 1974, Keenleyside 1962 .…”