Open AccessCOMPUTATION OF THE RELAXATION OSCILLATION PERIOD IN LOTKA‐VOLTERRA SYSTEMS
Author(s) -
SHIH SHAGIDI,
MATHEW TAREK P.
Publication year - 2000
Publication title -
natural resource modeling
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.28
H-Index - 32
eISSN - 1939-7445
pISSN - 0890-8575
DOI - 10.1111/j.1939-7445.2000.tb00040.x
Subject(s) - mathematics , gauss , laplace transform , relaxation (psychology) , range (aeronautics) , period (music) , mathematical analysis , representation (politics) , computation , oscillation (cell signaling) , physics , quantum mechanics , psychology , social psychology , materials science , algorithm , politics , biology , political science , acoustics , law , composite material , genetics
. This work surveys techniques of Grasman and Veling [1973], Vasil'eva and Belyanin [1988] and Shih [1996] for computing the relaxation oscillation period of singularly perturbed Lotka‐Volterra systems. Grasman and Veling [1973] used an implicit function theorem to derive an asymptotic formula for the period; Vasil'eva and Belyanin [1988] employed a method of matched asymptotic expansions to obtain an approximation to the period; Shih [1996] obtained two (exact) integral representations for the period in terms of two inverse functions W(–k, x) of xexp(x). These results are compared numerically and asymptotically. In particular, the integral representation of the period in Shih [1996] is computed numerically using a Gauss‐Tschebyscheff integration rule of the first kind, and is further investigated asymptotically by virtue of the asymptotics of W(–k, x) , Laplace's method, and a method of consequent representation. Computational results indicate that the Gauss‐Tschebyscheff approximation of the period in Shih [1996] is uniformly accurate for a wide range of the singular parameter (ɛ in the paper).